Yvan Martel

INTERACTION OF SOLITONS FROM THE PDE POINT OF VIEW

Keywords: 2010 Mathematics Subject Classification. Primary: 35B40; Secondary: 37K40, 35Q51, 35Q53, 35Q55, 35L71 soliton, multi-soliton, asymptotic behavior of solutions, elastic and non-elastic collisions, nonlinear Schrödinger equation, generalized Korteweg-de Vries equation, semilinear wave equation

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Introduction

Pioneering numerical experiments of Fermi, Pasta, Ulam [START_REF] Fermi | Studies of nonlinear problems, I[END_REF] in 1955, and of Zabusky, Kruskal [START_REF] Zabusky | Interaction of "solitons" in a collisionless plasma and recurrence of initial states[END_REF] in 1965, revealed unexpected phenomena related to the interactions of nonlinear waves 1 . Shortly thereafter, the inverse scattering theory and its generalizations, developed by many influential mathematicians [START_REF] Ablowitz | The inverse scattering transform-Fourier analysis for nonlinear problems[END_REF][START_REF] Gardner | Method for solving the Korteweg-de Vries equation[END_REF][START_REF] Gardner | Korteweg-de Vries equation and generalization. VI. Methods for exact solution[END_REF][START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF][START_REF] Miura | The Korteweg-de Vries equation, a survey of results[END_REF][START_REF] Miura | Existence of conservation laws and constants of motion[END_REF][START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF], provided a rigorous ground and a unified approach to these observations. It led very rapidly to an accurate and deep understanding of remarkable properties of several universal nonlinear models, referred to as completely integrable, such as for example, the Korteweg-de Vries equation, the one dimensional cubic Schrödinger equation and the sine-Gordon equation. It has created a very active and inspiring field of research since then 2 . Among the most notable achievements of this theory, we mention (i) the existence of infinitely many conservation laws; (ii) the purely elastic nature of the collision of any number of solitary waves, which means that the interacting solitary waves recover their exact shape and velocity after a collision. Solitary waves enjoying such remarkable property were called solitons 3 ; (iii) the decomposition into solitons, saying that from any solution should emerge in large time a sum of nonlinear states, such as solitons, plus a dispersive part.

These rigorous mathematical facts are known to be physically relevant in numerous contexts, though sometimes under less extreme forms. For example, in several practical applications or for more elaborate nonlinear models, the collision of nonlinear waves is not purely elastic and some loss of energy takes place during collisions 4 . This reveals that the inverse scattering theory is restricted to models with specific algebraic structure and despite many extensions to nearly integrable systems (see e.g. [START_REF] Kivshar | Dynamics of solitons in nearly integrable models[END_REF]), it cannot be applied to general nonlinear models.

In view of the beautiful achievements of the integrability theory but also of its inevitable limitations, it appeared necessary to investigate similar questions for general nonlinear models with solitary waves using tools from the theory of partial differential equations. In these notes, we review some results on interactions of solitary waves obtained for models that are not close to any known integrable equation, such as the generalized Korteweg-de Vries equation, the nonlinear Schrödinger equation in any space dimension, the φ 4 equation and the nonlinear wave equation.

Mainly in the 80s, the solitary wave theory, proving existence, uniqueness, symmetry and stability properties of nonlinear waves, was successfully developed using the elliptic theory, ODE analysis and general variational arguments, at least for ground states (see Section 3). More recently, asymptotic stability results appeared (see Section 4). Then, energy type arguments extending the elliptic theory have allowed to consider several solitary waves in weak interactions, i.e. cases where the soliton dynamics is only slightly perturbed by the interactions.

Pushing the perturbative analysis one step forward, some examples of strong interactions have also been exhibited; the solitons are still distant, but their dynamics is substantially modified by the interactions (see Section 5). Next, we review the few recent cases where a version of the soliton resolution conjecture was proved for non-integrable wave models in Section 6. Finally, we discuss in Section 7 some situations where collisions were proved to be inelastic.

This review points out that despite some impressive and surprizing recent progress, notably on the soliton resolution conjecture, most of the questions raised above on the interaction of solitary waves remain open for general nonlinear models.

Integrable equations

In this section, we briefly highlight some results from the inverse scattering theory that inspired mathematical research much beyond their range of applicability.

2.1.

KdV solitons and multi-solitons. For the Korteweg-de Vries equation5

∂ t u + ∂ x (∂ 2 x u + u 2 ) = 0, (t, x) ∈ R × R, (1)
the inverse scattering transform led to a very striking property which is the existence of exact multi-soliton solutions (see e.g. [START_REF] Hirota | Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[END_REF][START_REF] Miura | The Korteweg-de Vries equation, a survey of results[END_REF][START_REF] Whitham | Linear and Nonlinear Waves[END_REF]). Let Q(x) = 3 2 cosh -2 ( x 2 ) be the unique positive even solution of

Q ′′ + Q 2 = Q, and for c > 0, let Q c (x) = cQ( √ cx). Then, for any c > 0, σ ∈ R, the function defined by u(t, x) = Q c (x -ct -σ
) is a solution of (1), called soliton, traveling with speed c. Solutions containing an arbitrary number of such solitons (called multi-solitons) have been obtained by the inverse scattering theory.

Theorem 1 (Multi-solitons for KdV, [START_REF] Hirota | Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[END_REF][START_REF] Miura | The Korteweg-de Vries equation, a survey of results[END_REF]).

Let K ∈ N, K ≥ 2. Let 0 < c K < • • • < c 1 and σ - 1 , . . . , σ - K ∈ R. There exist σ + 1 , . . . , σ + K ∈ R and an explicit solution u of (1) such that lim t→±∞ u(t) - K k=1 Q c k • -c k t -σ ± k H 1 = 0.
The most remarkable fact is that all the solitons recover exactly the same sizes and speeds after the collision. Moreover, the values of σ + k are explicit. It it interesting to recall that the multi-soliton behavior, even in the simple case of two solitons, differs qualitatively according to the relative sizes of the solitons. We refer to Lemma 2.3 in [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF] for a definition of the three Lax categories of two-solitons and to [START_REF] Zabusky | Interaction of "solitons" in a collisionless plasma and recurrence of initial states[END_REF] for a previous formal discussion. In particular, if their sizes are close (i.e. c 1 ∼ c 2 ), the two solitons never cross, but rather repulse each other at a large distance (this is category (c) in [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF]). See §7.1-7.3.

2.2.

Decomposition into solitons for KdV. The multi-soliton behavior is fundamental for general solutions of the KdV equation as shown by the following decomposition result.

Theorem 2 (Decomposition into solitons, [START_REF] Eckhaus | The emergence of solutions of the Korteweg-de Vries equation from arbitrary initial conditions[END_REF][START_REF] Schuur | Asymptotic analysis of solitons problems[END_REF]). Let u 0 be a C 4 function such that for any j ∈ {0, ..., 4}, for all x ∈ R, (∂ j u 0 /∂x j )(x)

x -10 . Let u be the solution of (1) corresponding to u 0 . Then, there exist

K ∈ N, σ 1 , . . . , σ K ∈ R and c 1 > • • • > c K > 0 such that, for all x > 0, lim t→+∞ u(t, x) - K k=1 Q c k (x -c k t -σ k ) = 0.
This result has a rich history, see [START_REF] Ablowitz | The inverse scattering transform-Fourier analysis for nonlinear problems[END_REF][START_REF] Cohen | Existence and regularity for solutions of the Korteweg-de Vries equation[END_REF][START_REF] Dauxois | Physics of solitons[END_REF][START_REF] Dodd | Solitons and nonlinear wave equations[END_REF][START_REF] Eckhaus | The emergence of solutions of the Korteweg-de Vries equation from arbitrary initial conditions[END_REF][START_REF]The Korteweg-de Vries equation and related evolution equations[END_REF][START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF][START_REF] Schuur | Asymptotic analysis of solitons problems[END_REF][START_REF] Zabusky | Interaction of "solitons" in a collisionless plasma and recurrence of initial states[END_REF] and the references therein. Note that if some space decay is necessary to apply the inverse scattering transform, the decay assumption on the initial data in the above result is not optimal. Note also that the asymptotic behavior of the solution is described for x > 0 (see results in [START_REF] Schuur | Asymptotic analysis of solitons problems[END_REF] for slight improvement). For the region x < 0, see [START_REF] Deift | The collisionless shock region for the long-time behavior of solutions of the KdV equation[END_REF][START_REF] Eckhaus | The emergence of solutions of the Korteweg-de Vries equation from arbitrary initial conditions[END_REF][START_REF] Schuur | Asymptotic analysis of solitons problems[END_REF] and references therein.

Last, we mention that the modified KdV equation (i.e. the KdV equation with a cubic nonlinearity) is also an integrable model that enjoys most of the properties of the KdV equation, like the infinitely many conservation laws and the existence of pure multi-soliton solutions (see e.g. [START_REF] Miura | The Korteweg-de Vries equation, a survey of results[END_REF]). Actually, it even has a richer family of exceptional solutions: breather solutions (see [START_REF] Alejo | Nonlinear stability of mKdV breathers[END_REF][START_REF] Lamb | Element of soliton theory[END_REF][START_REF] Wadati | The modified Korteweg-de Vries Equation[END_REF]) and dipole solitons, i.e. special multi-solitons where solitons are distant like C log t (see [START_REF] Gorshkov | Interactions of solitons in nonintegrable systems: direct perturbation method and applications[END_REF][START_REF] Karpman | A perturbational approach to the two-soliton systems[END_REF][START_REF] Wadati | Multiple-pole solutions of modified Korteweg-de Vries equation[END_REF]). This complicates any possible soliton resolution conjecture on this equation (see [START_REF] Schuur | Asymptotic analysis of solitons problems[END_REF]).

2.3.

One dimensional cubic NLS. The 1D cubic nonlinear Schrödinger equation

i∂ t u + ∂ 2 x u + |u| 2 u = 0, (t, x) ∈ R × R, (2)
is also an integrable equation, widely studied for its numerous physical applications and remarkable mathematical properties. See e.g. [START_REF] Dauxois | Physics of solitons[END_REF][START_REF] Deift | Inverse scattering on the line[END_REF][START_REF] Deift | Long-time asymptotics for integrable nonlinear wave equations[END_REF][START_REF] Dodd | Solitons and nonlinear wave equations[END_REF][START_REF] Faddeev | Hamiltonian Methods in the Theory of Solitons[END_REF][START_REF] Novoksenov | Asymptotic behavior as t → ∞ of the solution to the Cauchy problem for a nonlinear Schrödinger equation[END_REF]124,[START_REF] Yang | Nonlinear waves in integrable and nonintegrable systems[END_REF][START_REF] Zabusky | Interaction of "solitons" in a collisionless plasma and recurrence of initial states[END_REF][START_REF] Zakharov | Asymptotic behavior of nonlinear wave systems integrated by the inverse scattering transform[END_REF][START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF].

Here, we denote Q(x) = √ 2 cosh -1 (x) the unique positive even solution of

Q ′′ + Q 3 = Q, and for any c > 0, Q λ (x) = √ cQ( √ cx). Then, for any c > 0, β ∈ R, σ ∈ R and γ ∈ R, u(t, x) = Q c (x -βt -σ)e iΓ(t,x) , Γ(t, x) = 1 2 (β • x) - 1 4 |β| 2 t + ct + γ,
is a solitary wave of (2), moving on the line x = σ + βt and also called soliton.

As the KdV equation, the 1D cubic NLS admits explicit multi-solitons. However, the possible behaviors of multi-solitons is richer for NLS. In addition to multi-solitons distant like Ct, which is the generic situation, the equation also admits multi-solitons where the distance between some solitons is C log t (see [124,[START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF]; this requires solitons of exactly the same size, like for mKdV) and solutions where some solitons are staying at a finite distance from each other for all time (see [START_REF] Yang | Nonlinear waves in integrable and nonintegrable systems[END_REF][START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF]). As for mKdV, the presence of such multi-solitons complicates any general decomposition result but does not prevent it. For such questions, we refer to the recent work [START_REF] Borghese | Long-time asymptotic behavior of the focusing nonlinear Schrödinger equation[END_REF] and its references.

2.4. The sine-Gordon equation. The sine-Gordon equation

∂ 2 t u -∂ 2
x u + sin u = 0, (t, x) ∈ R × R, was also widely studied as a physically relevant and completely integrable model (see e.g. [START_REF] Dauxois | Physics of solitons[END_REF][START_REF] Dodd | Solitons and nonlinear wave equations[END_REF][START_REF] Lamb | Element of soliton theory[END_REF]). This equation has an explicit kink solution S(x) = 4 arctan(e x ). It also has other exceptional solutions, like time-periodic wobbling kinks (see [START_REF] Cuenda | Sine-Gordon wobbles through Bäcklund transformations[END_REF][START_REF] Segur | Wobbling kinks in ϕ 4 and sine-Gordon theory[END_REF]), and breathers (see [START_REF] Lamb | Element of soliton theory[END_REF]).

2.5. Other integrable models and nearly integrable models. For the derivative NLS equation, we refer to [START_REF] Jenkins | Soliton resolution for the derivative non-linear Schrödinger equation[END_REF] and its references. For the KP-I equation, see Lamb [START_REF] Lamb | Element of soliton theory[END_REF]. For integrable models set on the torus, see [START_REF] Kuksin | Analysis of Hamiltonian PDEs[END_REF] and references therein.

Several nearly integrable equations have also been studied in the context of the theory of inverse scattering. We refer to [START_REF] Dauxois | Physics of solitons[END_REF][START_REF] Deift | Perturbation theory for infinite-dimensional integrable systems on the line. A case study[END_REF][START_REF] Kivshar | Dynamics of solitons in nearly integrable models[END_REF][START_REF] Yang | Nonlinear waves in integrable and nonintegrable systems[END_REF] and to the references therein.

2.6. Formal works and numerical simulations. Note that shortly after the development of the inverse scattering and the discovery of explicit multi-solitons, other approaches appeared, like in [START_REF] Ei | The motion of interacting pulses[END_REF][START_REF] Gorshkov | Interactions of solitons in nonintegrable systems: direct perturbation method and applications[END_REF][START_REF] Karpman | A perturbational approach to the two-soliton systems[END_REF], to investigate possible multi-soliton behaviors for integrable or non-integrable models. Such papers focus on the modulation equations of the parameters of the solitons and lack the analysis of the error terms, but they aim at justifying formally multisolitons behaviors beyond any integrability property or proximity to integrable equations. In particular, as for the rigorous results presented in Section 5 below, they are asymptotic results, restricted to cases where the distances between the various solitons are large enough.

Theoretical and numerical works have been developed in parallel. As mentioned in the Introduction, the subject started with two fundamental numerical experiments presented in [START_REF] Fermi | Studies of nonlinear problems, I[END_REF][START_REF] Zabusky | Interaction of "solitons" in a collisionless plasma and recurrence of initial states[END_REF]. Since then, there has been an intense activity on studying solitary waves interactions from the numerical point of view. We refer to [START_REF] Craig | Solitary water wave interactions[END_REF] which compares KdV multi-solitons, the water wave problem from the numerical point of view and real experiments on waves generated in water tanks. For Klein-Gordon equations, we refer to [START_REF] Ablowitz | Solitary wave collisions[END_REF]. We also refer to [START_REF] Bona | Solitary-wave interaction[END_REF][START_REF] Dauxois | Physics of solitons[END_REF][START_REF] Hammack | Solitary-wave collisions[END_REF][START_REF] Li | Soliton collisions in the ion acoustic plasma equations[END_REF][START_REF] Shih | Soliton-like interaction governed by the generalized Korteweg-de Vries equation[END_REF][START_REF] Yang | Nonlinear waves in integrable and nonintegrable systems[END_REF] and references therein. One of the main questions studied by numerical experiments is the elastic versus inelastic character of the collisions of nonlinear waves.

Nonlinear models with solitary waves

In these notes, we consider four typical nonlinear models and work with the notion of solution in the energy space.

3.1. The generalized Korteweg-de Vries equation. Consider the following 1D model, for any integer p ≥ 2,

∂ t u + ∂ x (∂ 2 x u + u p ) = 0, (t, x) ∈ R × R. (3)
As seen before, the case p = 2 corresponds to the KdV equation and p = 3 to the mKdV equation, which are both completely integrable. The mass and energy

u 2 (t), 1 2 u 2 x (t) - 1 p + 1 u p+1 (t)
are formally conserved for solutions of (3). We refer to [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] for the local well-posedness of the Cauchy problem in the energy space H 1 (see also [START_REF] Kato | On the Cauchy problem for the (generalized) Korteweg-de Vries equation[END_REF]). For 1 < p < 5, all solutions in H 1 are global and bounded, and the problem is called sub-critical. For p = 5, the problem is mass critical (blow up solution do exist, see [START_REF] Martel | Blow up for the critical generalized Korteweg de Vries equation. I: Dynamics near the soliton[END_REF] and references therein) and p > 5 correspond to the super-critical case. The notion of criticality corresponds to the scaling invariance of

equation (3): indeed, if u(t, x) is solution then for any c > 0, u c (t, x) = c 1 p-1 u(c 3 2 t, c 1 2 x) is also solution and u c (t) L 2 = c 1 p-1 -1 4 u(t) L 2 .
Let Q be the unique (up to sign change if p is odd) non-trivial even solution of Q ′′ +Q p = Q on R, explicitely given by

Q(x) = p + 1 2 1 p-1 cosh -2 p-1 p -1 2 x . For c > 0, let Q c (x) = c 1 p-1 Q(c 1 2
x). Note that these formulas for Q and Q c generalize the previous ones given for p = 2 and p = 3. As before, solitary waves (also called solitons by abuse of terminology) are solutions of (3) of the form u(t, x) = Q c (x -ct -σ), for any c > 0 and σ ∈ R.

The orbital stability of solitons with respect to small perturbations in the energy space H 1 is known in the sub-critical case.

Theorem 3 (Stability of the soliton for sub-critical gKdV [START_REF] Benjamin | The stability of solitary waves[END_REF][START_REF] Bona | On the stability theory of solitary waves[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF][START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF][START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF]). Let 1 < p < 5. For all ǫ > 0, there exists δ > 0, such that if

u 0 -Q H 1 ≤ δ, then the solution u of (3) with initial data u 0 satisfies, for all t ∈ R, u(t, . + σ(t)) -Q H 1 ≤ ǫ, for some function σ.
In contrast, solitons are unstable in the critical and super-critical case p ≥ 5. Note that the instability phenomenon is quite different in the critical case (linear stability holds and the nonlinear instability is related to the scaling parameter) and in the super-critical case (linear exponential instability). See [START_REF] Bona | Stability and instability of solitary waves of Korteweg-de Vries type[END_REF][START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Martel | A Liouville Theorem for the critical generalized Korteweg-de Vries equation[END_REF][START_REF] Pego | Eigenvalues, and instabilities of solitary waves[END_REF].

The nonlinear Schrödinger equation. Recall the nonlinear NLS equation

i∂ t u -∆u -|u| p-1 u = 0, (t, x) ∈ R × R d . (4)
We consider the case p > 1 for d = 1, 2, and 1 < p < d+2 d-2 for d ≥ 3. For d = 1 and p = 3, the model is completely integrable, as seen before. Note that, similarly as for the gKdV equation, p = 1 + 4 d corresponds to L 2 criticality, while for d ≥ 3, p = d+2 d-2 corresponds to Ḣ1 criticality. The mass, energy and momentum

|u(t)| 2 , 1 2 |∇u(t)| 2 - 1 p + 1 |u(t)| p+1 , ℑ ∇u(t)ū(t)
are formally conserved for solutions of (4). We refer to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Ginibre | On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case[END_REF][START_REF] Tao | Nonlinear dispersive equations. Local and global analysis[END_REF] for the local well-posedness of the Cauchy problem in the energy space H 1 . We denote by Q the unique positive radially symmetric

H 1 solution of ∆Q + |Q| p-1 Q = Q on R d (
the function Q is called the ground state; see existence and uniqueness results in [START_REF] Berestycki | Nonlinear scalar field equations. I. Existence of a ground state[END_REF][START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Kwong | Uniqueness of positive solutions of ∆u -u + u p = 0 in R n[END_REF][START_REF] Tao | Nonlinear dispersive equations. Local and global analysis[END_REF]). For c > 0, let

Q c (x) = c 1 p-1 Q(c 1 2 x).
Note that this is a further generalization of the notation for gKdV, for any space dimension d ≥ 1. For d ≥ 2, ground states are no longer explicit, but their properties are well-understood (see references above). Then, for any

c > 0, β ∈ R d , σ ∈ R d and γ ∈ R, the function u defined by u(t, x) = Q c (x -βt -σ)e iΓ(t,x) where Γ(t, x) = 1 2 (β • x) - 1 4 |β| 2 t + ct + γ,
is a traveling wave of (4), with speed β.

The stability and instability properties of solitary waves of NLS are similar: stability in the L 2 sub-critical case, and instability in the critical and super-critical cases. We refer to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF][START_REF] Grillakis | Analysis of the linearization around a critical point of an infinite dimensional hamiltonian system[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF] for details.

3.3. The φ 4 equation. We consider the φ 4 model (see e.g. [START_REF] Dauxois | Physics of solitons[END_REF][START_REF] Manton | Topological solitons[END_REF])

∂ 2 t φ -∂ 2 x φ -φ + φ 3 = 0, (t, x) ∈ R × R. (5)
Recall that the energy

E(φ, ∂ t φ) = 1 2 |∂ t φ| 2 + 1 2 |∂ x φ| 2 + 1 4 1 -|φ| 2 2
is formally conserved along the flow. The kink, defined by H(x) = tanh x/ √ 2 is the unique (up to sign change), bounded, odd solution of the equation -H ′′ = H -H 3 on R. We recall that the orbital stability of the kink with respect to small perturbations in the energy space has been proved in [START_REF] Henry | Stability theory for solitary-wave solutions of scalar field equations[END_REF] using mainly the energy conservation. This model is analogue to the sine-Gordon equation, but it is not completely integrable and breathers solutions or woobling kinks are not expected to exist.

3.4. The energy critical nonlinear wave equation. For space dimensions d ≥ 3, we consider the following nonlinear wave equation,

∂ 2 t u = ∆u + |u| 4 d-2 u, (t, x) ∈ R × R d . (6)
We denote

E(u, v) = 1 2 |∇u| 2 + 1 2 v 2 - 1 6 |u| 6
so that the energy of a solution (u, ∂ t u) of ( 6), defined by E(u, ∂ t u), is formally conserved by the flow. Concerning the Cauchy problem in Ḣ1 × L 2 for the energy critical wave equation, we refer to [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation[END_REF] and the references given therein. As before, the notion of criticality is related to the scaling invariance: if u(t, x) is a solution, then for any λ > 0,

u λ (t, x) = 1 λ d-2 2 u t λ , x λ is also solution and ∇u λ L 2 = ∇u L 2 .
Here, solitary waves are stationary solutions W ∈ Ḣ1 satisfying ∆W + |W | 4 d-2 W = 0, and traveling waves obtained as Lorentz transforms of such solutions. For ℓ ∈ R d , |ℓ| < 1, we denote [START_REF] Benjamin | The stability of solitary waves[END_REF]. As for the NLS equation, we consider only ground states solitary waves, i.e. solutions of the above elliptic equation explicitely given by

W ℓ (x) = W 1 1 -|ℓ| 2 -1 ℓ(ℓ • x) |ℓ| 2 + x , so that u(t, x) = W ℓ (x -ℓt) is solution of
W (x) = 1 + |x| 2 d(d -2) -d-2 2 .
As solutions of the evolution equation ( 6), they are unstable with respect to perturbation of the initial data with one direction of exponential instability (see [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF][START_REF] Grillakis | Analysis of the linearization around a critical point of an infinite dimensional hamiltonian system[END_REF]).

Asymptotic stability

We recall briefly some results of asymptotic stability of solitons.

Asymptotic stability for gKdV solitons.

Theorem 4 (Asymptotic stability of the gKdV soliton in H 1 , [START_REF] Martel | Asymptotic stability of solitons for subcritical generalized KdV equations[END_REF]). Let p = 2, 3, 4. For any β > 0, there exists δ = δ(β) > 0 such that the following is true. Let u 0 ∈ H 1 be such that u 0 -Q H 1 ≤ δ. Then, the global solution u of (3) with initial data u 0 satisfies

lim t→+∞ u(t) -Q c + (• -σ(t)) H 1 (x>βt) = 0,
for some c + > 0 with |c + -1| δ and some C 1 function σ such that lim +∞ σ ′ = c + .

We refer to [START_REF] Pego | Asymptotic stability of solitary waves[END_REF] for the first result of asymptotic stability of gKdV solitons. Theorem 1 claims strong convergence in H 1 in the region x > βt. Strong convergence in H 1 (R) is never true since it would imply by stability that u is exactly a soliton. The region where convergence is obtained in Theorem 1 is sharp since one can construct a solution which behaves asymptotically as t → +∞ as the sum Q(x -t) + Q c (x -ct), where 0 < c = 1 is arbitrary (see [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF][START_REF] Miura | The Korteweg-de Vries equation, a survey of results[END_REF][START_REF] Wadati | The exact N -soliton solution of the Korteweg-de Vries equation[END_REF]). In particular, choosing c ≪ 1, the H 1 norm of Q c (x -ct) is small, and this soliton travels on the line x = ct. This explains the necessity of a positive β in the convergence result. We also refer to the survey [START_REF] Tao | Why are solitons stable?[END_REF]. For p = 4, the result has been completed in [START_REF] Kenig | Asymptotic stability of solitons for the Benjamin-Ono equation[END_REF][START_REF] Tao | Scattering for the quartic generalised Korteweg-de Vries equation[END_REF] showing that the residue scatters in a Besov space close to the critical Sobolev space Ḣ-1/6 . For p = 3, we refer to [START_REF] Germain | Asymptotic stability of solitons for mKdV[END_REF] for a full asymptotic stability statement. 4.2. Asymptotic stability for NLS equations. In the context of the nonlinear Schrödinger equation, pioneering results on asymptotic stability of traveling waves are [START_REF] Buslaev | Scattering for the nonlinear Schrödinger equations: states close to a soliton[END_REF][START_REF] Buslaev | On the stability of solitary waves for nonlinear Schrödinger equations[END_REF][START_REF] Soffer | Multichannel nonlinear scattering for nonintegrable equations[END_REF][START_REF] Soffer | Multichannel nonlinear scattering ii. The Case of Anisotropic Potentials and Data[END_REF]. These papers initiated the method of separating modes and using dispersive estimates with potential, under assumptions on the spectrum of the linearized operator.

This question has then been extensively studied, for the NLS equation with or without potential and for various nonlinearities, see e.g. [START_REF] Buslaev | On asymptotic stability of solitary waves for nonlinear Schrödinger equations[END_REF][START_REF] Cuccagna | On asymptotic stability of moving ground states of the nonlinear Schrödinger equation[END_REF][START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF][START_REF] Nakanishi | Invariant manifolds and dispersive Hamiltonian evolution equations[END_REF]129,[START_REF] Rodnianski | Asymptotic stability of N -soliton states of NLS[END_REF][START_REF] Schlag | Spectral theory and nonlinear partial differential equations: a survey[END_REF][START_REF] Schlag | Dispersive estimates for Schrödinger operators: A survey[END_REF][START_REF] Schlag | Stable manifolds for an orbitally unstable nonlinear Schrödinger equation[END_REF] as typical papers, and their references. Most of these works require specific assumptions, like spectral assumptions or suitable dispersive estimates for equations with unknown potential, a suitable Fermi Golden Rule or flatness conditions on the nonlinearities at 0. It follows that no result of asymptotic stability is fully proved for any pure power NLS equation without potential with stable solitons, except for the integrable cubic 1D NLS treated in [START_REF] Cuccagna | The asymptotic stability of solitons in the cubic NLS equation on the line[END_REF].

In larger dimensions, or higher order nonlinearities, the solitons are unstable. The notion of conditional asymptotic stability and the construction of center stable manifolds then become relevant. For the focusing 3D cubic NLS equation (which is an Ḣ 1 2 critical equation with exponentially unstable solitons) the theory has been especially well-developed, at least in the radial case, in [START_REF] Beceanu | A centre-stable manifold for the focussing cubic NLS in R 1+3[END_REF][START_REF] Beceanu | A critical center-stable manifold for Schrödinger's equation in three dimensions[END_REF][START_REF] Costin | On the spectral properties of L± in three dimensions[END_REF][START_REF] Nakanishi | Invariant manifolds and dispersive Hamiltonian evolution equations[END_REF][START_REF] Schlag | Spectral theory and nonlinear partial differential equations: a survey[END_REF][START_REF] Schlag | Stable manifolds for an orbitally unstable nonlinear Schrödinger equation[END_REF]. In particular, spectral assumptions implying the desired dispersive estimates for the linearized equation around the soliton have been checked, first numerically and then rigorously by computer assisted proof (see [START_REF] Costin | On the spectral properties of L± in three dimensions[END_REF] and references therein). 4.3. Asymptotic stability of the φ 4 kink. The asymptotic stability of the kink H by the φ 4 flow ( 5) is known in the case of odd perturbations in the energy space. Note that for odd initial data, the corresponding solution of ( 5) is odd. Rewrite φ = H + u. Then, one has

∂ 2 t u -∂ 2 x u -u + 3H 2 u + 3Hu 2 + u 3 = 0, (t, x) ∈ R × R. (7)
Theorem 5 (Asymptotic stability of the kink by odd perturbations, [START_REF] Kowalczyk | Kink dynamics in the φ 4 model: asymptotic stability for odd perturbations in the energy space[END_REF]). There exists δ > 0 such that for any odd (u 0 , u 1 ) ∈ H 1 × L 2 with (u 0 , u 1 ) H 1 ×L 2 ≤ δ, the solution (u, ∂ t u) of ( 7) with initial data (u 0 , u 1 ) satisfies, for any bounded interval I ⊂ R,

lim t→±∞ (u, ∂ t u)(t) H 1 (I)×L 2 (I) = 0.
As for gKdV, if a solution u of (7) satisfies lim t→+∞ (u, ∂ t u)(t) H 1 ×L 2 = 0, then by orbital stability [START_REF] Henry | Stability theory for solitary-wave solutions of scalar field equations[END_REF], u(t) ≡ 0, for all t ∈ R. Thus the local result is in some sense optimal.

For previous related results, we refer to [START_REF] Kopylova | On asymptotic stability of kink for relativistic Ginzburg-Landau equations[END_REF][START_REF] Kopylova | On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation[END_REF] where the asymptotic stability of the kink is studied for the 1D equation ∂ 2 t u -∂ 2 x u + F (u) = 0, under specific assumptions on F (not including the φ 4 model) and to [START_REF] Cuccagna | On asymptotic stability in 3D of kinks for the φ 4 model[END_REF], where the stability and asymptotic stability of the one dimensional kink for the φ 4 model, subject to localized three dimensional perturbations is studied. We also refer the references in [START_REF] Cuccagna | On asymptotic stability in 3D of kinks for the φ 4 model[END_REF][START_REF] Kopylova | On asymptotic stability of kink for relativistic Ginzburg-Landau equations[END_REF][START_REF] Kopylova | On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation[END_REF][START_REF] Kowalczyk | Kink dynamics in the φ 4 model: asymptotic stability for odd perturbations in the energy space[END_REF] for related works on scattering of small solutions to Klein-Gordon equations. See also the review [START_REF] Kowalczyk | On asymptotic stability of nonlinear waves[END_REF] and references therein. 4.4. Blow up profile for the critical wave equation. Recall that [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation[END_REF] provides a classification of all possible behaviors (blow up or scattering) of solutions of (6) whose initial data (u 0 , u 1 ) satisfies E(u 0 , u 1 ) < E(W, 0). Next, [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF] studies the threshold case E(u 0 , u 1 ) = E(W, 0) and constructs the stable manifold around W . Then, [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF][START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation : the non-radial case[END_REF] proved the following result for solutions slightly above the threshold.

Theorem 6 (Blow up profile for 3D critical NLW, [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF][START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation : the non-radial case[END_REF]). Let d = 3. There exists δ > 0 such that if u is a solution of (6) blowing up in finite time T > 0 and satisfying the bound

sup [0,T ) ∇u(t) L 2 + 1 2 ∂ t u(t) L 2 ≤ ∇W L 2 + δ, then lim t↑T (u(t), ∂ t u(t)) -(v 0 , v 1 ) ∓ 1 λ 1 2 (t) W ℓ •-σ(t) λ(t) , - 1
λ 3 2 (t) (ℓ • ∇W ℓ ) •-σ(t) λ(t) L 2 = 0 for some σ, λ and ℓ ∈ R 3 , |ℓ| < 1 and (v 0 , v 1 ) ∈ Ḣ1 × L 2 .
We see that the family {±W ℓ } is the universal blow up profile. We refer to the original papers for more results and details.

We refer to [START_REF] Krieger | Threshold phenomenon for the quintic wave equation in three dimensions[END_REF][START_REF] Krieger | Global dynamics away from the ground state for the energycritical nonlinear wave equation[END_REF] for classification results of solutions with energy at most slightly above the one of the ground state, and to [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF][START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF] for contructions of solutions with prescribed blow up rates (type II blow up). We also refer to [START_REF] Martel | Stability of blow up profile and lower bounds for blow up rate for the critical generalized KdV equation[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg de Vries equation. I: Dynamics near the soliton[END_REF][START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation[END_REF] and references therein for previous results of blow up profile in the case of mass critical gKdV and NLS equations. Concerning blow up, see also the review [START_REF] Raphaël | On singularity formation in Hamiltonian evolution equations[END_REF] and the references therein.

Asymptotic multi-solitons

In this section, we discuss results of existence of asymptotic multi-solitons for non-integrable models, inspired by Theorem 1 and other explicit constructions of multi-solitons for integrable models, but limited to one direction of time. In particular, these results are valid in asymptotic situations where the distances between all the solitary waves are large enough. 5.1. Multi-solitons with weak interactions. As a rough idea, weak interaction means that the trajectories of the solitary waves are not affected asymptotically.

Theorem 7 (Existence and uniqueness of gKdV multi-solitons, [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]). Let p = 2, 3, 4 or 5.

Let K ≥ 2, 0 < c K < • • • < c 1 , and σ 1 , . . . , σ K ∈ R. Let R = K k=1 R k where R k (t, x) = Q c k (x -σ k -c k t).
There exists a unique H 1 solution of (3) such that lim t→-∞ u(t) -R(t) H 1 = 0. Moreover, there exists κ > 0 such that, for all t ≤ 0, u(t) -R(t) H 1 e -κ|t| .

Such result shows that the multi-soliton behavior is universal, at least in one direction of time. Observe that it says nothing about the behavior the solution as t → +∞. The uniqueness statement in the energy space is relevant even in the integrable case since the inverse scattering theory requires a priori more space decay. The stability of the multi-soliton structure was studied in the sub-critical case in [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations[END_REF].

We observe that a similar existence result also holds for the gKdV super-critical equation (p > 5), with a specific classification related to the exponential instability, see [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF].

For the NLS equation, we recall the following existence result.

Theorem 8 (Existence of NLS multi-solitary waves, [START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF][START_REF] Martel | Multi-solitary waves for nonlinear Schrödinger equations[END_REF][START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF]). Let d ≥ 1. Let p > 1 for d = 1, 2 and 1 < p < d+2 d-2 for d ≥ 3. Let K ≥ 2 and for any k ∈ {1, . . . , K}, let c k > 0,

β k ∈ R d , σ k ∈ R d and γ k ∈ R. Assume that, for any k = k ′ , β k = β k ′ . Let R = K k=1 R k where R k (t, x) = Q c k (x -σ k -β k t)e iΓ k (t,x) and Γ k (t, x) = 1 2 (β k • x) - 1 4 |β k | 2 t + c k t + γ k .
Then, there exist T 0 ∈ R, κ > 0 and an H 1 solution u of (4) such that, for all t ≤ T 0 ,

u(t) -R(t) H 1 e -κ|t| .
Uniqueness (for critical and sub-critical nonlinearities) or classification (for super-critical nonlinearity) is an open problem. See [START_REF] Combet | Multi-existence of multi-solitons for the super-critical nonlinear Schrödinger equation in one dimension[END_REF] for multi-existence in the 1D super-critical case.

Note that the construction of multi-solitons and the study of the stability of the sums of multi-soliton has been extended to several other models, see e.g. [START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF][START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF] for the case of the nonlinear Klein-Gordon equation, and [START_REF] Ming | Multi-solitons and related solutions for the water-waves system[END_REF] for the water wave model.

For the 5D energy critical wave equation, the following existence result is proved in [START_REF] Martel | Construction of multi-solitons for the energy-critical wave equation in dimension 5[END_REF].

Theorem 9 (Existence of NLW multi-solitary waves, [START_REF] Martel | Construction of multi-solitons for the energy-critical wave equation in dimension 5[END_REF]). Let d = 5. Let K ≥ 2, and for any k ∈ {1, . . . , K}, let

λ k > 0, σ k ∈ R 5 , ι k ∈ {-1, +1} and ℓ k ∈ R 5 with |ℓ k | < 1. Assume that, for any k ′ = k, ℓ k = ℓ k ′ . Let R = K k=1 R k where R k (t, x) = ι k λ 3 2 k W ℓ k x -ℓ k t -σ k λ k .
Assume that one of the following assumptions holds (1) Two-solitons: K = 2 (2) Collinear speeds: For all k ∈ {1, . . . , K}, ℓ k = ℓ k e 1 where ℓ k ∈ (-1, 1). Then, there exist T 0 ∈ R and a solution u of (6) on (-∞, T 0 ] in the energy space such that lim t→-∞ ∇ x,t (u(t) -R(t)) L 2 = 0. 5.2. Multi-solitons with strong interactions. We state a typical result where the strong interactions between the traveling waves indeed affect their trajectories.

Theorem 10 (Two-solitary waves with logarithmic distance, [START_REF] Nguyen | Existence of multi-solitary waves with logarithmic relative disctances for the NLS equation[END_REF]). Let d ≥ 1. Let

1 < p < d + 2 d -2 (p > 1 for d = 1, 2) and p = 1 + 4 d .
There exists a solution u of (4) such that |z

1 (t) -z 2 (t)| ∼ 2 log t as t → -∞ and lim t→-∞ u(t) -e -iγ(t) k=1,2 Q(• -z k (t)) H 1 = 0.
As discussed in §2.3, such solutions were already known in the integrable case by the inverse scattering theory. The above result means that this behavior is universal for general NLS equations, under the same restriction that the traveling waves have equal scaling. The mass critical case p = 1 + 4 d is excluded since it displays a special behavior related to blow up and where the above behavior is visible only in rescaled variables, as previously described in [START_REF] Martel | Strongly interacting blow up bubbles for the mass critical NLS[END_REF]. For the gKdV equation, a result similar to Theorem 10 is given in [START_REF] Nguyen | Strongly interacting multi-solitons with logarithmic relative distance for gKdV equation[END_REF].

We mention a few other previous results of strong interactions: for the Hartree equation [START_REF] Krieger | Two-soliton solutions to the three-dimensional gravitational Hartree equation[END_REF], for the energy critical wave equation [START_REF] Jendrej | Construction of two-bubble solutions for energy-critical wave equations[END_REF][START_REF] Jendrej | Two-bubble dynamics for threshold solutions to the wave maps equation[END_REF], for the mass critical gKdV equation [START_REF] Combet | Sharp asymptotics for the minimal mass blow up solution of the critical gKdV equation[END_REF][START_REF] Combet | Construction of multi-bubble solutions for the critical gKdV equation[END_REF], and for the half-wave equation [START_REF] Gérard | A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line[END_REF].

5.3. Soliton interaction with the background. Several papers deal with the question of the interaction of a soliton with a changing background or an impurity. See [START_REF] Holmer | Slow soliton interaction with delta impurities[END_REF][START_REF] Holmer | Soliton interaction with slowly varying potentials[END_REF][START_REF] Holmer | Soliton splitting by external delta potentials[END_REF][START_REF] Holmer | Fast soliton scattering by delta impurities[END_REF] for the interaction of a soliton of NLS with a Dirac mass or a slowly varying potential, and [START_REF] Muñoz | Inelastic character of solitons of slowly varying gKdV equations[END_REF][START_REF] Muñoz | Dynamics of soliton-like solutions for slowly varying, generalized KdV equations: refraction versus reflection[END_REF] for the interaction of a gKdV soliton with a slowing variable bottom.

Decomposition into solitons for the energy critical wave equation

Here, we recall the few existing results of decomposition in solitons in non-integrable cases. First, a complete result of decomposition into solitons for equation ( 6) was proved in [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF] for the radial 3D case.

Theorem 11 (Soliton resolution for the 3D radial critical wave equation, [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF]). Let d = 3. Let u be a global radial solution of (6). Then, there exist a solution v L of the linear wave equation,

K ∈ N, ι k ∈ {-1, 1}, λ k > 0, such that lim t→+∞ (u(t), ∂ t u(t)) -   v L (t) + K k=1 ι k λ 1 2 k (t) W • λ k (t) , ∂ t v L (t)   Ḣ1 ×L 2 = 0,
and λ 1 (t) ≪ λ 2 (t) ≪ • • • ≪ λ K (t) ≪ t, as t → +∞.
Note that the above result is in some sense more complete than for gKdV ( §2.1), since the residue is proved to scatter. A similar result holds for blow up solutions, provided they exhibit type II blow up. The soliton resolution conjecture was later proved in the non-radial case for a subsequence of time for the 3, 4 and 5D energy critical wave equation in [START_REF] Duyckaerts | Solutions of the focusing nonradial critical wave equation with the compactness property[END_REF][START_REF] Duyckaerts | Soliton resolution along a sequence of times for the focusing energy critical wave equation[END_REF]. Note that a fundamental idea in the approach of [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF] is the introduction of the method of channels of energy for the linear wave equation (see Theorem 16 for a typical result in 5D).

See similar results for the wave map problem in [START_REF] Côte | On the soliton resolution for equivariant wave maps to the sphere[END_REF][START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF][START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: II[END_REF].

Collision problem

Concerning the collision problem, we recall the discussion in [START_REF] Craig | Solitary water wave interactions[END_REF] on inelastic collisions. To study the collision problem, it is natural to study the behavior as t → +∞ of the solutions constructed in Theorems 7, 8, 9. See [START_REF] Craig | Solitary water wave interactions[END_REF], page 057106-4 for suggesting this approach which seems more canonical than to study initial data with the sum of two solitons initially distant. 7.1. Collision for the quartic gKdV equation I. We consider the quartic gKdV equation

∂ t u + ∂ x (∂ 2 x u + u 4 ) = 0 (t, x) ∈ R × R. (8)
The article [START_REF] Martel | Description of two soliton collision for the quartic gKdV equation[END_REF] (see also [START_REF] Muñoz | On the inelastic two-soliton collision for gKdV equations with general nonlinearity[END_REF] for generalization to any gKdV equation) gives the first rigorous results concerning collision of solitons for a non-integrable equation, and in particular the first proof of non-existence of pure two-soliton solutions, in the case where one soliton is much smaller than the other one.

Theorem 12 (Collision of solitons with very different size, [START_REF] Martel | Description of two soliton collision for the quartic gKdV equation[END_REF]). Assume 0 < c ≪ 1. Let u be the solution of (8) such that

lim t→-∞ u(t) -Q(• -t) -Q c (• -ct) H 1 → 0.
(i) Global stability of the 2-soliton behavior. There exist

c + 1 ∼ c→0 1, c + 2 ∼ c→0 c, ρ 1 , ρ 2 such
that the function w + defined by

w + (t, x) = u(t, x) -Q c + 1 (x -ρ 1 (t)) -Q c + 2 (x -ρ 2 (t)) satisfies lim t→+∞ w + (t) H 1 (x≥ c 10 t) = 0 and sup t∈R w + (t) H 1 c 1 3 .
(ii) Inelasticity of the collision. Moreover, for t ≫ 1,

c + 1 -1 c 17 6 , 1 - c + 2 c c 8 3 , c 17 12 w + x (t) L 2 + c 1 2 w + (t) L 2 c
11 12 .

The first part of the theorem means that the two solitons are preserved through the collision, even the smaller one. Indeed, for c small, sup t w + (t) H 1 c

1 3 ≪ Q c H 1 ∼ c 1 12 .
The second part of the theorem says that the sizes of the final solitons as t → +∞ are slightly changed with respect to their original sizes as t → -∞, and that the residue does not converge to zero. In particular, the solution is not a pure 2-soliton as t → +∞ in this regime. Thus, the collision is not elastic.

Collision for the quartic gKdV equation II.

A first intuition on the general problem of two solitons with almost same sizes comes from the explicit multi-solitons of the integrable case. From [START_REF] Leveque | On the interaction of nearly equal solitons in the KdV equations[END_REF], we have a sharp description of the behavior of the multi-soliton of (1) satisfying lim

t→±∞ u(t) -Q c 1 (• -c 1 t -σ ± 1 ) -Q c 2 (• -c 2 t -σ ± 2 ) H 1 = 0,
in the special asymptotics where 0 < µ 0 = c 2 -c 1 c 1 +c 2 ≪ 1, i.e. for nearly equal solitons. Indeed, the following global in time estimate is proved for some explicit functions c k (t), σ k (t):

sup t,x∈R u(t, x) -Q c 1 (t) (x -σ 1 (t)) -Q c 2 (t) (x -σ 2 (t)) µ 2 0 .
Moreover, it is proved that min t∈R (σ 1 (t) -σ 2 (t)) ∼ 2| ln µ 0 |. This means that the minimum separation between the two solitons is large when µ 0 ≪ 1. In particular, the two solitons never cross and the solution has two maximum points for all time. The interaction is repulsive, the solitons exchange their sizes and speeds at large distance and consequently avoid the collision. We now recall results from [START_REF] Mizumachi | Weak interaction between solitary waves of the generalized KdV equations[END_REF] for the quartic gKdV equation. Let u be a solution of (8) for which the initial data is close to the sum Q(x) + Q c (x + Y 0 ), where Y 0 > 0 is large and 0 ≤ c -1 ≤ exp(-1 2 Y 0 ), so that the quicker soliton can be initially on the left of the other soliton. It follows from [START_REF] Mizumachi | Weak interaction between solitary waves of the generalized KdV equations[END_REF] that the interaction of the two solitons is repulsive: the two solitons remain separated for all positive time and eventually u(t) behaves as

u(t, x) = Q c + 1 (x -c + 1 t -σ + 1 ) + Q c + 2 (x -c + 2 t -σ + 2 ) + w(t, x),
for large time, for some c + 1 > c + 2 close to 1 and w small in some sense. The situation for almost equal solitons of the quartic (gKdV) is thus at the main order similar to the one described in the integrable case in [START_REF] Leveque | On the interaction of nearly equal solitons in the KdV equations[END_REF]. The analysis part in [START_REF] Mizumachi | Weak interaction between solitary waves of the generalized KdV equations[END_REF] relies on techniques from [START_REF] Hayashi | Large time asymptotics of solutions to the generalized Korteweg-de Vries equation[END_REF][START_REF] Hayashi | On the modified Korteweg-de Vries equation[END_REF] and on the use of spaces introduced in this context in [START_REF] Pego | Eigenvalues, and instabilities of solitary waves[END_REF].

Before presenting the main result from [START_REF] Martel | Inelastic interaction of nearly equal solitons for the quartic gKdV equation[END_REF], for simplicity, we change variables. For c 2 -c 1 > 0 small, and any σ 1 , σ 2 , let u(t) be the unique solution of [START_REF] Bona | On the stability theory of solitary waves[END_REF]

such that lim t→-∞ u(t) -Q c 1 (• -c 1 t -σ 1 ) -Q c 2 (• -c 2 t -σ 2 ) H 1 = 0. Let c 0 = c 1 + c 2 2 , µ 0 = c 2 -c 1 c 1 + c 2 , y 1 = σ 1 √ c 0 , y 2 = σ 2 √ c 0 . Then the function U (t, x) = c -1/3 0 u(c -3/2 0 t, c -1/2 0 (x + t)) solves ∂ t U + ∂ x (∂ 2 x U -U + U 4 ) = 0, (t, x) ∈ R × R, (9)
and is the unique solution of ( 9) satisfying

lim t→-∞ U (t) -Q 1-µ 0 (• + µ 0 t -y 1 ) -Q 1+µ 0 (• -µ 0 t -y 2 ) H 1 = 0.
The next result concerns the asymptotics µ 0 > 0 small. Theorem 13 (Inelastic interaction of two nearly equal solitons, [START_REF] Martel | Inelastic interaction of nearly equal solitons for the quartic gKdV equation[END_REF]). Assume 0 < µ 0 ≪ 1.

Let U be the unique solution of (9) such that

lim t→-∞ U (t) -Q 1-µ 0 (• + µ 0 t + Y 0 + ln 2) -Q 1+µ 0 (• -µ 0 t -Y 0 -ln 2) H 1 = 0, where Y 0 = 1 2 | ln(µ 2 0 /α)| and α = 12(10) 2/3 ( Q 2 ) -1 .
Then the following holds. (i) Global stability of the 2-soliton behavior. There exist µ 1 , µ 2 , y 1 , y 2 such that

w(t, x) = U (t) -Q 1+µ 1 (t) (x -y 1 (t)) -Q 1+µ 2 (t) (x -y 2 (t)) satisfies | min t∈R (y 1 (t) -y 2 (t)) -2Y 0 | µ 0 and lim t→+∞ w(t) H 1 (x>-9 10 t) = 0, sup t∈R |w(t) H 1 µ 3 2 0 .
(ii) Inelasticity of the interaction.

lim +∞ µ 1 -µ 0 µ 5 0 , µ 0 -lim +∞ µ 2 µ 5 0 , lim inf t→+∞ w(t) H 1 µ 3 0 .
It follows that no pure 2-soliton exists also in this regime. The proofs of Theorems 12 and 13 are based on the construction of a refined approximate solution of the two-soliton problem for all t and x.

Collision for the quartic gKdV equation III.

Still concerning the collision of two solitons for the quartic gKdV equation, we recall from [START_REF] Martel | On the nonexistence of pure multi-solitons for the quartic gKdV equation[END_REF] the following negative result.

Theorem 14 (Inelasticity of collision for gKdV, [START_REF] Martel | On the nonexistence of pure multi-solitons for the quartic gKdV equation[END_REF]).

Let K ≥ 2, 0 < c K < • • • < c 1 = 1 and σ 1 , . . . , σ K ∈ R. Let u be the solution of (8) satisfying lim t→-∞ u(t) - K k=1 Q c k (• -c k t -σ k ) H 1 = 0. Assume that K k=2 (1 -c k ) 2 < 1 16
. Then, u(t) is not an asymptotic multi-soliton as t → +∞. In particular, there exists no pure multi-soliton of (1) with the speeds c 1 , c 2 , . . . , c K .

In the case of two solitons, the condition on the speeds reduces to 3 4 c 1 < c 2 < c 1 . In contrast with Theorems 12 and 13, the result in Theorem 14 is not perturbative and the explicit condition on the speeds seems technical. The strategy of the proof of Theorem 14 is to study the asymptotic behavior of u(t, x) for any t and for any large x (i.e. far from the collision region, which seems impossible to describe in the general case) and to find a contradiction with the fact that u is an asymptotic two-soliton in the two directions of time. Being a proof by contradiction, it does not give further information on the collision. 7.4. Collision for the perturbed integrable NLS equation. Let β > 0 and 0 < c ≪ 1. Under the following assumptions for the perturbation |f (u)| 0 u 2 , f (u) |u| q (q < 2), it is proved in [START_REF] Perelman | Two soliton collision for nonlinear Schrödinger equations in dimension 1[END_REF], that there exists a solution u of

i∂ t u + ∂ 2 x u + |u| 2 u + f (|u| 2 )u = 0, (t, x) ∈ R × R, satisfying lim t→-∞ u(t) -e it Q -e iΓ β (t,•) Q c (• -βt) H 1 → 0,
and for which the small soliton splits in two after the collision, in the following sense

u(t, x) ∼ e iΓ(t,x) Q(x -σ(t)) + ψ + (t, x) + ψ -(t, x),
where ψ ± are solutions of (2) corresponding to the transmitted part and the reflected part of the small soliton. The above estimate holds on large time intervals after the collision depending on 1/c. The splitting means some strong form of inelasticity. 7.5. Collision for the 5D energy critical wave equation. In view of the results described so far, it was natural to search a situation where a non-perturbative approach would allow to treat all two-soliton collisions. In the case of the 5D energy critical wave equation, we now recall from [START_REF] Martel | Inelasticity of soliton collisions for the 5D energy critical wave equation[END_REF] a result showing the inelastic nature of the collision of any two solitons, except the special case of same scaling and opposite signs.

W ∞ k (t, x) = ǫ k (λ ∞ k ) 3 2 W ℓ k x -ℓ k t -y ∞ k λ ∞ k .
Assume that ℓ 1 = ℓ 2 and ǫ 1 λ ∞ 1 + ǫ 2 λ ∞ 2 = 0. Then, there exists a solution u of (6) in the energy space such that (i) Two-soliton as t → -∞ lim t→-∞ ∇ t,x u(t) -∇ t,x (W ∞ 1 (t) + W ∞ 2 (t)) L 2 = 0.

(ii) Dispersion as t → +∞. For all A > 0 large enough,

lim inf t→+∞ ∇u(t) L 2 (|x|>t+A) A -5 2 . ( 10
)
Note first that the solution constructed in Theorem 15 is a two-soliton asymptotically as t → -∞ and that it does not necessarily exist for all t ∈ R. However, by finite speed of propagation and small data Cauchy theory, it is straightforward to justify that it can be extended uniquely as a solution of (6) for all t ∈ R in the region |x| > |t| + A, provided that A is large enough. Thus, the limit in (10) makes sense. Since the estimate [START_REF] Bona | Stability and instability of solitary waves of Korteweg-de Vries type[END_REF] gives an explicit lower bound on the loss of energy as dispersion as t → +∞, the solution u is not a two-soliton asymptotically as t → +∞ and the collision is inelastic. Note that the two-soliton could have any global behavior, like dislocation of the solitons and dispersion, blow-up or a different multi-soliton plus radiation, but the property obtained is universal.

The only case left open by Theorem 15 corresponds to the dipole case. It is the first result proving inelasticity rigorously without restriction on the relative sizes or speeds of the solitons except an exceptional case.

The strategy of the proof is to construct a refined approximate solution of the two-soliton problem for large negative times that displays an explicit dispersive radial part at the leading order and then to propagate the dispersion for any positive time at the exterior of large cones by finite speed of propagation and the method of channels of energy from [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF][START_REF] Kenig | Relaxation of wave maps exterior to a ball to harmonic maps for all data[END_REF]. To finish, we recall such a typical result of channel of energy for the radial linear wave equation in 5D from [START_REF] Kenig | Relaxation of wave maps exterior to a ball to harmonic maps for all data[END_REF], Proposition 4.1 (see also [START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation : the non-radial case[END_REF][START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF][START_REF] Kenig | Channels of energy for the linear radial wave equation[END_REF]).

Theorem 16 (Exterior energy estimates for the 5D linear wave equation, [START_REF] Kenig | Relaxation of wave maps exterior to a ball to harmonic maps for all data[END_REF]). Any radial energy solution U L of the 5D linear wave equation where π ⊥ R (U 0 , U 1 ) denotes the orthogonal projection of (U 0 , U 1 ) onto the complement of the plane span{(|x| -3 , 0), (0, |x| -3 )} in ( Ḣ1 × L 2 )(|x| > R).

∂ 2 t U L -∆U L = 0, (t, x) ∈ R × R 5 , U L|t=0 = U 0 ∈ Ḣ1 , ∂ t U L|t=0 = U 1 ∈ L 2 ,

Theorem 15 (

 15 Inelasticity of soliton collisions for NLW,[START_REF] Martel | Inelasticity of soliton collisions for the 5D energy critical wave equation[END_REF]).Let d = 5. For k ∈ {1, 2}, let λ ∞ k > 0, y ∞ k ∈ R 5 , ǫ k ∈ {±1}, ℓ k ∈ R 5 with |ℓ k | < 1, and

  U L (t, x)| 2 + |∇U L (t, x)| 2 dx π ⊥ R (U 0 , U 1 ) 2 ( Ḣ1 ×L 2 )(|x|>R)

satisfies, for any R > 0, ± lim sup t→±∞ |x|>|t|+R |∂ t

We refer to Chapter 1 of[START_REF] Dauxois | Physics of solitons[END_REF] for historical facts on this equation and its applications to Physics.


A Short History of the Investigation of the Origin of the Terrestrial

Biosphere's Homochirality (Some Personal Remarks, Recalling Historical Milestones) Wolfram Hans-Peter Thiemann * University of Bremen, Germany *Correspondance : wolframthiemann@googlemail.com DOI : XXXX -000 Submitted: July 20, 2018 -Published: August 12, 2018 Volume : 4 -Année : 2018 Titre du numéro: Asymmetry Editors: Giovanni Fusco, Alice Guyon, Ulrich Kuhl, Uwe Meierhenrich and Frédéric Patras Résumé A description of the history of the investigation of the origins of homochirality in our terrestrial biosphere over the last half decade is given here. It tells about the moment, when the discovery of the hitherto rather untold puzzle of the fact that biology is so biased in the structure of its chemical elements (biology produces almost exclusively left-or right-handed molecules!): In stark contrast to the laboratory experience, that whenever a (strictly abiotic) synthesis of the basic chemicals making up biology was undertaken-like in particular amino acids and sugars -it always resulted in the production of a racemic (exact 1:1) mix of left-and right-handed molecules. Is it due to an accident ("Was God a lefthander?"), is it a physical law, and is it really necessary that living nature had to consist exclusively of the left-(or the right-) handed basic molecules to grow spontaneously from symmetric simple molecules during the evolution up to the subtle complex homochiral structures making up biochemistry today? The essay attempts to give credit to some of the pioneers of this scientific research, who quite often have been overlooked in spite of their essential contribution to the field. The importance of the tool of an analysis of homochirality in probing for traces of life beyond Earth is underlined and supports a further systematic investigation on extraterrestrial bodies in future space missions.

Mots-clés

chirality; enantiomers; homochirality; symmetry breaking

I THE VERY FIRST FLASH

It happened, I guess, very early in my scientific career, entering my university in München, -it was the Ludwig-Maximilian-Universität with its full name in the Department of Chemistry's newly opened laboratories -, where I listened eagerly to my professors in my second semester in the year 1957. Personally almost everything was completely new for me, having left my "Gymnasium" focusing on old languages like Latin and Old Greek and neglecting pretty much the natural science courses like physics and chemistry. I had thought there must be more in it than the little boring things being taught there, so it was exactly my curiosity for this almost completely forgotten field chemistry that attracted me to learn more about it in my following university years. So in this fall 1957 it was the peculiar subject of Organic Chemistry when Prof. R. Huisgen lectured about the biological important group of amino acids. Among others he taught us that a (non-biological!) synthesis of these compounds was rather easily accomplished mixing the necessary ingredients together, such as water, acetaldehyde, cyanic acid (or ammonia), heating, -and finished was the product of alanine for instance, called after its inventor, the Strecker Synthesis (after the famous chemist Adolph Strecker, who succeeded to synthesize the basic amino acid alanine from its abiotic precursors) showing that the synthesis of biochemical compounds could be simply facilitated. The lecturer stressed however the point that principally the product of such an "abiotic" synthesis was always the appearance of 2 so called "enantiomer" amino acids, the one of the geometrically left, the other one of the corresponding right conformation. Both forms, the L-(sometimes also called S-) α-alanine and the D-(sometimes also called R-) α-alanine appearing in the product at exactly equal stoichiometric concentrations or masses, being exactly identical except for the fact that they were the mirror images of each ot her making them non-superimposable (in spite of their otherwise identical properties), in a similar manner like our two hands behave to each other, being almost identical, but of course not superimposable. That, lectured "ex cathedra", had to be accepted from us students in our first year. But at the very end of the semester, at the point when it came to the chapter "Natural Compounds", the domain of Biochemistry, Huisgen drew a disturbing, yet fascinating conclusion to the basic course in Organic Chemistry: He claimed that the alanines, -the simplest amino acid to show this stereochemical enantiomer property of appearing in two antagonistic forms, an n-amino propionic acid -and all the other existing amino acid homologues identified in the biological tissue everywhere on Earth were surprisingly only of the (arbitrarily defined) L-form! Similarly, one could show that not only the amino acids making up living matter showed this strange property, but also all the other of the myriads of different specific biomolecules showed this very peculiar "Broken Symmetry"; next to the amino acids it is particularly the group of biological sugars which demonstrate this broken symmetry, -only the D-enantiomers of sugars 1 show up in our terrestrial biology. This observation, first heard in the great auditory hall of the University München, struck me hard,and never let me go. What could be or could have been the hidden reason for this very confusing symmetry breaking in nature ? Huisgen gave no answer as I remember, of course in these days a first-year-student did not dare to ask the professor at this desk in front of about 1000 students in the big auditory an obviously a bit critical question on a generally accepted textbook wisdom, which needed a special explanation. This episode of my first year in chemistry haunted me over the years, WHY for heaven's sake was the nature around us so symmetry-biased in view of the beautifully symmetrical laws in physics and chemistry. It does not matter at all whether we believe in creation of the world around us by a sort of GOD, such as the conventional wisdom of our Bible, Koran or Thora tries to teach us or whether we rather believe in Darwin's Evolution Theory, which of course is much more widely accepted among scientists, but it is difficult to assume a left-handed creator 1 L-amino acids and D-sugars are strictly complimentary to each other as seen in particular in the interction of DNA/RNA with the proteins at work during biosynthesis. The connotation that the L-amino acid enantiomer connects to the D-sugar enantiomer, is only an accidental « mirror image » of each other, caused by the rather arbitrary definition of L-and D-molecular geometry.

or "intelligent design" or any sort of physical law which threw life into existence with this preference of one side only, the so called "homochiral" feature…

II HOW IT WENT ON WITH THE STRANGE PUZZLE OF MY OWN BEGINNINGS IN THE VAST FIELD OF CHEMISTRY

It took me in fact almost eleven more years of waiting when a unique chance opened for me in the year 1968 to focus my scientific curiosity on the study of the origins of biochemical Homochirality. I joined the recently opened Nuclear Research Institute (KFA, now Forschungszentrum FZ) Jülich in Germany, where my assignment was "to do some innovative research of strictly non-nuclear character" of my own choice. What an opportunity, the 60's were a period of economic boom, young and ambitious scientists were offered numerous careers in research areas, without too many limitations, not being forced to guarantee economic profits at all stages of their projects without being haunted by the prospect of getting fired after the end of time-limited research grants like the situation appears for young scientists momentarily. I was allowed to build a small working group devoted exclusively to the subject of Homochirality. Of course I learned quickly,-searching the available literature -that I was by no means alone in the field. There were great people before me who occasionally devoted their interest to this peculiar topic, but their number was small and they left the subject quite often far from any final solution. Actually it was the great Louis Pasteur of course who opened the field by observing that the basic organic element in grapes was tartaric acid (or rather the salts of the acid), existing in 3 different forms, a mixture of the L-and the D-enantiomerdistinguishable by their unique property to rotate a linearly polarized beam of light either to one or the other direction -and a meso-tartaric form, which showed no such phenomenon of any such rotation. As we know Pasteur believed in a special "force de vie", a particular "force of life" reserved to biochemical matter which could not be synthesized in the laboratory. In the long time after Pasteur's fundamental discovery it was firmly established that this phenomenon of Homochirality -that is the break of molecular symmetry -holds in general for (almost!) all the organic elements in biology, of which all the amino acids and carbohydrates are the most pronounced examples.

The very nice thing in this project -we are talking here about the late 60's to 70's, where there was not even the slightest idea of a world wide web enabling one to search the literature in fractions of a second! -was the fact, that obviously the number of living researchers working in this peculiar, rather exotic field of chemistry, was rather small. A few names to be remembered may suffice at this point only: G. Wyrouboff, P. Curie, Y. Yamagata, W. Bonner, T.L.V. Ulbricht, F. Vester, A. S. Garay, F. F. Seelig, C. Ponnamperuma, G. Spach, A. Brack, and some others were among those, who devoted their energy to study this outstanding phenomenon of Homochirality enthusiastically.

Our Jülich Approach consisted of a model to be described as such: Consider theoretically that the mirror images of the D-and the L-enantiomers of any amino acids making up the body of proteins in the biological organisms were not exactly identical, but only nearly identical, the energy of formation perhaps distinguishable perhaps by an extremely minute amount only, so small that hitherto available analytical instruments were not able to prove such a difference of perhaps the L-and the D-alanine; then it may have been overlooked so far that in the end of a slowly evolving biosphere from the beginning some few billion years ago on Earth the Lalanine had an extremely slight, yet definite advantage of growing finally into a selfreproducing cell, overriding ultimately the energetically inferior D-alanine in the end. The phase of the Chemical Evolution on early Earth could be imagined as an amplification process which eventually would enrich the energetically favored enantiomer up to an end product of ISSN: xxxxxxx, ©JIMIS, Creative Commons Revue en libre accès : j i m i s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000 100 % homochirality. Such an amplification -enriching a favored enantiomer, small as the initial advantage over the competing compound would be -could be simulated in our laboratory. We had at this point yet no idea whatsoever from which such a minute difference of the enantiomer compounds would eventually stem. We at Jülich designed 2 different experimental lines, the first one using crystallization as the amplification machine, with the aim to prove or disprove such an energetic preference.( 1) We looked at the potassium ammonium salt of the tartaric acid, the very same compound which lead L. Pasteur to the discovery of chirality. A racemic mixture was crystallized from its saturated aqueous solution, for several times repeatedly collecting the solid crystals and dissolving freshly the mixture at higher temperatures and letting it crystallize at lower temperatures again. At the end of some cycles we could measure a small optical rotation by a sensitive measurement, which allowed us to calculate the "elementary energy difference" of L-and D-tartrates.

The second experimental procedure was as follows: We subjected some exactly racemized mixtures of D-and L-alanine, -lysine, and alpha-aminobutyric acids to a several weeks long polycondensation reaction resulting in the production of the corresponding polymers. These polymers were investigated for any preferred optical rotation, below or above zero. Our hypothesis was that, if the rotation of the polymer showed any significant rotation, then the inherent energetic bias -expressed in the kinetic preference in building the polymer -would be amplified to such a value, which could be measured by our conventional spectropolarimeter. Logic? And the final result published by us, together with my co-worker, the late W. Darge (2) was in fact a confirmation of a very small, but significant excess of the D-enantiomers of the investigated amino acids. The problem was of course that the re-calculated energy difference of the monomeric enantiomers was too big -in the order of a few per mille in relation to the total value of 1-2 kJ or so, that the work was generally not well received in the scientific community. The even more important point of criticism was the open question how such a hypothetical energy difference of enantiomer carbon compounds should be explained at all; such a statement was in stark contrast to all hitherto common knowledge, published in textbooks of chemistry as a sort of well-established principal dogma of its kind.

III A TRUE REVELATION

There were two things discovered in literature which appeared like a sort of a true enlightenment supporting our experimental design: First it was a fundamental short essay by Y. Yamagata (3) in distant Japan, who apparently argued along similar lines like ours in Jülich, small energy differences in enantiomer molecules could be "amplified" in the process of chemical evolution to such an excess that we observe today in our 100 % pure homochiral biosphere. This very short contribution helped strengthening our peculiar view. And it was him, who pointed to a so called "Parity Violation in the Weak Interaction", a fundamental statement for which formulation 2 Chinese-born US-scientists, namely T. D. Lee and C. N. Yang (4) were awarded the Nobel Prize in Physics in 1957. They reported a peculiar phenomenon in physics, quite revolutionary at this point more than 60 years back in time from now: The normally little known so called Weak Interaction showed a strong "Asymmetry":

We observe here a similar phenomenon as in our electromagnetic forces ruling biochemistry, where we observe only one enantiomer, the other one missing! In Lee and Yang's Weak Interaction -the experiment run by Ms. Wu confirmed the theoretical prediction -we see that the atom leaving ß-particle possesses only one spin along the path, while the other mirror image of this spin cannot be observed. Strange! Immediately it comes to one's mind, that perhaps the one phenomenon in physics could be related to the other one in biochemistry, could it? The first researchers following this idea directly after Lee and Yang's discovery were T. L. V. Ulbricht and F. Vester (5) attempting a link between these phenomena and proving a connection by some experimental set-up. The second part of enlightenment was a personal meeting in Moscow with my late friend and excellent scientist Andras Garay, (6) who was also busy proving the basic idea of transferring the parity violation of the weak interaction to the realm of chemistry, -could it be that the fact that life is so much biased goes back basically to the nuclear property of the atom?

In September 1973 we decided to call these rather few colleagues in the field around the world to Jülich to exchange the latest ideas and results touching this fascinating subject of "Broken Symmetry in Chemistry" in order to stimulate a much broader interest. It was the prominent scientist A. S. Garay

IV A NEW AREA OPENED, WHICH SUPPORTED EARLY HYPOTHESES

At the University of Oxford there arouse a group of theoretical chemists, who calculated abinitio the effect, which the "Parity Violation of the Weak Interaction" would have on the molecular energy content of chiral organic chemicals, in particular on amino acids. And, -alas, they found that this nuclear property would indeed, as speculated by several authors before, affect the entire molecular structure in such a way, that the L-enantiomer would have a small difference to its corresponding enantiomer, the D-enantiomer. The Oxford group around S. F. Mason consisted of A. J. MacDermott, G. E.. Tranter, P. A. Hegstrom, (9a -d) who confirmed our earlier thesis, that enantiomers are distinguishable by a certain difference, called from then on the PVED -the Parity Violation Energy Difference. We first celebrated this new knowledge supporting our earlier starting point, against all conventional textbook wisdom, "Enantiomer compounds are just the exact mirror image of each other" was not true anymore:

We learned that the true enantiomer of the L-alanine for example was the D-alanine from antimatter! And vice versa! Yet our joy was a bit too early: First of all the PVED was so immensely small, something in the order of 10 -14 eV, that most people doubt whether this extremely small effect could ever have directed chemical evolution into the 100 % result of Damino acids in the present biosphere at all. Any advantage of the L-amino acids would have been overruled in the billions of years of evolution on Earth by a simple randomizing racemisation (induced by heat, radiation, pressure, etc.). Another result led to the general scepticism that the surprisingly found energetic advantage of the "biologically just fitting", namely the L-amino acid enantiomer depended on the exact conformation of the 3-dimensional structure of the compound, which is rather variable in its structure in vacuo, solid state, or dissolved (in water) state. Anyway the activities around the value of the effect of parity violation in weak interactions inducing possibly the geometrical structure of the compounds triggered an enormous interest throughout academic institutions around the world. The problem is now discussed at many occasions, throughout international conferences, papers in respected journals and monographs.

V THE SEARCH FOR HOMOCHIRALITY'S ORIGINS EXTENDED AT AN ENORMOUS RATE

In all these years since our first cautious hesitating steps in the early 70's into this rather novel research field the search for the rather mysterious origin of the chemical homochirality gained an enormous scientific popularity: My own rough guess would be that around the globe a few ISSN: xxxxxxx, ©JIMIS, Creative Commons Revue en libre accès : j i m i s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000 hundreds of scientists have been engaged investigating this problem. And with the number of scientists the number of theories and more or less weird theories have accumulated attempting to demonstrate numerous physical reasons for the rise of this truly striking phenomenon of our terrestrial biological homochirality. Earth rotation was simulated in the laboratory influencing the outcome of stereoselective chemical products, Earth magnetic field operating jointly with specific electric fields were tested to affect the stereoselective product, even Earth gravitation as well as sunlight being broken through the Earth early atmosphere was proposed to have influenced the first asymmetric prebiotic molecules of the early Earth some billion of years ago during the chemical evolution.

Out of all these rather phantastic ideas a strong factor, competing seriously with the PVED effect as the initiator of the stereoselective result of terrestrial evolution of the biosphere, stands out as an influence: A circularly polarized UV radiation -called "stereoselective photochemical reaction" -may have pushed the origin of the first (pre-) biological cells during or after the Hadrosphere phase of Earth into the homochiral direction as we observe it today. It is well known, since the 30's of the last century by W. Kuhn and E. Knopf, (10) that a circularly polarized UV beam could interfere with a suitable chemical reaction in such a way, that the resulting product would contain an excess of one or the other enantiomer, depending on the direction left or right of the polarized light beam. In the last 30 or so years it has been speculated that the young planet Earth got in contact with some strong UV light sources emitted from a nearby neutron star, which was to some extent polarized. And this strong light source interacted with the chemical reagents forming the first organic compounds of some biological use in a stereoselective way. Principally we must assume though that there might have existed only one strong event of such an influence from outside Earth, and this must have been of the preferred polarisation direction forming preferentially the first excess of L-amino acids and D-sugars as we see them in the biosphere. Numerous are laboratory experiments and theories supporting this speculation apparently confirming such a theory on the origin of the fitting homochiral products. Among the first authors was the group of W. A. Bonner (11) reporting such effects in the late 70's and early 80's. Up until now, it was in particular the Nice University Group of U. Meierhenrich, [START_REF] Good | overview of the research going on in Nice can be gained from the monograph: Meierhenrich U[END_REF] who published a great number of results supporting the polarized UV-radiation-directed origin of homochirality in an impressive amount of papers published in numerous international journals and monographs. I must admit that the vast material so far published up to today would likely support convincingly the process of polarized UV radiation having induced stereoselectivity on Earth, although of course the above discussed PVED effect is up to this date not yet completely out of the game. (To my personal opinion the PVED, being a universal property of matter, would work in the same way throughout the whole universe, while on the other hand, the direction of the polarized UV radiation would be restricted to our own Solar System only! So in the end, the PVED effect would deserve the attribute of possessing more "beauty" in its own…)

Moreover, not to be forgotten at all, there are of course many scientists around the world, who do not believe in a sort of "directed" chiral decision, a number of arguments could be raised, too, in favour of a random choice of the presently preferred L-amino acids and the D-sugars: Pure accidence could have stirred the evolution as we see its products today, an accidentally chosen beginning of the "first dissymmetric cell" could have triggered the homochiral solution, leaving the other enantiomer design no chance anymore, once self-reproduction, etc. had started in favour of the L-amino acid and D-sugar world at a considerable rate. "God creating the living world must have been a left-hander..." as somebody had expressed this scenario in a beautiful short way.

As this essay should focus on the first steps into the realm of the studies on the origin of homochirality only, not ever only attempting to describe the enormously expanding (hi-)story of the field up to today, a few thoughts should be devoted here to highlight the very latest developments and applications of the field.

VI RECENT CONSEQUENCES OF OUR STUDIES INTO THE FIELD OF HOMOCHIRALITY

Wherever the true origins of the homochirality of our terrestrial biosphere lie and whether any "Eureka" claims could ever be proven at all, the whole scientific community seems to agree that homochirality is perhaps the unique mark of anything which claims to be "living/biological" on Earth and beyond Earth. A biological entity built up from basically symmetrical chemical elements seems to be extremely implausible at all. All the complex macromolecular structures of our biochemistry can be designed by nature only on the ground of homochiral elements, just imagine a DNA, RNA, or a polymer carbohydrate like starch/lignin built from monomer racemic molecules alone. One would never ever find any complex, functional, and beautiful structures like specific helices and the various well-defined tertiary and quaternary structures which make up our biochemical architecture. This unique property of "life" seems to distinguish living from non-living materials. It looks as if this homochiral precondition would even hold for any imagined living entities built from some alien non-carbon based chemicals.

It should be allowed here to roam a bit on the thermodynamics of a living cell in sharp contrast to a sort of mineral (non-living) cell. The majority of researchers would allow a living entity to break for a limited amount of time and space the law of entropy production, the so called Second Law of Thermodynamics, stating that in any natural and spontaneous process the total entropy of the system is forced to increase. The lone exception from the rule is the territory of life (13): A living organism is allowed to break this law for a certain amount of time and space only, -in other words, biology alone allows to reduce the entropy of an open (semi-isolated) system for a certain time regime! Examples of this phenomenon is the well accepted theory of (Darwinian) evolution, where we observe that systems of higher order evolve spontaneously from systems of lower order, -as well as the simple observation of the development of a cell or a multicellular organism from the fertilization of a rather simple egg, truly a process from less to higher order, or from larger to lower entropy of the system, proceeding without any help from outside the system! From here it follows logically that in the search for life outside planet Earth we should focus our attention to systems of a -more or less -homochiral nature, independent of the problem, whether the observed system is of a familiar carbonaceous nature (like ours! Alas, we have no other living system than our own terrestrial, built from carbonaceous material, to compare with!) or not. This line of argumentation has long since taken up into the design of space vehicles exploring the universe around us for traces of non-terrestrial lives. Many years ago a fundamental paper was published along this lines of thoughts by a consortium of researchers proposing such an ambitious project SETH (stands for Search for Extraterrestrial Homochirality), cf. A. J. MacDermott et al.( 14) Spectacular was the rather recent success of the ESA sponsored space mission ROSETTA/PHILAE, when a module landed on the comet 67P/Churyumov-Gerasimenko, which aimed at analyzing the surface for organic compounds. (15) For the first time the vehicle carried -among many other instruments -onboard a gas chromatograph coupled with a mass spectrometer equipped to identify (homo-?)chiral organic chemicals like enantiomer amino and other acids. While the identification of some 20 organic compounds by mass spectrometer succeeded, unfortunately the gas chromatograph instrument was not able to do its job so far properly, there were no results satisfying our curiosity for the potential existence of chiral amino acids on this comet. Yet, other projects aiming on the analysis of chiral compounds on solid objects within our solar system are well under way, such as the ExoMars mission sponsored by the European Space Agency ESA having on board similar analytical equipments as on our ROSETTA mission. Japanese researchers are aiming to study within a return mission potentially chiral compounds on some asteroids, the rover CURIOSITY from the last DISCOVERY Mars mission organized by NASA have similar analytical tools on board and have reported already some, yet hitherto not confirmed, data produced by some similar chirality detecting devices like ours.

Of course the run for detecting any living features beyond Earth seems to bear fascinating results on the existence of life in the Universe in the next future. Moreover the ever increasing number of newly discovered exoplanets outside our own Solar System is the target of interesting speculations on a potential biological habitat farther away from our home solar system. Will be ever able to discover living species there? Perhaps an answer could lie in the more systematic search for homochirality there; we do not necessarily have to go there and pick up some samples, which seems rather hopeless for the next future due to the immense distances separating us from these targets. But we could try to get some information through spectroscopic investigations looking for some alien biospheres emitting or reflecting some polarized light telling us about possible homochiral structures on their home planets.

As a hopeful young scientist, attending as a humble guest a NASA sponsored meeting back in the early 70's preparing the 2 spectacular VIKING missions to planet Mars, I dared to propose cautiously to carry a simple polarimeter (measuring the optical rotation of a chiral compound) onboard the mission to look for any(homo-)chirality of organics, and was not surprised -of course! -that my remark was not taken seriously, on the contrary I earned a rather arrogant smile from the audience. This very first idea of applying a polarimeter searching for life anywhere in this Universe giving us perhaps some hints for receiving an answer to our question "Is there anybody out there?" as to where exactly to direct our sensors, was possibly after all not so naive…

  ISSN: xxxxxxx, ©JIMIS, Creative Commons Revue en libre accès : j i m i s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

  ISSN: xxxxxxx, ©JIMIS, Creative Commons Revue en libre accès : j i m i s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

  ISSN: xxxxxxx, ©JIMIS, Creative Commons Revue en libre accès : j i m i s . e p i s c i e n c e s . o r g Volume : N -Année : AAAA, DOI : XXXX -000

  (Director of the Biology Research Center Szeged, Hungary), C. Ponnamperuma (Prof. Univ. Maryland and Director of the Laboratory of Chemical Evolution USA), K. Wagener (Inst. Phys. Chemistry, KFA Jülich) and W. Thiemann who succeeded to assemble about 40 participants to discuss and argue about the latest ideas, theories and experiments on this subject. The whole Symposium was published (7) -a total of 540 pages inclusive the accompanying completely recorded discussions; the central contributions to this Jülich Conference were published in the Journal of Molecular Evolution. Another international symposium on "Generation and Amplification of Chirality in Chemical Systems" was organised in 1980 at the University of Bremen by the author jointly with C. Ponnamperuma, published as a separate issue[START_REF]Generation and Amplification of Chirality in Chemical Systems[END_REF], promoting even a greater interest in this topic within the international scientific community.