domain: math nlin

text:
Sergey Gavrilyuk
email: sergey.gavrilyuk@univ-amu.fr

Boniface Nkonga
email: boniface.nkonga@unice.fr

Keh-Ming Shyue
email: shyue@ntu.edu.tw

The conduit equation : hyperbolic approximation and generalized Riemann problem

Keywords: Mathematics Subject Classification numbers: 35L40, 35Q35, 35Q74

The conduit equation is a dispersive non-integrable scalar equation modeling the flow of a low-viscous buoyant fluid embedded in a highly viscous fluid matrix. This equation can be written in a special form reminiscent of the famous Godunov form proposed in 1961 for the Euler equations of compressible fluids. We propose a hyperbolic approximation of the conduit equation by retaining the Godunov-type structure. The comparison of solutions to the conduit equation and those to the approximate hyperbolic system is performed : the wave fission of a large initial perturbation of a rectangular or of gaussian form. The results are in good agreement. New generalized solutions to the conduit equation are discovered composed of a finite set of waves of the same period and linked with a constant solution by generalized Rankine-Hugoniot relations. Such multi-hump structures interact which each other almost as solitary waves : they collide, merge and reconstruct after the interaction. This partly indicates the stability of such multi-hump solutions under small perturbations. Both the exact and approximate hyperbolic system describe such an interaction with good accuracy.

Introduction

Consider the conduit equation :

u t + u 2 + u x u t -uu tx x = 0, (1)
involving one dependent variable u(t, x) and two independent variables t (time) and x (space coordinate). Physically, the equation (1) represents the mass conservation law written in dimensionless variables for the magnitude u(t, x) of the non-dimensional circular cross-section of a low-viscosity buoyant fluid embedded in a highly viscous fluid at rest (cf. [START_REF] Olson | Solitary wave propagation in a fluid conduit within a viscous matrix[END_REF][START_REF] Lowman | Dispersive hydrodynamics in viscous fluid conduits[END_REF][START_REF] Lowman | Dispersive shock waves in viscously deformable media[END_REF][START_REF] Lowman | Interactions of large amplitude solitary waves in viscous fluid conduits[END_REF][START_REF] Maiden | Observation of dispersive shock waves, solitons, and their interactions in viscous fluid conduits[END_REF][START_REF] Maiden | Modulations of viscous fluid conduit periodic waves[END_REF][START_REF] Maiden | Solitary wave fission of a large disturbance in a viscous fluid conduit[END_REF]). Hence, only positive solutions u(t, x) are physically admissible. Another conservative form which, a priori, has no physical meaning can be found for the conduit equation :

1 u + u xx u t -(2 ln (u)) x = 0. ( 2
)
Its mathematical importance will be shown later. The dispersion relation for the conduit equation linearized on the solution u = u 0 = const > 0 can be written as

c p = 2u 0 1 + u 0 k 2 , ( 3
)
where k is the wave number, and c p is the corresponding phase velocity. It has the same dispersive properties as the Benjamin-Bona-Mahony (BBM) equation [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]. The derivation of the modulation equations to the conduit equation and their stability study for small amplitude waves has been performed in [START_REF] Maiden | Modulations of viscous fluid conduit periodic waves[END_REF][START_REF] Johnson | Modulational instability of viscous fluid conduit periodic waves[END_REF].

In [START_REF] Favrie | A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves[END_REF][START_REF] Dhaouadi | Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation[END_REF][START_REF] Busto | On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems[END_REF][START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF][START_REF] Besse | Perfectly matched layers methods for mixed hyperbolic-dispersive equations[END_REF][START_REF] Tkachenko | Extended Lagrangian approach for the numerical study of multi-dimensional dispersive waves: applications to the Serre-Green-Naghdi equations[END_REF], a general method of hyperbolic regularization of dispersive equations that are the Euler-Lagrange equations for a "master" Lagrangian has been proposed: the original high order derivative dispersive equations were approximated by a first order hyperbolic system of the Euler-Lagrange equations for a one or two parameter family of "extended" Lagrangians. The "master" Lagragian is obtained from the "extended" Lagrangian in some limit. Thus, the variational structure of the governing equations was conserved. In a particular case of long gravity surface waves described by the Serre-Green-Naghdi equations, the method of "extended Lagrangian" was mathematically justified in [START_REF] Duchêne | Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model[END_REF]. The advantage of such an approach is obvious: one can use for dispersive equations the full range of finite volume methods developed for hyperbolic equations. Furthermore, some non-linear dispersive equations admit shock-type solutions: the dispersion cannot always prevent the formation of such singularities [START_REF] Gavrilyuk | Stationary shock-like transition fronts in dispersive systems[END_REF][START_REF] Gavrilyuk | Singular solutions of the BBM equation: analytical and numerical study[END_REF]. Thus, the study of such singularities can be done much more easily when the hyperbolic approximation is used.

Despite a large number of works on the conduit equation, unlike the almost similar BBM equation (at least in the linear limit), neither other linearly independent conservation laws for this equation, nor the existence of a Lagrangian allowing this equation to be considered as the Euler-Lagrange equation, are known [START_REF] Johnson | Modulational instability of viscous fluid conduit periodic waves[END_REF]. Having in mind to approximate the conduit equation by a system of hyperbolic equations, we then ask the following question: what is the mathematical structure (different from a classical variational structure) of the conduit equation, and should it be retained when the equation is approximated by a system of hyperbolic equations?

In this paper, we will exhibit such a structure and will formulate an approximating hyperbolic system of equations conserving this structure. The compar-iason of numerical solutions to the exact conduit equation and to its "structure conserving" hyperbolic approximation show a very good convergence results.

2 Mathematical structure of the conduit equation

In 1961 [START_REF] Godunov | An interesting class of quasi-linear systems[END_REF] S. K. Godunov proposed the following abstract form of a system of conservation laws for the vector variable v = (v 1 , ..., v n ) T :

∂L 0 (v) ∂v t + m i=1 ∂L i (v) ∂v xi = 0, (4)
with given functions (potentials) L i (v), i = 0, ..., m. This system admits an additional conservation law

∂L 0 (v) ∂v • v -L 0 (v) t + m i=1 ∂L i ∂v • v -L i xi = 0. ( 5
)
If the Hessian matrix of L 0 is positive definite, the equations can be written in the symmetric form of Friedrichs. Denoting the variable t by x 0 , we can rewrite the system (4) and its consequence (5) in a compact form :

∂ ∂x β ∂L β ∂v α = 0, ∂E β ∂x β = 0, E β = v α ∂L β ∂v α -L β , (6)
with β = 0, ..., m, α = 1, ..., n. Here the summation is taken over repeated indexes. A number of reversible models of continuum mechanics can be written in Godunov's form (6) [START_REF] Godunov | Elements of Continuum Mechanics and Conservation Laws[END_REF].

A generalization of such a class of models with multiple examples coming from the reversible continuum mechanics was proposed in [START_REF] Gavrilyuk | Media with equations of state that depend on derivatives[END_REF], with potentials L β depending not only on uknowns but also on their first derivatives. More precisely, let us denote v α , γ = ∂v α ∂x γ . Consider functions L β (v α , v α , γ ) (we will use the same notations as for the old potentials L β depending only on v β ) and a conservative system in the form

∂ ∂x β δL β δv α = 0. ( 7
)
Here we used usual notation for the variational derivatives :

δL β δv α = ∂L β ∂v α - ∂ ∂x γ ∂L β ∂v α , γ
. Equations ( 7) also admit an additional conservation law

∂E β ∂x β = 0, E β = v α δL β ∂v α -L β + v α , γ ∂L β ∂v α , γ . (8)
Let us rewrite the conduit equation in the form [START_REF] Duchêne | Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model[END_REF]. For this, we will use the conservative form (2) with x 0 = t and x 1 = x . Let us consider the change of variables u = √ 1 + 2v [START_REF] Gavrilyuk | Singular solutions of the BBM equation: analytical and numerical study[END_REF] and potentials L(v, v x ) and M (v) (instead of generic potentials L 0 and L 1 ) defined as :

L(v, v x ) = √ 1 + 2v - v 2 x 2(1 + 2v) , M (v) = - 1 2 (1 + 2v)(ln(1 + 2v) -1). ( 10
)
Then, one obtains :

δL δv = 1 √ 1 + 2v + v xx 1 + 2v - v 2 x (1 + 2v) 2 = 1 + u xx u = 1 u + u xx u . ( 11
)
∂M (v) ∂v = -ln(1 + 2v) = -2 ln (u). ( 12
)
Hence, the equation ( 2) is written of the form [START_REF] Duchêne | Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model[END_REF]:

δL(v, v x ) δv t + ∂M (v) ∂v x = 0, (13)
The equation ( 13) admits an additional conservation law :

v δL δv -L t + v t L vx + v ∂M ∂v -M x = 0. ( 14
)
Unfortunately, it is not a new conservation law, but just a linear combination of (1) and [START_REF] Besse | Perfectly matched layers methods for mixed hyperbolic-dispersive equations[END_REF]. The equations (13) also admit a "symmetric" form. Indeed, consider the partial Legendre transform of L :

L (v, w) = wv x -L(v, v x ), where w = L vx . (15)
It implies, by using the implicit function theorem,

L v = -L v , L w = v x . (16)
It allows thus to rewrite [START_REF] Godunov | An interesting class of quasi-linear systems[END_REF] as

(L v ) t -(M (v)) x + w tx = 0, (L w ) t -v tx = 0. ( 17
)
Or

L vv L vw L vw L ww v w t - M (v) 0 0 0 v w x + 0 1 -1 0 v w tx = 0. ( 18
)
The form [START_REF] Ketcheson | High-order wave propagation algorithm for hyperbolic systems[END_REF] contains also a skew symmetric matrice multiplied by a vector of mixed derivatives.

Extended hyperbolic system

The idea of the hyperbolic approximation of the equation ( 1) is to replace it by an "extended" parametric family of reversible hyperbolic systems having also the Godunov-type form [START_REF] Duchêne | Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model[END_REF]. The word "extended" means that the governing equations contain an extra unknown z(t, x) which is asymptotically close to the unknown u(t, x) when parameters of the model go to infinity. Such a "penalization" method was already used in [START_REF] Favrie | A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves[END_REF][START_REF] Dhaouadi | Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation[END_REF][START_REF] Busto | On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems[END_REF][START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF][START_REF] Besse | Perfectly matched layers methods for mixed hyperbolic-dispersive equations[END_REF][START_REF] Bourgeois | Dynamics of a regularized and bistable Ericksen bar using an extended Lagrangian approach[END_REF] for mathematical models admitting a variational formulation. The "extended" system of equations was obtained as the Euler-Lagrange equations for an "extended" Lagrangian. Equation [START_REF] Godunov | An interesting class of quasi-linear systems[END_REF] does not correspond to any Euler-Lagrange equation. Indeed, it involves two potentials, L and M , while the Euler-Lagrange equations are written in terms of a single potential (Lagrange function). Doing so, we expect a better approximation of the conduit equation by the corresponding hyperbolic system of equations. Consider a two-parameter family of potentials

L(v, z, z x , z t ) = √ 1 + 2v + z 2 t 2c 2 - z 2 x 2 - λ 2 z - √ 1 + 2v 2 , (19)
where λ and c are large parameters. Let us replace the equation ( 13) by a system of equations for two unknowns v and z:

∂L ∂v t + ∂M ∂v x = 0, δL δz = 0, (20)
with

δL δz = L z -(L zt ) t -(L zx ) x .
It admits the conservation law :

(z t L zt + vL v -L) t + (z t L zx + vM v -M ) x = 0, (21)
i.e., it conserves the structure of the system [START_REF] Godunov | An interesting class of quasi-linear systems[END_REF]. Indeed,

(z t L zt + vL v -L) t + (z t L zx + vM v -M ) x = -z t δL δz = 0.
Let us also remark that if one introduces a function ϕ(t, x) instead of variable z(t, x) : z = ϕ t , the equation for z can be written in equivalent form as

δL δz t = δL δϕ = 0.
Now we will write the equations [START_REF] Leveque | Finite Volume Methods for Hyperbolic Problems[END_REF] in explicit form by using the expressions for derivatives :

∂L ∂v = 1 √ 1 + 2v + λ z - √ 1 + 2v √ 1 + 2v , (22a)
∂L ∂z = -λ z - √ 1 + 2v , (22b)
∂L ∂z x = -z x , (22c)
∂L ∂z t = z t c 2 . ( 22d
)
The first equation of ( 20) becomes :

1 √ 1 + 2v + λ z - √ 1 + 2v √ 1 + 2v t - 2 1 + 2v v x = 0. ( 23
)
The second equation of ( 20) becomes :

- 1 c 2 z tt + z xx = λ(z - √ 1 + 2v). (24)
Finally, we return back to u-variable (u = √ 1 + 2v):

1 u + λ z -u u t - 2u x u = 0, - 1 c 2 z tt + z xx = λ(z -u). (25)
Similar to the conduit equation, this system is reversible in the sense that it is invariant under the change of independent variables t → -t , x → -x. Its first order quasi-linear formulation can then be written as :

1 u + λ z -u u t - 2u x u = 0, (26a)
- 1 c z t + z x = p, (26b)
1 c p t + p x = λ(z -u). (26c)
The eigenvalues are ±c and 2u/(1 + λz). Hence, for large enough c and λ, the equations are hyperbolic. The initial conditions for (26) are :

u(0, x) = u 0 (x), z(0, x) = u 0 (x), p(0, x) = du 0 (x) dx . ( 27
)
These are the conditons we used for the benchmark tests in Section 7. In the following, we will take λ = c 2 . The system (25) admits the conservation law [START_REF] Leveque | Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems[END_REF] which is equivalent to the following one :

z 2 t c 2 + z 2 x + λz 2 -u(1 + λz) t -(u 2 + 2z t z x ) x = 0. ( 28

Dispersion relation

The dispersion relation for the hyperbolic system (26) linearized on the solution u = u 0 , z = u 0 , p = 0 is :

1 k 2 = 1 + λu 0 λ (c 2 p /c 2 -1)(c p -2u 0 /(1 + λu 0 )) c p -2u 0 . (29)
For any wave number k the real root c p approximating the exact dispersion relation (3) satisfies the inequality

2u 0 1 + λu 0 < c p < 2u 0 . (30)
For large λ and c 2 the approximate dispersion relation can be written as :

1 k 2 = c p u 0 2u 0 -c p + O 1 λ + 1 c 2 . (31)
Figure [START_REF] Busto | On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems[END_REF] shows the "quality" of the approximate dispersion relation [START_REF] Olson | Solitary wave propagation in a fluid conduit within a viscous matrix[END_REF]. The asymptotic formula [START_REF] Sprenger | Discontinuous shock solutions of the Whitham modulation equations as dispersionless limits of travelling waves[END_REF] suggests a natural choice of λ = c 2 to pass to one parameter family of penalty functions L(v, z, z t , z x ) defined by [START_REF] Métayer | A numerical scheme for the Green-Naghdi model[END_REF].

Periodic solutions

Periodic and solitary wave solutions to the conduit equation can be found, for example, in [START_REF] Olson | Solitary wave propagation in a fluid conduit within a viscous matrix[END_REF]. Here a quick overview of the solutions is given. Looking for traveling solutions to (2) depending only on ξ = x -Dt, D = const > 0 is the wave velocity, one obtains the ODE : admitting the first integral

Du = (C -1)u -2u ln(u) -D, C = const, (32)
P(u) u 1 u 2 u 3 u
Du 2 = P (u) = Cu 2 -2u 2 ln(u) -2Du -Q, Q = const. (33)
Here "prime" means the derivative with respect to ξ. Since u ln(u) is convex for u > 0, the function P (u) has maximum two critical points for u > 0, and hence maximum three roots. In the last case we denote them u i , P (u i ) = 0, i = 1, 2, 3, 0 < u 1 < u 2 < u 3 . A typical behavior of P (u) is shown in Figure 5. One can construct periodic solution oscillating between u 2 (minimum of the wave amplitude) and u 3 (maximum of the wave amplitude). The case u 1 = u 2 gives the solitary wave solutions. The wave velocity D, and the constants C and Q are thus calculated from the linear system :

Cu 2 i -2u 2 i ln(u i ) -2Du i -Q = 0, i = 1, 2, 3. ( 34
)
Its solution is unique if u 1 = u 2 = u 3 . The wave length L and averaged over the wave length the periodic solution u are given by the following expressions coming directly from (33):

L = 2 √ D u3 u2 du P (u) , u = u3 u2 udu P (u) u3 u2 du P (u) . ( 35
)
To study traveling wave solutions to (26), we will use the conservative form of equations :

1 u + λ z -u u t -(ln(u 2 )) x = 0, z 2 t c 2 + z 2 x + λz 2 -u(1 + λz) t -(u 2 + 2z t z x ) x = 0.
Using again the sign "prime" for the derivative with respect to the traveling wave coordinate ξ = x -Dt, one gets the following ODE system

-D 1 u + λ z -u u -ln(u 2 ) = c 1 , (36a)
D 1 - D 2 c 2 z 2 = D(λz 2 -u(1 + λz)) + u 2 + c 2 , (36b)
where c i , i = 1, 2 are constants. It can be reduced to only one equation for u.

Indeed, one has

z = u - u Dλ c 1 + ln(u 2 ) + D u , (36c)
that yields

dz du = 1 - 1 Dλ (c 1 + 2 + 2 ln(u)) . ( 36d
)
Then the equation (36b) becomes

D 1 - D 2 c 2 1 - 1 Dλ (c 1 + 2 + 2 ln(u)) 2 (u ) 2 = u 2 Dλ Dλ -c 1 -2 ln(u) - D u 2 -Dλu 2 + c 1 u 2 + u 2 + 2u 2 ln(u) + c 2
Finally, a compact form of this equation is :

(u ) 2 = F (u) G(u) , (37a)
where

F (u) = u 2 Dλ Dλ -c 1 -2 ln(u) - D u 2 + u 2 (1 + c 1 -Dλ + 2 ln(u)) + c 2 , (37b)
G(u) = D 1 - D 2 c 2 1 - 1 Dλ (c 1 + 2 + 2 ln(u)) 2 . ( 37c
)
To find the solution of (37) numerically using an ODE solver, we need to determine the parameters c 1 , c 2 , and D first. Given three constant states u 1 , u 2 , and u 3 , 0 < u 1 < u 2 < u 3 , that are the equilibrium solutions of F (u), this amounts solving the system of nonlinear equations:

F (u 1 ) = 0, F (u 2 ) = 0, F (u 3 ) = 0;
we do this by employing a quasi-Newton method (cf. [START_REF] Nocedal | Numerical Optimization[END_REF]) using the coefficients from the periodic solution of the conduit equation as the initial guess, see (33), achieving the convergent results after 1 or 2 iterative steps, depending on the convergence tolerance. Once we get u, we may set z and p = 1 -D c z based on (36c) and (36d), respectively.

Numerical methods

As in [START_REF] Favrie | A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves[END_REF][START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF], we use a fractional-step approach for the numerical resolution of the hyperbolic conduit system: at each time step, we alternate between by solving the homogeneous (hyperbolic) part of the system (26)

  1/u + λ (z -u) /u -z/c p/c   t +   -ln u 2 z p   x = 0 (38a)
over a time step ∆t, and the ODEs

  1/u + λ (z -u) /u -z/c p/c   t =   0 p λ (z -u)   (38b)
using the inital data from the previous step and the same time step. Here the numerical method we employed for (38a) is the same as for the conduit equation, see Appendix A. To update the solution of the ODEs (38b), we need to solve the linear second-order ODE:

z tt + λc 2 E 0 E 0 + λ z = λc 2 E 0 + λ (39a)
with the initial conditions

1 u + λ z -u u t=0 = E 0 , z(0) = z 0 , p(0) = p 0 . (39b)
If E 0 > 0, its exact solution is:

z = 1 E 0 1 + (E 0 z 0 -1) cos (ωt) - cp 0 E 0 ω sin (ωt) , (40a)
where ω 2 = λc 2 E 0 /(E 0 + λ). We then have

p = - 1 c z t = p 0 cos (ωt) + ω (E 0 z 0 -1) cE 0 sin (ωt) . (40b)
If E 0 < 0 (E 0 + λ > 0 for large λ), we find the exact solution:

z = 1 2E 0 2 + E 0 z 0 -1 - cp 0 E 0 µ e µt + E 0 z 0 -1 + cp 0 E 0 µ exp -µt , (40c)
p = - 1 c z t = - µ 2cE 0 E 0 z 0 -1 - cp 0 E 0 µ e µt -E 0 z 0 -1 + cp 0 E 0 µ exp -µt , (40d)
where µ 2 = -λc 2 E 0 /(E 0 + λ). Recall that E 0 , z 0 and p 0 are the solution of the homogeneous system (38a).

Numerical results

For the tests below, we take a uniform mesh size ∆x = 0.05, and a time step ∆t determined from the Courant-Friedrich-Lewy (CFL) condition for the stability of the hyperbolic solver. The non-reflecting boundary condition was employed on the left and right of the boundaries during the computations. For comparision, we will present results obtained using four different schemes for the homogeneous system (38a): MUSCL, WENO3, WENO5, and BVD35, see Appendix A for the details. The ODEs (38b) is solved using the exact solution (40) in all cases.

Our first test is an example studied in [START_REF] Maiden | Solitary wave fission of a large disturbance in a viscous fluid conduit[END_REF] for solitary wave fission of a large disturbance in a viscous fluid conduit. In this test, the initial condition for the conduit equation is the box:

u(0, x) = 1 + 1 2 α tanh x -x 0 β -tanh x -x 0 -L β , (41)
where α = 0.88, β = 2.5, and x 0 = 300 for x ∈ [0, 1500].

For the hyperbolic model, the parameter values we set for c and λ are 30 and 900, respectively.

Figure 3 shows numerical results for L = 48 and 96 at time t = 350 obtained using BVD35 case of the algorithm, observing good agreement of the state variable u between the conduit equation and its hyperbolic variant. In addition, we observe the similar solution structure between u and z which confirms the validity of our formal approach. For comparison, we repeat the computations using MUSCL, WENO3, and WENO5 cases. In Figure 4 we show snapshots of the state variable u at time t = 350 only partially in the region x ∈ [1000, 1500] (for completeness, the BVD35 results are included). It is clear that among them WENO5 and BVD35 give better solutions than WENO3 and MUSCL. For the MUSCL case, in particular, it is surprising to see the nonconvergence on the phase and amplitude for the foregoing solitary waves; this may mean that the third-order truncation (dispersive) error is too large for this problem, when discretizing the hyperbolic conduit equation based on the MUSCL approach.

Our second test is an example studied in [START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF] for the BBM equation. In this test, for the conduit equation, we take the Gaussian profile: -200, 600]. For the hyperbolic model, we use the same initialization procedure as before under (42), and the same parameter values for c and λ during the computations.

u(0, x) = 1 + 2 √ π exp(-x 2 /L 2 ) (42) for x ∈ [
Figure 5 shows numerical results in the case of L = 20 and 50 at time t = 150 obtained using BVD35 case of the algorithm. We again observe good agreement of the state variable u between the conduit equation and its hyperbolic variant, and also the same solution behavior between u and z. As in the previous test, we perform the computations using MUSCL, WENO3, and WENO5 cases also, Figure 3: Numerical results for the first test. Snapshots of solutions are obtained using BVD35 case of the method are shown at time t = 350 for L = 48 and L = 96. In both cases, parameter values c = 30 and λ = 900 were used in the computations. The blue line legend "conduith" means the results obtained using the hyperbolic approximation of the conduit equation, while the red line legend "conduit" is for the exact conduit equation. 6; only the partial solutions in the region x ∈ [200, 600] are shown. We find sensible good agreement of the solutions, even in the MUSCL case.

To show the convergence of the hyperbolic conduit solution to the conduit one, we perform a parameter study on c and λ = c 2 for c = 20 and 100. In Figure 7, the solutions of u for the first test in the case of L = 96 and the second test in the case of L = 50 are shown at times t = 350 (the first row) and t = 150 (the second row), respectively. Here, for clarity, only the partial solutions in the region x ∈ [1210, 1450] and x ∈ [300, 530] are drawn. It is clear that the solution is more accurate when a larger parameter is used in the computations. Table 1 gives the timing study in CPU (sec) for the results shown in Fig. 7, where the tests were performed in a Mac mini M2 Pro with 32GB RAM. We observe the higher computational cost when the hyperbolic model ( 26) is used as compared to the dispersive conduit equation (2).

Generalized Riemann problem

We call a generalized Riemann problem (GRP) the Cauchy problem

u(0, x) = u L (x), x < 0, u R (x), x > 0, (43)
where u L and u R (x) are different periodic travelling wave solutions of the corresponding dispersive equations (in particular, of the conduit equation). Such a problem was studied in [START_REF] Gavrilyuk | Stationary shock-like transition fronts in dispersive systems[END_REF] for the Serre-Green-Naghdi and Boussinesq equations with linear dispersion, in [START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF] for the BBM equation, and in [START_REF] Sprenger | Discontinuous shock solutions of the Whitham modulation equations as dispersionless limits of travelling waves[END_REF] for the fifth order KdV equation. In particular, in the first reference new stable shocklike travelling wave solutions were found linking a constant solution (denoted further by u ) to a periodic wave train. The shock-like transition zone between the constant state and the wave train was well described by the half of solitary wave having the wave crest at the maximum of the nearest periodic wave. Such a configuration was stable under certain conditions. For example, for the BBM equation such a shock-like structure is stable if the phase velocity of the periodic wave train is not less than the solution wave averaged representing indeed the characteristic velocity of a dispersionless homogeneous state [START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF]. In our case, the characteristic velocity of the dispersionless equation (u t + (u 2 ) x = 0) is 2u.

Since the dispersive properties of the BBM equation are similar to those of the conduit equation, we expect that the stable configuration linking a constant state u to a periodic wave train having the velocity D can be also realized for D > 2u (see the definition (35) of the wave averaged.) The aim of this section is thus to reveal the analogous solutions for the conduit equation numerically.

We begin by looking into a modified version of (43) in the form

u(0, x) =    u, x < x 0 , u(x), x 0 < x < x 1 , u, x > x 1 , (44)
where u(x) is a wave profile that consists of N periodic waves in the interval (x 0 , x 1 ), and u is the average value of a single periodic wave over a wavelength.

In the numerical experiments performed here, the parameters we take for the initial periodic solution are u 1 = 1, u 3 = 2 and u 2 = 2 -m 0 , m 0 = 0.999. Then with Wolfram Mathematica, Version 12, one gets the phase speed D ≈ 2.546, the average state u ≈ 1.216, and the wave length L ≈ 42.72. The initial wave train is formed by introducing N = 48 of such a periodic solution into one.

In Figure 8, we show the pseudo-color plot of the solution in (x, t)-plane, observing clearly the formation of a constant state u on the left of the primary periodic wave train and on the right of the left rarefaction wave. This is as expected, because as in [START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF] we have the phase speed D ≈ 2.546 larger than the characteristic speed 2u ≈ 2.432, a necessary condition for the existence of the stable shock-like travelling structure. The snapshot of the solution for the problem at time t = 600 is shown in Figure 9, where the solution shown on the left is obtained using the conduit equation, and on the right is obtained using the hyperbolic model. We observe good agreement of the results qualitatively.

To determine analytically the state u , we use the Rankine-Hugoniot relation coming from the conservative form (1) (the mass conservation law). We consider the jump relation for (1) on the travelling wave solutions for a shock having the same velocity D as that of the travelling wave train, and linking the maximum amplitude u 3 of the wave train with the constant state u (see [START_REF] Gavrilyuk | Stationary shock-like transition fronts in dispersive systems[END_REF][START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF] for details) :

-

D(u 3 -u ) + (u 2 3 -u 2 + u 3 Du | u=u3 ) = 0 (45)
We have used the fact that in the u 3 state and extremal state "star" the derivative u vanishes. The states u 2 (minimum value of u), u 3 (maximum value of u) and the velocity D are related through the first integral :

Cu 2 2 -2u 2 2 ln(u 2 ) -2Du 2 = Cu 2 3 -2u 2 3 ln(u 3 ) -2Du 3 (46)
It allows us to find C :

C = 2u 2 3 ln(u 3 ) + 2Du 3 -2u 2 2 ln(u 2 ) -2Du 2 u 2 3 -u 2 2 . (47)
Hence, one can estimate Du at the maximum u 3 from (32) :

Du | u=u3 = (C -1)u 3 -2u 3 ln(u 3 ) -D. (48)

Conclusion

We have proposed a hyperbolic approximation of the conduit equation preserving, in particular, invariance properties of the conduit equation (reversibility in time and space) and approximating the solutions of the conduit equation with good accuracy. The advantage of the hyperbolic approximation is that it allows all the numerical tools developed for hyperbolic equation systems to be applied to the study of dispersive equations.

We have constructed new solutions to the conduit equation representing an assemblage of many waves of the same period linked to a constant solution by the generalized Rankine-Hugonit relation, also taking into account the curvature of periodic waves. The generalized shock linking the maximum of the lateral periodic waves to a constant state has the same velocity as that of the periodic wave train. Such a multi-hump solitary wave is stable if the wave velocity is twice as great as u. This condition means that the phase velocity of such a structure must be supercritical with respect to the homogeneous state u having the characteristic slope 2u. The hyperbolic approximation of the conduit equation also admits such stable solutions.

for u with prescribed boundary conditions based on a finite-difference scheme [START_REF] Leveque | Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems[END_REF].

It should be mentioned that from (1) one can also find numerical solutions to the conduit equation when we apply the algorithm to solve the following hyperbolic-elliptic system:

u t + u 2 + x = 0, (53a)
- x u x + u 2 = 2u xx , (53b)
separately for u and during each time step. More precisely, in the hyperbolic step, we use the semi-discrete finite volume method written in a wave-propagation form as before [START_REF] Gavrilyuk | Stationary shock-like transition fronts in dispersive systems[END_REF], but employ a different solution reconstruction technique, the BVD (boundary variation diminishing) principle, which is more robust than the classical one for the interpolated states (K for (52) or u for (53)) at cell boundaries (cf. [START_REF] Deng | High fidelity discontinuity-resolving reconstruction for compressible multiphase flows with moving interfaces[END_REF] and the references cited therein). These reconstructed variables form the basis for the initial data of the Riemann problems, where the solutions of the Riemann problems are then used to construct the fluctuations in the spatial discretization that gives the right-hand side of the system of ODEs (cf. [START_REF] Leveque | Finite Volume Methods for Hyperbolic Problems[END_REF][START_REF] Ketcheson | Wenoclaw: A higher order wave propagation method, in: Hyperbolic problems: Theory, numerics, applications[END_REF][START_REF] Ketcheson | High-order wave propagation algorithm for hyperbolic systems[END_REF]). To integrate the ODE system in time, the strong stability-preserving (SSP) multistage Runge-Kutta scheme [START_REF] Gottlieb | Strong stability-preserving highorder time discretization methods[END_REF][START_REF] Shu | High order weighted essentially nonoscillatory schemes for convection dominated problems[END_REF] is used. In particular, for the numerical results presented in this paper, the third-order SSP scheme was employed together with the pair of third-and fifth-order WENO (weighted essentially non-oscillatory) scheme in the BVD reconstruction process.

) 2 Figure 1 :

 21 Figure 1: Comparison of the exact dispersion relation for u 0 = 1 (black dashed curve) and approximate one (red curve for c = 15, λ = c 2 = 225) is shown.

Figure 2 :

 2 Figure 2: A typical behavior of the function P (u) is shown. In a domain of parameters C, D > 0, Q < 0 it has three roots 0 < u 1 < u 2 < u 3 . The periodic solution oscillates between u 2 and u 3 .

Figure 4 :

 4 Figure 4: Numerical methods comparison for the first test. Snapshots of the state variable u obtained using four different hyperbolic solvers are shown at time t = 350 in the case of L = 48 and L = 96; only partial solutions in the region x ∈ [1000, 1500] are shown. In both cases, parameter values c = 30 and λ = 900 were used in the computations. 13

Figure 5 :

 5 Figure 5: Numerical results for the second test. Snapshots of solution obtained using BVD35 case of the method are shown at time t = 150 for L = 20 and L = 50. In both cases, parameter values c = 30 and λ = 900 were used in the computations.

Figure 6 :

 6 Figure 6: Numerical methods comparison for the second test. Snapshots of the state variable u obtained using four different hyperbolic solvers are shown at time t = 150 for L = 20 and L = 50; only the partial solutions in the region x ∈ [200, 600] are shown. In both cases, parameter values c = 30 and λ = 900 were used in the computations. 16

Figure 7 :

 7 Figure 7: A parameter study of the solutions on c and λ = c 2 for the hyperbolic model. On the first row, the solutions are for the first test in the case of L = 96 at time t = 350, and on the second row, the solutions are for the second test in the case of L = 50 at time t = 150; only the snapshots of the state variable u are shown together with the conduit solution. In both cases, we used parameter values c = 20 and c = 100 in the computations.

Figure 8 :

 8 Figure 8: The general solution structure of Cauchy problem (44) for the conduit equation is shown in (x, t)-plane. The wave train in the middle consists of 48 periodic waves, where the initial parameters for each of them are: u 1 = 1, u 3 = 2 and u 2 = 2 -m 0 , m 0 = 0.999. The wave average on the left and right of the wave train is u ≈ 1.216.

Figure 9 :

 9 Figure 9: The snapshot of the solution for the Cauchy problem (44) at time t = 600. The solution shown on the left is obtained using the conduit equation, and on the right is obtained using the hyperbolic model with the parameters c = 30 and λ = 900. 19

Figure 10 :

 10 Figure 10: Illustration of the initial conditions for the multi-hump problems. The plot shown on the left is for the single hump problem (50), and on the right is for the double hump problem (51).

Figure 11 :

 11 Figure 11: The snapshot of the conduit solution for the multi-hump problem (50) at time t = 1000. The solution on the left is for the conduit equation, and on the right is for the hyperbolic model. The computation domain is x ∈ [0, 4000]; only the partial solutions in the region x ∈ [3130, 3730] are shown.

Figure 12 :

 12 Figure 12: The snapshot of the conduit solution for the two multi-hump problem (51) at times t = 500, 2000, 2500, 3000, 4000, 5000. The plots are displayed from the left top to bottom and continues from the right top to bottom. The computation domain is adjusted in time to have the multi-hump solution stayed in the domain; only the partial solutions in the neighborhood of the multi-hump solitons are shown.

Figure 13 :

 13 Figure 13: The snapshot solutions of the hyperbolic conduit model for the two multi-hump problem (51) at times t = 500, 2000, 2500, 3000, 4000, 5000. The plots are displayed in the same manner as Fig. 13.

Table 1 :

 1 The CPU time (sec) taken for the numerical results shown in Fig.7

		conduit	hyperbolic model
			c = 20	c = 100
	first test	1021.528 3057.535 14572.31
	second test 247.996 1238.719 6150.387
	and show numerical results in Figure

Acknowledgments SG was partially supported by Agence Nationale de la Recherche, France (SNIP ANR-19-ASTR-0016-01). KMS thanks the Institut Mécanique et Ingénierie of Marseille and INRIA -Sophia-Antipolis -Méditerranée for financial support during his stay at IUSTI, Marseille.

Data Availability Statement

The computational codes developed for this research are available from the authors by request.

Declaration of competing interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

We can now replace all into (45) to obtain the equation for u :

As for the BBM equation [START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF], one can prove that u 1 < u < u 2 .

In particular, if we take the same u 2 , u 3 , and D as in the Cauchy problem (44), we find the value u ≈ 1.000499. Then we may construct a multi-hump structure in the form

with u M as a periodic wave train linked to u . Fig. 10, the left column, shows u M composed of N = 11 periodic wave solutions. This multi-hump solution structure is a stable one as we can see in Fig. 11. Here we present the snapshot solutions obtained using the conduit equation and the hyperbolic model at time t = 1000. We are next concerned with a double multi-hump problem for the interaction of two periodic wave trains. The initial condition is:

where ũL and ũR are having analogous structure to (50), see the right column of Fig. 10 for an illustration. To be specific, for each wave train it consists of N = 11 periodic waves together with a hybrid half wavelength periodic and solitary waves, and x 0 = 1600. The state values we take for ũL are u L 1 = 0.9, u L 2 = 0.907, u L 3 = 1.7, and that give D L ≈ 2.25, u L ≈ 1.123, and u L ≈ 0.903477. For ũR , we have u R 1 = 0.9, u R 2 = 0.907, u R 3 = 1.25, and get D R ≈ 2.01871, u R ≈ 1.00894, and u R ≈ 0.903467. Since the states u L and u R are approximately the same, and D L > D R , we can study the overtaking multi-hump soliton problem propagation on the same level u .

Figure 12 shows the numerical solutions for the conduit equation at times t = 500, 2000, 2500, 3000, 4000, 5000, observing the wave interaction, merging, and their full reconstruction. Here without introducing a large domain size, the computation domain is adjusted in time by the method to have the multi-hump solution stayed inside the region. The solutions for the hyperbolic model are shown in Fig. 13, we again observe good qualitative agreement of the solution, and the validation of the numerical solutions.

The stability of multi-hump solutions created "artificially" by combining periodic solutions and constant states related by the generalized Rankine-Hugoniot relations show that they are stable weak solutions to the conduit equation. In particular, they are stable under a "perturbation" of the conduit equation by a hyperbolic system conserving its original Godunov type form. Ethics

The work is original, has not been published before, and is not currently being considered for publication elsewhere.

A Numerical methods for the conduit equation

To find approximate solutions to the conduit equation ( 2), we use the hyperbolicelliptic splitting approach developed previously in [START_REF] Métayer | A numerical scheme for the Green-Naghdi model[END_REF][START_REF] Gavrilyuk | Stationary shock-like transition fronts in dispersive systems[END_REF][START_REF] Gavrilyuk | Hyperbolic approximation of the BBM equation[END_REF]. This algorithm consists of two steps. In the first step, the hyperbolic step, we employ the stateof-the-art method for hyperbolic conservation laws for the numerical resolution of the equation K t -ln u 2

x = 0 (52a) over a time step ∆t. In the second step, the elliptic step, using the approximate solution K computed during the hyperbolic step, we invert numerically the elliptic operator:

-u xx + Ku = 1 (52b)


Romain Ageron

Paul Casteras

Thibaut Pellerin

Yann Portella

Arpad Rimmel

Joanna Tomasik

New lower bounds for Schur and weak Schur numbers

Keywords: Schur number, Weak Schur number, Ramsey theory, Sum-free partition MSC 2020: 05D10, 05A17, 11B75, 11P81

This article provides new lower bounds for both Schur and weak Schur numbers by exploiting a "template"-based approach. The concept of "template" is also generalized to weak Schur numbers. Finding new templates leads to explicit partitions improving lower bounds as well as the growth rate for Schur numbers, weak Schur numbers, and multicolor Ramsey numbers Rn(3). The new lower bounds include S(9) 17 803, S(10) 60 948, WS (6) 646, WS (9) 22 536 and WS (10) 71 256.

Introduction

We are interested in partitioning the set of integers {1, ..., p} into n subsets such that there is no subset containing three integers x, y and z verifying x + y = z. We say these subsets are sum-free. If we lessen the above constraint by only considering integers x, y such that x = y, we say the subsets are weakly sum-free. The greatest p for which there is a partition into n sum-free subsets is called the n th Schur number and is denoted S(n) [START_REF] Schur | Über Kongruenz x[END_REF]. Likewise for weakly sum-free partitions we define WS (n) the n th weak Schur number [START_REF] Irving | An extension of Schur's theorem on sum-free partitions[END_REF]. Values of S(n) and WS (n) are known for small n only.

State of the art

Up to recently, the most efficient generic construction for Schur numbers was given by Abbott and Hanson [START_REF] Abbott | A problem of Schur and its generalizations[END_REF] in 1972 with a recursive construction. It gave the best lower bounds for all sufficiently large numbers. No equivalent was known for weak Schur numbers and, as a result, the best known partitions for large weak Schur numbers did not use the weakly sum-free property.

As for smaller weak Schur numbers, the best lower bounds were obtained by computations. Eliahou [START_REF] Eliahou | Investigating Monte-Carlo Methods on the Weak Schur Problem[END_REF], Bouzy [START_REF] Bouzy | An Abstract Procedure to Compute Weak Schur Number Lower Bounds[END_REF] and Rafilipojaona [START_REF] Rafilipojaona | Nombres de Schur classiques et faibles[END_REF] improved the lower bounds with Monte-Carlo methods. This was the main approach during the past decade.

In 2020, Rowley introduced the notion of templates for Schur and Ramsey numbers [START_REF] Rowley | A generalised linear Ramsey graph construction[END_REF] which generalizes Abbott and Hanson's construction and produces new lower bounds (and inequalities) for Schur numbers. He also provided two inequalities for weak Schur numbers [START_REF] Rowley | New lower bounds for weak Schur partitions[END_REF]. Besides, these two inequalities do use the weakly sum-free property.

The hereunder tables recap the lower bounds for Schur and weak Schur numbers, respectively. Gray cells indicate the exact values.

Organization of this article

Our main contribution is a generalization of the concept of template to weak Schur numbers. Our templates provide new lower bounds (and inequalities) for weak Schur numbers. As a special case, our construction also includes a construction similar to Abbott and Hanson's [START_REF] Abbott | A problem of Schur and its generalizations[END_REF], but for weak Schur numbers this time.

In Section 2, we explain Rowley's template-based construction for Schur numbers. Then, we give new templates, thus providing new lowers bounds and inequalities as well as showing that the growth rates for both Schur and Ramsey numbers R n (3) exceed 3.28.

In Section 3, we generalize the concept of template to weak Schur numbers and provide new lower bounds for weak Schur numbers. Then, we use a different approach and give a new lower bound for WS [START_REF] Rafilipojaona | Nombres de Schur classiques et faibles[END_REF]. Now, we introduce notations and definitions used throughout this article.

Definitions and notations

We start by defining sum-free and weakly sum-free subsets to introduce regular and weak Schur numbers. The set of positive natural numbers is denoted by N * = N\{0}.

Definition 1.1. A subset A of N is said to be sum-free if

∀(a, b) ∈ A 2 , a + b / ∈ A.
Definition 1.2. A subset B of N is said to be weakly sum-free if

∀(a, b) ∈ B 2 , a = b =⇒ a + b / ∈ B.
Let us notice that a sum-free subset is also weakly sum-free, hence justifying the name of weakly sumfree subsets. Given p and n two integers, we are interested in partitioning the set of integers {1, 2, ..., p}, denoted by [ [1, p]], into n (weakly) sum-free subsets.

Schur proved [START_REF] Schur | Über Kongruenz x[END_REF] that given a number of subsets n, there is a value of p such that [ [1, q]] cannot be partioned into n sum-free subsets for q p. A similar property holds for weakly sum-free subsets [START_REF] Irving | An extension of Schur's theorem on sum-free partitions[END_REF]. These observations lead to the following definitions written for n ∈ N * : Definition 1.3. There is a largest integer denoted by S(n) such that [[1, S(n)]] can be partitioned into n sum-free subsets. S(n) is called the n th Schur number.

Definition 1.4.

There is a largest integer denoted by WS (n) such that [[1, WS (n)]] can be partitioned into n weakly sum-free subsets. WS (n) is called the n th weak Schur number.

Subsets of an n-partition are denoted A 1 , ..., A n . The smallest element of A i is denoted m i = min(A i ). By ordering the subsets, we mean assuming that m 1 < ... < m n . We will make clear in the text when we use a subset ordering.

A partition may be seen as a number coloring.

Definition 1.5. The coloring associated to a partition

A 1 , ..., A n of [[1, p]] is the function f such that ∀x ∈ [[1, p]], x ∈ A f (x) . Likewise, the partition associated to a coloring f of [[1, p]] with n colors is ∀c ∈ [[1, n]], A c = f -1 (c).

Templates for Schur numbers

We use Rowley's template-based constructions [START_REF] Rowley | A generalised linear Ramsey graph construction[END_REF] for Schur numbers. To improve lower bounds for Schur and Ramsey numbers, he has introduced special sum-free partitions verifying some additional properties which can be extended using a method generalizing Abbott and Hanson's construction [START_REF] Abbott | A problem of Schur and its generalizations[END_REF]. He called these partitions "templates", and we keep this nomination. Then, we propose new templates and use them to produce new lower bounds for Schur numbers.

Definition of [[1, S + ]]

We begin by introducing S-templates, standing for "Schur templates". The idea is to consider the first line of Figure 1 not as a combination of two blocs, like in [START_REF] Abbott | A problem of Schur and its generalizations[END_REF] but as a whole, single construction. An S-template is then defined as a new object playing the role of the first line but with less, yet sufficient, constraints which allow for an expansion of the partition using a sum-free partition.

Definition 2.1. Let (p, n) ∈ (N * ) 2 . An S-template with n colors is defined as a partition of [[1, p]] into n sum-free subsets A 1 , A 2 , ..., A n verifying for all subsets but A n ∀(x, y) ∈ A 2 i , x + y > p =⇒ x + y -p / ∈ A i . (1)
The integer p is called the width of the template.

Here n can be seen as a "special" color in the sense that it is not constrained by Assertion (1). Number p is necessarily colored with color n. However, note that n is not necessarily the last color by order of appearance.

Proposition 2.2. Let n ∈ [[2, +∞[[. We define S + (n) as the greatest integer w such that there is an S-template with width w and n colors. S + (n) is well defined and verifies

2S(n -1) + 1 S + (n) S(n).
Proof. The upper bound comes from the fact that an S-template with width p and n colors is also a sum-free n-partition of [ [1, p]]. The lower bound comes from Abbott and Hanson's construction [START_REF] Abbott | A problem of Schur and its generalizations[END_REF].

Construction of Schur partitions using S-templates

We start by reminding the explicit construction of a sum-free partition with the use of an S-template and a sum-free partition. We rephrase in terms of Schur numbers this construction stated by Rowley in the context of Ramsey numbers [START_REF] Rowley | A generalised linear Ramsey graph construction[END_REF].

Theorem 2.3. Let (p, k), (q, n) ∈ (N * ) 2 .
If there are an S-template with width q and n + 1 colors and a sum-free k-partition of [ [1, p]] then there is a partition of [[1, pq + m n+1 -1]] into n + k sum-free subsets. m n+1 denotes the minimum element colored with the special color in the S-template.

The idea lying beneath this theorem is similar to Abbott and Hanson's construction [START_REF] Abbott | A problem of Schur and its generalizations[END_REF]. First, they put the integers from [[1, S(k)(2S(n) + 1) + S(n)]] in a table with width 2S(n) + 1 and height S(k) + 1 and they numbered both the rows and the columns, starting from one upwards. Then, they extended a sum-free partition vertically by repeating it. Next, they used another sum-free partition to color the other half according to the line number, as in Figure 1. We give an example for p = 4, q = 9, n = 2 and k = 2 showing that S(2 + 2) S(2) (2S(2) + 1) + S(2), both with Abbott and Hanson's construction (Figure 1) and with an S-template which is not included in Abbott and Hanson's construction (Figure 2). In both cases, the special color is grey. In the general construction with S-templates, the special color may not paint consecutive numbers any longer. However, the special color is still replaced by the colors of the sum-free partition according to the line number and the other colors are still vertically extended. Proof. Let f and g be colorings associated to the S-template with width q and the sum-free partition of [ [1, p]], respectively:

f : [[1, q]] -→ [[1, n + 1]] and g : [[1, p]] -→ [[1, k]].
NB: In the following three predicates, the conditions x + y q and x + y p are omitted for readability. The sum-free condition is expressed as:

∀(x, y) ∈ [[1, q]] 2 , f (x) = f (y) =⇒ f (x + y) = f (x), ∀(x, y) ∈ [[1, p]] 2 , g(x) = g(y) =⇒ g(x + y) = g(x).
The additional constraint for the S-template is:

∀(x, y) ∈ [[1, q]] 2 , f (x) = f (y) n x + y > q =⇒ f (x + y -q) = f (x). For x ∈ [[1, pq+m n+1 -1]],
we write x = (α-1)q+u for certain integers α ∈ N * and u ∈ [ [1, q]]. Obviously, this decomposition unique. Coefficients α and u may be interpreted as line and column coordinates of x in the template. We define a coloring h of [ [1, pq

+ m n+1 -1]] using [[1, n + k]] colors: h : [[1, pq + m n+1 -1]] -→ [[1, n + k]], x -→ f (u), if f (u) n, n + g(α), if f (u) = n + 1. Function h is well-defined since, by definition of m n+1 , ∀x ∈ [[pq + 1, pq + m n+1 -1]], f (u) n and therefore ∀x ∈ [[1, pq + m n+1 -1]], f (u) = n + 1 =⇒ α ∈ [[1, p]].
Now, we prove that h is a sum-free coloring. Let x, y ∈ [[1, pq + m n+1 -1]] such that h(x) = h(y) and x + y pq + m n+1 -1. We claim that h(x + y) = h(x). We write x = (α -1)q + u and y = (β -1)q + v where α, β ∈ Z and u, v ∈ [ [1, q]]. Two cases are to be distinguished according to the value of h(x).

Case 1: h(x) n

Let us assume that h(x + y) n, otherwise h(x + y) = h(x) obviously holds. By definition of function h and given that h(u) = h(v), we conclude f (u) = f (v). Two cases are to be distinguished according to the value of x + y.

If u + v > q, we write w = u + v -q ∈ [[1, q]]. Consequently x + y = (α + β -1)q + w. By definition, h(x + y) = f (w). Given that f (u) = f (v) n, the additional constraint on f implies f (w) = f (u), that is h(x + y) = h(x). If u + v q, we write w = u + v ∈ [[1, q]]. Consequently x + y = (α + β -2)q + w. By definition, h(x + y) = f (w). Given that f (u) = f (v) n, the sum-free property of f implies f (w) = f (u), that is h(x + y) = h(x). Case 2: h(x) n + 1 Now we have h(x) = n + g(α) = n + g(β) = h(y), hence g(α) = g(β).
As above, we distinguish between two cases according to the value of x + y.

If u + v > q, write w = u + v -q ∈ [[1, q]]. Then x + y = (α + β -1)q + w. Assume that h(x + y) n + 1, otherwise h(x + y) = h(x) obviously holds. By definition, h(x + y) = n + g(α + β). Given that g(α) = g(β), the sum-free property of g implies g(α + β) = g(α) that is h(x + y) = h(x). If u + v q, write w = u + v ∈ [[1, q]]. Then x + y = (α + β -2)q + w. The sum-free property of f implies f (w) = f (u). Therefore f (w)
n and thus h(x + y) n. In particular, given that

h(x) n + 1, h(x + y) = h(x).
Setting q = S + (n + 1) and p = S(k) in Theorem 2.3 yields the following corollary.

Corollary 2.4. Let n, k ∈ N * . Then S(n + k) S + (n + 1)S(k) + m n+1 -1.
The following proposition may improve the additive constant of Corollary 2.4.

Proposition 2.5. Let (q, n) ∈ (N * )
2 and let f be a coloring associated to an S-template with width q and n + 1 colors. Let b ∈ N and assume there is a coloring g of [ [1, b]] with n + 1 colors such that:

∀(x, y) ∈ [[1, q]] 2 , f (x) = f (y) (x + y) mod q b =⇒ g((x + y) mod q) = f (x), ∀(x, y) ∈ [[1, q]] × [[1, b]], f (x) = g(y) x + y b =⇒ g(x + y) = f (x).
Then, for every n ∈ N * , by using on the last row the coloring x -→ g(x -qS(n)), we have

S(n + k) qS(k) + b.
This proposition corresponds to the fact that the hypotheses made on the coloring of the last row can be weakened because the constraints in the definition of an S-template prevent a number from: being the sum of two of numbers of its own color. creating a sum in its own color.

The later is not always relevant on the last row given that it is not completely filled. As a result, we can change the coloring on [[qS(k) + 1, qS(k) + b]] (the last row) to extend the previous partitions.

There is a construction theorem for S-templates as well.

Theorem 2.6. Let (p, k), (q, n) ∈ (N * ) 2 . If there are an S-template with width q and n + 1 colors, and an S-template with width p and k colors, then there also is an S-template with width pq and (n + k) colors.

Proof. The idea is the same as in the Theorem 2.3. The only difference is the S-template property inherited from the second S-template.

The inequality associated with this theorem is given by:

Corollary 2.7. Let n, k ∈ N * . Then S + (n + k) S + (n + 1)S + (k).

New lower bounds for Schur numbers

We now give the inequalities corresponding to the current best S-templates. Definition 2.8. A sum-free partition A 1 , ..., A n of [ [1, p]] is said to be symmetric if for all x ∈ [[1, p]], x and p + 1 -x belong to the same subset (except if x = p + 1 -x).

An S-template with n colors is said to be symmetric if the sum-free n-partition derived from this template by applying the extension procedure of Theorem 2.3 with a sum-free partition with length one is symmetric.

We produced S-templates using the lingeling SAT solver [START_REF] Biere | CaDiCal, Lingeling, Plingeling, Treengeling and YalSAT entering the SAT competition 2017[END_REF], hence providing lower bounds on S + and inequalities of the type S(n + k) aS(n) + b. We sought templates providing us with the largest value of (a, b) (in the lexicographic order). In order to reduce the search space when the number of colors exceeded five we only looked for symmetric S-templates, we assumed that the special color was the last color to appear and we constrained the m c 's out of being too small. Details concerning the encoding as a SAT problem can be found in [START_REF] Heule | Schur number five[END_REF].

The following six inequalities are given by the current best S-templates with n 7 colors.

S(n + 1) 3 S(n) + 1 (2) S(n + 2) 9 S(n) + 4 (3) S(n + 3) 33 S(n) + 6 (4)
Inequality (2) comes from Schur's original article [START_REF] Schur | Über Kongruenz x[END_REF]. Inequality (3) is due to Abott and Hanson [START_REF] Abbott | A problem of Schur and its generalizations[END_REF] and inequality (4) to Rowley [START_REF] Rowley | A generalised linear Ramsey graph construction[END_REF]. The three following inequalities are our result:

S(n + 4) 111 S(n) + 43 (5)
S(n + 5) 380 S(n) + 148 ( 6)

S(n + 6) 1 160 S(n) + 536 (7)
The templates corresponding to inequalities (4), ( 5), and ( 6) are listed in A. Inequalities (2), (3), and (4) cannot be further improved (with this definition of S-template). Inequality (5) cannot be further improved by only searching for symmetric S-templates whose special color is the last in the order of apparition (and with a multiplicative factor less than or equal to 118). Inequality ( 6) can most likely be further improved but the improvement probably will not be substantial. Finally, an S-template corresponding to inequality [START_REF] Rowley | A generalised linear Ramsey graph construction[END_REF] was found by extending into a S-template the Schur 6-partition used in [START_REF] Rowley | An improved lower bound for s(7) and some interesting templates[END_REF] (owing to the size of this template, its interest is limited and since it can easily be derived from above mentioned partition, it is not given in A). Although we could not find a better S-template with seven colors, inequality [START_REF] Rowley | A generalised linear Ramsey graph construction[END_REF] can definitely be improved by a wide margin.

The previous inequalities give new lower bounds for S(n) for n 9. We compute the lower bounds for n ∈ [ [START_REF] Rowley | New lower bounds for weak Schur partitions[END_REF]15]] using the four different inequalities. The best lower bounds are highlighted. Proof. It is a mere consequence of the inequality S(n + 5) 380 S(n) + 148. As for Ramsey numbers, the following inequality holds S(n) R n (3) -2 (see [START_REF] Abbott | A problem of Schur and its generalizations[END_REF]) hence the result.

Conclusion on S-templates

First, we formalized Rowley's template-based constructions [START_REF] Rowley | A generalised linear Ramsey graph construction[END_REF] in the context of Schur numbers by introducing S-templates as well as a new sequence, S + . We found new S-templates allowing us to obtain new lower bounds for schur numbers. One may notice that we mostly gave only lower bounds for S + . It should be possible to find better S-templates by making different assumptions or using a different method (Monte-Carlo methods, for instance).

In the next section, we provide similar results for weak Schur numbers. We introduce WS-templates and a corresponding sequence, WS + . Then, we derive similar relations and a construction method allowing us to find new lower bounds for weak Schur numbers.

Templates for weak Schur numbers

We generalize Rowley's constructions for weak Schur numbers [START_REF] Rowley | New lower bounds for weak Schur partitions[END_REF]. We give an analogous for weak Schur numbers of Abbott and Hanson's construction for Schur numbers [START_REF] Abbott | A problem of Schur and its generalizations[END_REF]. Then, as in the previous section, we introduce WS-templates, standing for "weak Schur templates", as well as an associated sequence WS + (n).

We find templates and use them to provide new lower bounds for weak Schur numbers. Finally, we give a short explanation for the new lower bound WS (6) 646 which was not directly obtained with a template, contrary to the other lower bounds given in this article.

Inequality for weak Schur numbers using Schur and weak Schur numbers

Up to now, no equivalent for weak Schur numbers of Abbott and Hanson's construction for Schur numbers [START_REF] Abbott | A problem of Schur and its generalizations[END_REF] was known. Here we give a general lower bound for weak Schur numbers as a function of both regular and weak Schur numbers. Theorem 3.1. Let (p, k), (q, n) ∈ (N * ) 2 . If there are a weakly sum-free n-partition of [ [1, q]] and a sum-free k-partition of [ [1, p]] then there is a partition of [[1, p(q + q 2 + 1) + q]] into n + k weakly sum-free subsets. As Theorem 3.1 is actually a particular case of a more general theorem (Theorem 3.17) which will be formulated after the introduction of templates for weak Schur number, we only outline the demonstration. Its complete proof using templates for weak Schur numbers is in Subsection 3.3.

Let (p, k), (q, n) ∈ (N * ) 2 such that there are a weakly sum-free n-partition of [ [1, q]] and a sum-free k-partition of [ [1, p]]. Let a ∈ N with a > q and let us try to build a coloring of [[1, ap + q]] into n + k weakly sum-free subsets. Let l = a -q -1, r ∈ [ [1, q]] and w = a -l -r -1 = q -r.

First, we construct the following table with a columns and p + 1 rows containing the integers from [[1, ap + q]]. The columns are numbered from -l to +q. This table is made of three different blocks.

T (the "tail"): in the row indexed by 0: numbers from [ [1, q]]. R (the "rows"): the integers in columns from -l to +r. C (the "columns"): the integers in the last w columns.

2 ... r r + 1 ... q -1 q a -l a -l + 1 ... a -1 a a + 1 ... a + r -1 a + r a + r + 1 ... a + q -1 a + q 2a -
R                        C T block
We color this block using the weakly sum-free coloring of [ [1, q]] with colors 1, ..., n.

R block

We use the colors n + 1, ..., n + k in this block. We color an integer x according to its line number λ(x). For any x ∈ R, we color x with n + c where c is the color of λ(x) in the sum-free coloring of [ [1, p]]. Let (x, y) ∈ R 2 . The cases are twofold. In this case, we do not have information about the color of λ(x+y). Thereby, we want to have x+y ∈ C. A simple solution is to limit the horizontal displacement made by summing the two integers. That is, when the sum's row number is different from λ(x) + λ(y), to upper bound |x + y -(λ(x) + λ(y))a| so that x + y stays in C. There, the maximal displacement to the left (resp. to the right) is 2l (resp. 2r). Not crossing entirely C by going to the left is then expressed as -2l > -a + r. Likewise, not going to far to the right is expressed as 2r < a -l. It can then be written as max(l, r) w.

λ(x + y) = λ(x) + λ(y) If x + y ∈ R,

C block

In this block, we use colors 1, ..., n. We color an integer x according to its projection on the first row π(x) offeset by a, that is its column number. A simple solution is to color x with the same color as π(x) -a in the weakly sum-free coloring of [ [1, q]]. As long as 2q a + r (not going two far to the right) and there is no x ∈ π(C) -a such that 2x ∈ π(C) -a, we do not have a sum in C when taking two numbers in the same colum. Finally, the weakly sum-free property of the coloring of [ [1, q]] guarantees that if x ∈ [ [1, r]] and y ∈ C are colored the same, then x + y is colored differently.

Setting w = l = q 2 and r = q 2 satisfies these constraints: it corresponds to Theorem 3.1.

By setting q = WS (n) and p = S(k) in Theorem 3.1, one obtains the following corollary.

Corollary 3.2. WS (n + k) S(k) WS (n) + WS (n) 2 + 1 + WS (n) for all (n, k) ∈ (N * ) 2 .
This can be seen as an equivalent for weak Schur numbers of Abott and Hanson's construction for Schur numbers [START_REF] Abbott | A problem of Schur and its generalizations[END_REF]. This formula also includes the results from [START_REF] Rowley | New lower bounds for weak Schur partitions[END_REF] as a special case. For n > 2, the formula from Corollary 3.2 does not give new lower bounds.

Remark 3.3. The inequality from Corollary 3.2 can be improved by adding 1 to the lower bound if WS (n) is odd (more generally if q is odd in Theorem 3.1). However, it lengthens the proof and it is never useful in practice. Now, we introduce WS-templates and the sequence WS + in order to generalize the above construction.

Definition of [[1, WS + ]]

In this subsection, we introduce WS-templates and prove calculative results for the general construction theorem on templates for weak Schur numbers. Definition 3.4. Let (a, b) ∈ (N * ) 2 such that a > b. We define:

π a,b : x -→ (Id +a1 [[0,b]] )(x mod a).
If there is no confusion on the a and b to use, π a,b is denoted by π. Notice that for all x ∈ Z, π(x) = x mod a and for all

x ∈ [[b + 1, a + b]], b + 1 π(x) a + b.
Function π is the projection on the first line mentioned in the intuitive explanation. The following four propositions are calculative properties on π reflecting the behaviour of an element's column number in Figure 3 and that we will use later when we introduce WS-templates. Proof. It is a direct consequence of π(x) = x mod a.

Proposition 3.5. ∀x ∈ [[b + 1, a + b]], π(x) = x. Proof. Let x ∈ [[b + 1, a + b]]. If x < a then x mod a = x / ∈ [[1, b]]. Hence π(x) = x. Otherwise, x mod a = x -a ∈ [[1, b]]. Hence π(x) = x -a + a = x.
After defining the function related to the column number of each element in Figure 3, we introduce the function related to its line number.

Definition 3.9. Let (a, b) ∈ (N * ) 2 such that a > b. Define λ a,b : x -→ 1 + x -b -1 a .
If there is no confusion on the a and b to use, λ a,b is denoted by λ. Function λ maps an element x to its line number as mentioned in the intuitive explanation. As we have just done with π, we prove three more calculative results on both π and λ that are used in Subsection 3. Proof. By Proposition 3.10, x + y = a(λ(x) + λ(y)) + π(x) + π(y) -2a. Then

λ(x + y) = x + y -b -1 a + 1 = a(λ(x) + λ(y)) + π(x) + π(y) -2a -b -1 a + 1 = λ(x) + λ(y) -1 + π(x) + π(y) -b -1 a = λ(x) + λ(y) -1 + 1 since π(x) + π(y) ∈ [[a + b + 1, 2a + b]] = λ(x) + λ(y). Definition 3.13. Let (a, n, b) ∈ (N * ) 3 with a > b. Let (A 1 , ..., A n ) a partition of [[1, a + b]].
This partition is said to be a b-weakly-sum-free template (b-WS-template) with width a and n colors when:

∀i ∈ [[1, n]], A i is weakly-sum-free, ∀i ∈ [[1, n]], A i \[[1, b]] is sum-free,
For A n (the special subset):

∀(x, y) ∈ A 2 n , x + y > b + 2a =⇒ x + y -2a / ∈ A n ,
For the others subsets

∀i ∈ [[1, n -1]], ∀(x, y) ∈ A 2 i , x + y > a + b =⇒ π(x + y) / ∈ A i .
Number a is necessarily colored with color n. Note that the special color n is not necessarily the last color by order of appearance. Now, we introduce the number WS + (n) that plays the same role as S + (n) for S-templates. Definition 3.14. Let (n, b) ∈ (N * ) 2 . If there is a such that there is a b-WS-template with width a and n colors, we define:

WS + b (n) = max{a ∈ N * /
there is a b-WS-template with width a and n colors}.

If there is no such a, we set WS + b (n) = 0. Definition 3.15. Let n ∈ N * . We define:

WS + (n) = max b∈N * WS + b (n).
The following proposition shows how WS + (n) compares to weak Schur numbers.

Proposition 3.16. Let n ∈ [[2, +∞]]. Then: 3 2 WS (n -1) + 1 WS + (n) WS (n).
Proof. The lower bound comes from Corollary 3.2. The upper bound comes from the fact that a WStemplate with width a and n colors is in particular a partition of [ [1, a]] into n weakly sum-free subsets.

Having established these properties, we prove our main result.

Construction of weak Schur partitions using WS-templates

Theorem 2.3 describes a manner to extend an S-template into a larger Schur partitions using other Schur partitions. Likewise, a WS-template can be extended into larger weak Schur partitions by using Schur partitions. This is the object of the following theorem.

f : [[1, a + b]] -→ [[1, n + 1]] and g : [[1, p]] -→ [[1, k]].
Moreover, we assume that the sum-free coloring of [ [1, p]] is ordered. NB: To keep the notation short, the conditions x + y p and x + y a + b are omitted in the following five predicates.

The sum-free conditions are expressed as:

∀(x, y) ∈ [[b + 1, a + b]] 2 , f (x) = f (y) =⇒ f (x + y) = f (x), ∀(x, y) ∈ [[1, p]] 2 , g(x) = g(y) =⇒ g(x + y) = g(x).
The weakly sum-free condition is expressed as:

∀(x, y) ∈ [[1, a + b]] 2 , f (x) = f (y) x = y =⇒ f (x + y) = f (x),
The additional constraints for the WS-template are expressed as:

∀(x, y) ∈ [[1, a + b]] 2 , f (x) = f (y) n x + y > a + b =⇒ f (π(x + y)) = f (x), ∀(x, y) ∈ [[1, a + b]] 2 , f (x) = f (y) = n + 1 x + y > 2a + b =⇒ f (x + y -2a) = f (x).
We

T = [[1, b]], C = π -1 (f -1 ([[1, n]])), R = π -1 (f -1 ({n + 1})). We define a coloring h of [[1, pa + b]] using [[1, n + k]] colors: h : [[1, pa + b]] -→ [[1, n + k]], x -→    f (x), if x ∈ T , f (π(x)), if x ∈ C, n + g(λ(x)), if x ∈ R. Function h is well defined since sets T , C, R partition [[1, pa + b]]
. Now, we prove that h is a weakly sum-free coloring. Let x, y ∈ [[1, pa + b]] be such that x = y, h(x) = h(y), and x + y pa + b. We claim that h(x + y) = h(x). Nine cases are to be distinguished according to which of the sets T , C, R x and y belong to. It is sufficient to check only six cases out of nine since x and y play symmetric roles.

Case 1: (x, y) ∈ T 2 If x + y b then h(x + y) = f (x + y). Otherwise, b < x + y < a + b since b < a and therefore π(x + y) = x + y (Proposition 3.5). Hence in both cases h(x + y) = f (x + y). Given that f is a weakly sum-free coloring, f (x + y) = f (x) since f (x) = h(x) = h(y) = f (y) and x = y. That is h(x + y) = h(x).

Case 2: (x, y) ∈ T × C

Given that h(x) = h(y) and by definition of h, f (x) = f (π(y)). Besides, f (π(y)) n since y ∈ C. Two cases are to be distinguished according to the value of x + π(y).

If x + π(y) a + b then f (x + π(y)) = f (π(x + y)) (Proposition 3.7). Given that f is a weakly sum-free coloring, f (x + π(y)) = f (x) since f (x) = f (π(y)) and x = π(y) since x b < π(y). If x + π(y) > a + b then given that f is a WS-template and since f (x) = f (π(y)) n, f (π(x + π(y))) = f (x). Furthermore f (π(x + π(y))) = f (π(x + y)) (Proposition 3.6), such that f (π(x + y)) = f (x). In both cases f (π(x + y)) = f (x). If f (π(x + y)) n then h(x + y) = f (π(x + y)). Therefore h(x + y) = h(x) since f (x) = h(x). Otherwise, f (π(x + y)) = n + 1 and thus h(x + y) > n. In particular, h(x + y) = h(x) since h(x) = h(y) n. Case 3: (x, y) ∈ T × R Necessarily h(x) = h(y) = n + 1.
Two cases are to be distinguished according to the value of λ(x + y).

If λ(y) = λ(x + y) then π(x + y) = x + π(y) (Proposition 3.11). By definition of h, f (x) = f (π(y)).
Given that f is a weakly sum-free coloring, f

(x + π(y)) = f (x) since f (x) = f (π(y)) and x = π(y) since x b < π(y). Hence h(x + y) = h(x). If λ(y) = λ(x + y) then λ(x + y) = λ(y) + 1 since x b < a. Besides, n + 1 = h(y) = n + g(λ(y)).
Hence g(λ(y)) = 1. Moreover g(1) = 1 since g is an ordered coloring. Therefore, given that g is sum-free, g(λ(y)

+ 1) = 1. If π(x + y) ∈ A n+1 then h(x + y) = n + g(λ(x + y)) = n + 1. Otherwise, h(x + y) n.
In both cases h(x

+ y) = h(x). Case 4: (x, y) ∈ C 2
By definition of h and since h(x) = h(y), f (π(x)) = f (π(y)). Two cases are to be distinguished according to the value of π(x) + π(y).

If π(x) + π(y) a + b then π(x) + π(y) = π(x + y). Hence f (π(x + y)) = f (π(x)) since f is sum-free for x > b If π(x) + π(y) > a + b then given that f is a WS-template, f (π(π(x) + π(y))) = f (π(x)) since f (π(x)) = f (π(y)).
Besides, f (π(π(x) + π(y))) = f (π(x + y)) (Proposition 3.8). Hence f (π(x + y)) = f (π(x)).

In both cases f (π(x + y)) = f (x). If f (π(x + y)) n then h(x + y) = f (π(x + y)). Therefore h(x + y) = h(x) since f (x) = h(x). Otherwise, f (π(x + y)) = n + 1 and thus h(x + y) > n. In particular,

h(x + y) = h(x) since h(x) = h(y) n. Case 5: (x, y) ∈ C × R By definition of function h, h(x) = h(y). Case 6: (x, y) ∈ R 2 In particular f (π(x)) = f (π(y)) = n + 1.
Three cases are to be distinguished according to the value of π(x) + π(y).

If π(x) + π(y) ∈ [[a + b + 1, 2a + b]] then λ(x + y) = λ(x) + λ(y) (Proposition 3.

12). By definition of h and since

h(x) = h(y), g(λ(x)) = g(λ(y)). Hence, h(λ(x + y)) = h(λ(x)) since h is a sum-free coloring. If f (x + y) n + 1 then h(x + y) = n + g(λ(x + y)). And h(x) = n + g(λ(x)). Therefore, h(x + y) = h(x). Otherwise h(x + y) n < h(x). In particular h(x + y) = h(x). If π(x) + π(y) > 2a + b then f (π(π(x) + π(y))) = f (π(x)) = n + 1 since f is a b-WS template and f (π(x)) = f (π(y)). Given that π(π(x) + π(y)) = π(x + y) (Proposition 3.8), f (π(x + y)) = n + 1. If π(x) + π(y) b + a then, given that π(x) + π(y) b and f |[[b,a+b]] is sum-free, f (π(x) + π(y)) = f (π(x)) = n + 1. That is f (π(x + y)) = n + 1 (Proposition 3.5).
In both of the last two cases, f (π(x + y)) = n + 1 that is x + y ∈ C. Therefore h(x + y) < n h(x). In particular, h(x + y) = h(x).

We go back to Theorem 3.1 which can be seen as a particular case of Theorem 3.17 in the same way Abott and Hanson's construction [START_REF] Abbott | A problem of Schur and its generalizations[END_REF] can be seen as a particular case of Theorem 2.3. Its deferred proof is given below.

Proof of Theorem 3.1. Let (q, n) ∈ (N * ) 2 such that there is a partition of [ [1, q]] into n weakly sum-free subsets. Let f : [ [1, q]] → []1, n]] be a weakly sum-free colouring. Let b = q and a = q + q 2 + 1. A new colouring g is defined as follows:

g : [[1, a + b]] -→ [[1, n + 1]], x -→    f (x), if x ∈ [[1, b]], n + 1, if x ∈ [[b + 1, 2b + 1]], f (x -a), if x ∈ [[2b + 2, a + b]].
We claim that g is a b-WSF-template with width a and n + 1 colours. The result is then obtained by applying Theorem 3.17.

By setting p = S(k) and a = WS + (n + 1) in Theorem 3.17, one obtains: Remark 3.19. In the S-template construction for Schur numbers, the additive constant comes from the fact that the special color does not necessarily appear at the very begining of the repeating pattern. Likewise, b max can actually be replaced by

max b∈N * min(A n+1 \[[1, b]]) -1 | WS + b (n + 1) = WS + (n + 1) .
As in Corollary 2.4, the additive constant Theorem 3.17 can be improved by weakening the hypotheses made on the last row. The principle behind it is the same as in Proposition 2.5.

c ∈ [[1, k]], ∀(x, y) ∈ [[1, a + b]] × [[b + 1, a + b]], f (x) = f (y) π(x + y) b + c =⇒ g(π(x + y)) = f (x), ∀(x, y) ∈ [[1, a + b]] × [[b + 1, b + c]], f (x) = g(y) π(x + y) b + c =⇒ g(π(x + y)) = f (x).
Then, for every n ∈ N * , by using on the last row the coloring x -→ g(x -pS(n)), we have

WS (n + k) WS + (k + 1)S(n) + b + c.
The WS-templates can actually be fine-tuned further. However, it only gives minor improvements (most likely only an additive constant) while increasing the size of the search space.

These modifications rely on one main observation: the first row (excluding the "tail") has constraints that other rows do not have because of the tail, especially if the special color appears in the tail as well. Indeed, the line number of the sum of two numbers in the tail is never greater than one. Therefore, having a coloring dedicated to the first row and another coloring dedicated to the repeating pattern would weaken the constraints on the repeating pattern.

These additional constraints on the first row imply additional constraints on the rows of the its color (compared to other colors). However, one may notice that there are never two consecutive rows of this color. Moreover these additional constraints mostly concern two consecutive rows. Having a special coloring for rows of this color as well as another color dedicated to the first row avoids enforcing these constraints specific to one color on the other colors and thus on the main repeating pattern.

Finally, given that the color of the first row is different from the special color of the main repeating pattern, coloring g in Proposition 3.20 can exploit this difference.

There is a construction theorem for WS-templates as well. Proof. The idea is the same as in theorem 3.17

New lower bounds for Weak Schur numbers

We produced WS-templates using the lingeling SAT solver [START_REF] Biere | CaDiCal, Lingeling, Plingeling, Treengeling and YalSAT entering the SAT competition 2017[END_REF], hence providing lower bounds on W S + and inequalities of the type W S(n + k) aS(n) + b. We sought templates providing the greatest value of (a, b) (in the lexicographic order). Details concerning the encoding as a SAT problem can be found in [START_REF] Heule | Schur number five[END_REF].

Here are the inequalities given by the current best WS-templates. The template corresponding to the third inequality can be found in B.

W S(n

+ 1) 4 S(n) + 2 (8)
W S(n + 2) 13 S(n) + 8

Inequalities ( 8) and (9) were found by Rowley, they are detailed in [START_REF] Rowley | New lower bounds for weak Schur partitions[END_REF].

W S(n + 3) 42 S(n) + 24 [START_REF] Fredricksen | Symmetric sum-free partitions and lower bounds for Schur numbers[END_REF] W S(n + 4) 132 S(n) + 26 [START_REF] Rowley | An improved lower bound for s(7) and some interesting templates[END_REF] Inequality [START_REF] Fredricksen | Symmetric sum-free partitions and lower bounds for Schur numbers[END_REF], found with a SAT solver [START_REF] Biere | CaDiCal, Lingeling, Plingeling, Treengeling and YalSAT entering the SAT competition 2017[END_REF], cannot be improved (with this definition of WS-template). It uses the first sophistication explained in Subsection 3.3 in order to add the last number in the first color. As for inequality [START_REF] Rowley | An improved lower bound for s(7) and some interesting templates[END_REF], it was obtained by combining an S-template with width 33 with a WStemplate with width 4. The best template we could find with a computer search gives the inequality WS (n + 4) 127S(n) + 68. It was also found with the SAT solver. In order to reduce the search space, we only looked for WS-templates with five colors which start with a near-optimal WS (4) partition and we assumed that the special color was the last by order of appearance. We expect that better WS-templates with n 5 colors can be found but one would have not to use the above assumptions.

Like in Subsection 2.3, we compute the lower bounds given by Inequalities ( 8), [START_REF] Heule | Schur number five[END_REF], and [START_REF] Fredricksen | Symmetric sum-free partitions and lower bounds for Schur numbers[END_REF] for n ∈ [ [START_REF] Rowley | New lower bounds for weak Schur partitions[END_REF]15]]. The best lower bound for each value of n is highlighted. The weak Schur partitions obtained with WS-templates have an extremely regular structure. Therefore, one may expect to find larger partitions by allowing more freedom in the structure of the partitions while still preserving a structure somewhat close to a template-based partition thus reducing the search space to a manageable size. When applying the inequality WS (n + 1) 4 S(n) + 2, one may realize that it is possible to build a weakly sum-free partition of length 4 S(n) + 3 for small values of n (n 4) by using the construction of Corollary 3.18 for the integers 1, 2, 4i and 4i + 1 for i ∈ [[1, S(n)]] but not constraining the other integers. We did the same for the Schur partitions corresponding to Schur number five. More precisely, we imposed these constraints only for i 50 in order to have more degrees of freedom. The number 50 was chosen arbitrarly so that all of the Schur number 5-partitions could be tested in a few hours.

However, trying out all of the 2 447 113 088 Schur number 5-partitions [START_REF] Heule | Schur number five[END_REF] one by one would not result in a reasonable computation time: it is necessary to test the construction on several partitions at once. We encoded the problem as a satisfiability problem and used the 1616 backdoors that were used in [START_REF] Heule | Schur number five[END_REF] in order to encode a group of partitions in a compact and efficient way. Among all of the 1616 backdoors, only the 911 th backdoor gave a weakly sum-free partition of length 643. This backdoor gave a weakly sum-free partition of length 646 and cannot give a weakly sum-free partition of length 647. A weakly sum-free partition of length 646 can be found in C.

Conclusion on WS-templates

We started by giving a new construction which can be seen as an equivalent for weakly sum-free partitions of Abott and Hanson's construction for sum-free partitions. We then generalized this construction by introducing WS-templates. This allows us to find new lower bounds and new inequalities for weak Schur numbers. One may notice the significant gap between the former lower bounds for weak Schur numbers obtained by conducting a computer search and the new lower bounds obtained with WS-templates; thus questioning the methods used and the assumptions made when obtaining these former lower bounds. We reckon better WS-templates with n 5 colors can be found by making different assumptions and using a different method (Monte-Carlo methods for instance).

Conclusion and future work

We have produced new templates for Schur and Ramsey numbers. We also have generalized the concept of template to weak Schur numbers. These templates provide new general inequalities of the form S(n + k) aS(n) + b and WS (n + k) aS(n) + b as well as new lower bounds for both Schur and weak Schur numbers. We have given a new lower bound WS (6) 646 with a method that can yield slight improvements for fixed values of n over lower bounds obtained with the inequality WS (n + 1) 4S(n) + 2.

Given that the introduction of templates is quite recent, we expect the bounds to be improved as this special type of partition becomes better understood and larger templates are found. One may try to produce better templates by using heuristics or approximation algorithms, such as Monte-Carlo algorithms for instance.

The best lower bound WS (6) 583 achieved with a computer search using Monte-Carlo methods [START_REF] Eliahou | Investigating Monte-Carlo Methods on the Weak Schur Problem[END_REF] is significantly lower than those obtained with a structure: WS (6) 642 [START_REF] Rowley | New lower bounds for weak Schur partitions[END_REF], WS (6) 646. We have evidence that 583 is the maximal value in the search space considered by [START_REF] Eliahou | Investigating Monte-Carlo Methods on the Weak Schur Problem[END_REF][START_REF] Bouzy | An Abstract Procedure to Compute Weak Schur Number Lower Bounds[END_REF][START_REF] Rafilipojaona | Nombres de Schur classiques et faibles[END_REF]. Therefore, it questions the assumption that large partitions for WS (n + 1) start with large partitions for WS (n). It also indicates that WS (5) might need further investigation since the current lower bound WS (5) 196 [START_REF] Eliahou | Weak Schur numbers and the search for G.W. Walker's lost partitions[END_REF] was obtained by considering the same type of search space.

Figure 1 :

 1 Figure 1: Visualisation of an Abbott and Hanson construction

Figure 2 :

 2 Figure 2: Visualisation of a template-based construction

Corollary 2 . 9 . 5 √

 295 The growth rate for Schur numbers (and Ramsey numbers R n (3)) satisfies γ 380 ≈ 3.28.

Figure 3 :

 3 Figure 3: Construction of the weakly sum-free coloring

1

 1

  we use the sum-free property of the coloring of [[1, p]]. Otherwise, x + y ∈ C which is fine because we only use colors 1, ..., n in block C. λ(x + y) = λ(x) + λ(y)

Proposition 3 . 6 .Proposition 3 . 7 . 5 Proposition 3 . 8 .

 3637538 Let x ∈ [[1, b]] and y ∈ N * . Thenπ(x + π(y)) = π(x + y). Proof. It is a direct consequence of π(x) = x mod a. Let x ∈ [[1, b]] and y ∈ N * such that x + π(y) a + b. Then π(x + y) = x + π(y).Proof. π(y) b + 1 and thus b + 1 x + π(y) a + b. π(x + y) = π(x + π(y)) by Proposition 3.6 = x + π(y) by Proposition 3.Let (x, y) ∈ (N * ) 2 . Then π(π(x) + π(y)) = π(x + y).

3 . 3 . 10 .Proposition 3 . 11 .

 3310311 Proposition Let x ∈ N * . Then x = aλ(x) + π(x) -a.Proof. Let (a, b) ∈ (N * ) 2 such that a > b and let x ∈ N * . We have aλ(x) + π(x) -a = a x -b -1 a + (x mod a) + a1 [[0,b]] (x mod a). If x mod a > b then aλ(x) + π(x) -a = a x a + x mod a = x. If x mod a b then aλ(x) + π(x) -a = a x a -1 + x mod a + a = x. Let x, y ∈ Z such that λ(x + y) = λ(y). Then π(x + y) = x + π(y).Proof. By applying Proposition 3.10 twice, we get aλ(x + y) + π(x + y) -a = x + y = x + aλ(y) + π(y) -a.The result is then obtained by simplifying the equality. Proposition 3.12. Let x, y ∈ Z such that π(x) + π(y) ∈ [[a + b + 1, 2a + b]]. Then λ(x + y) = λ(x) + λ(y).

Theorem 3 .

 3 17. Let (a, n, b) ∈ (N * ) 3 with a > b and (p, k) ∈ (N * ) 2 . If there are a sum-free k-partition of [[1, p]] and a b-WS-template (A 1 , ..., A n+1 ) with width a and n + 1 colors, then there is a partition of [[1, pa + b]] into k + n weakly sum-free subsets. Proof. Let (a, n, b) ∈ (N * ) 3 and (p, k) ∈ (N * ) 2 . Let f and g be colorings associated to the b-WS-template and to the sum-free partition of [[1, p]], respectively:

  split [[1, pa + b]] into three subsets. NB: To keep the notation short, the restriction to [[b + 1, pa + b]] of π defined in Subsection 3.2 is denoted by π in the hereunder equations:

Function

  g |[[b+1,a+b]] is a sum-free colouring. Indeed, let (x, y) ∈ [[b + 1, a + b]] 2 such that g(x) = g(y). If g(x) = n + 1 then z = x + y > 2b + 1 and therefore, either z > a + b or f (z) = n + 1. Otherwise, x + y > a + b. Function g is a weakly sum-free colouring. Indeed, let (x, y) ∈ [[1, a + b]] 2 such that z = x + y a + b and g(x) = g(y). Given that x and y have symmetric roles, we can assume that x y. If x > b then g(z) = g(x) as seen above. If y b then f (x) = g(x) = g(y) = f (y) n and either z a + b and f (z) = f (x) since f is a weakly sum-free colouring or a + b + 1 z 2b and g(z) = n + 1; therefore g(z) = g(x). If x b and y > b then g(x) = f (x), g(y) = f (y -a) and g(z) = f (z -a).We have x = y -a (otherwise, we would have a + b z = 2y -a 4b + 4 -a > a + b) and thusf (x + y -a) = f (x), that is g(z) = g(x).Colour n + 1 verifies the additional constraints for the special colour. Indeed, b + 2a 4b + 2. Hence, ∀(x, y) ∈ g -1 ({n + 1}), x + y b + 2a. Colours 1, ..., n verify the additional constraints for the regular colours. Indeed, let (x, y) ∈ g -1 ([[1, n]]) such that x+y > a+b. Given that x and y have symmetric roles, we can assume that x y. Necessarily y 2b + 2. If x b then z = x + y ∈ [[a + b + 1, a + 2b]] and therefore π(z) ∈ [[b + 1, 2b]]. Otherwise, z = x + y ∈ [[4b + 4, 2a + 2b]] and therefore π(z) ∈ [[b + 1, 2b]]. Hence, in both cases, g(π(z)) = g(x).

Corollary 3 .

 3 18. Let n, k ∈ N * and set b max = max{b ∈ N * /WS + b (n + 1) = WS + (n + 1)}. Then: WS (n + k) S(k)WS + (n + 1) + b max .

Proposition 3 .

 3 20. Let (b, k, a) ∈ (N * ) 3 and let f be a coloring associated to a b-WS-template with width p and k colors. Let c ∈ N and assume there is a coloring g of [[b + 1, b + c]] with k colors such that for all

Theorem 3 .

 3 21. Let (k, p) ∈ (N * ) 2 and (a, n, b) ∈ (N * ) 3 . If there are an S-template with width p and k + 1 colors and a b-WS-template with width a and n colors, then there is a pb-WS-template with width pq and (n + k) colors.

  . The properties specific to WS-templates are inherited from both the S-template and the WS-template. Theorem 3.21 yields the following corollary. Corollary 3.22. Let n, k ∈ N * . Then WS + (n + k) S + (k + 1)WS + (n).

Table 1 :

 1 Comparison of lower bounds for Schur numbers

	n	1 2 3 4 5	6	7	8	9	10	11	12
	State of the art	1 4 13 44 160 536 1 696 5 286 17 694 60 320 201 696 637 856 [9] [10] [11] [7] [7] [7] [7] [7]
	Our results					17 803 60 948 203 828 644 628

Table 2 :

 2 Comparison of lower bounds for weak Schur numbers

	n	1 2 3 4	5	6	7	8	9	10	11	12
	State of the art	2 8 23 66 196 642 2 146 6 976 22 056 70 778 241 282 806 786 [12] [8] [8] [8] [8] [8] [8] [8]
	Our results			646			22 536 71 256 243 794 815 314

Table 3 :

 3 New lower bounds for S(n) with n ∈ [[8, 15]]

	n	8	9	10	11
	33 S(n -3) + 6	5 286	17 694	55 974	174 444
	111 S(n -4) + 43	4927	17 803	59 539	188 299
	380 S(n -5) + 148	5 088	16 868	60 948	203 828
	n	12	13	14	15
	33 S(n -3) + 6	587 505 2 011 290	6 726 330 21 272 730
	111 S(n -4) + 43 586 789 1 976 176	6 765 271 22 624 951
	380 S(n -5) + 148 644 628 2 008 828	6 765 288 23 160 388

Except for S

[START_REF] Rowley | New lower bounds for weak Schur partitions[END_REF]

, S

[START_REF] Heule | Schur number five[END_REF]

, and S(13), the best lower bounds are obtained thanks to the fifth inequality S(n + 5) 380 S(n) + 148. The table does not go any further, but the same inequality allows one to improve the lower bounds for n 15.

Table 4 :

 4 New lower bounds for WS (n) with n ∈ [[8, 15]]

	n	8	9	10	11
	4 S(n -1) + 2	6 786	21 146	71 214	243 794
	13 S(n -2) + 8	6 976	22 056	68 726	231 447
	42 S(n -3) + 24	6 744	22 536	71 256	222 036
	n	12	13	14	15
	4 S(n -1) + 2	815 314	2 578 514	8 045 162	27 061 154
	13 S(n -2) + 8	792 332	2 649 772	8 380 172	26 146 778
	42 S(n -3) + 24 747 750	2 559 840	8 560 800	27 074 400
	3.5 Another approach: WS (6) 646

These authors contributed equally.

These authors supervised the work of the main authors. *Corresponding author (joanna.tomasik@centralesupelec.fr)

Acknowledgements

This study results from a long-term project performed within the framework of the Project Cluster "Training for Research" of CentraleSupélec. We would like to address warm thanks to Fred Rowley who kindly answered our many questions and with who it was a pleasure to exchange. Last but not least, we would like to thank Marijn Heule for providing the backdoors used in the computation of the new lower bound for WS (6).

A S-templates

This template provides the inequality WS (n+3) 42 S(n)+24 by placing one last number, here represented by (N + 1), in the first subset (see Proposition 3.20).