Sébastien Boisgérault
email: sebastien.boisgerault@mines-paristech.fr.

A Core Theory of Delay Systems

We introduce a framework for the description of a large class of delay-differential algebraic systems, in which we study three core problems: first we characterize abstractly the well-posedness of the initial-value problem, then we design a practical test for well-posedness based on a graph-theoretic representation of the system; finally, we provide a general stability criterion. We apply each of these results to a structure that commonly arises in the control of delay systems.

Introduction

Systems of delay-differential algebraic equations (DDAE) combine differential and algebraic equations with delayed variables in the right-hand side. Among them, we represent linear and time-invariant systems with a finite memory r as

ẋ(t) = Ax t + By t y(t) = Cx t + Dy t (1)
where x(t) ∈ R n , y(t) ∈ R m and the notation z t stands for the memory of the variable z at time t: the function defined by z t (θ) = z(t + θ) for any θ ∈ [-r, 0]. The symbols A, B, C and D denote delay operators: linear and bounded operators from the space of continuous functions C([-r, 0], C j ) to C i for some integers i and j. Specifically, since (x(t), y(t)) ∈ R n+m , we require that the compound operator

[ A B C D ] (2)
is linear and continuous from C([-r, 0], C n+m ) to C n+m .

This description is general enough to encompass systems of seemingly different nature -such as ordinary differential equations (e.g. ẋ(t) = x(t)), delay systems of retarded type (e.g. ẋ(t) = x(t -1)) or neutral type (e.g. ẏ(t) = ẏ(t -1), rewritten as ẋ(t) = 0 and y(t) = x(t) + y(t -1)), difference equations (e.g. y(t) = y(t -1)) but also different types of delays: systems with single delays (e.g. ẋ(t) = x(t) + x(t -1)), multiple delays (e.g. ẋ(t) = x(t -1) + x(t -2)), distributed delays (e.g. ẋ(t) = ∫ t t-1 x(θ) dθ) or a combination thereof. We refer the reader to (Bensoussan et al. 2006, chap. 4) for a more comprehensive collection of examples.

However, competing theories of delay systems have been developped and used over time [START_REF] Hale | Theory of Functional-differential Equations[END_REF]M. Delfour and Karrakchou 1987;[START_REF] Salamon | Control and observation of neutral systems[END_REF]; they may use different definitions of delay operators, of the solutions to the initialvalue problem and may be restricted to systems of a certain type. For the newcomer in the field, this is an unintended source of complexity. To make the subject more widely accessible, we lay out in this paper a simple but general framework for the description of delay systems, based on a combination of linear algebra and measure theory. Then we develop on this foundation a core theory: we characterize the well-posedness of the initial-value problem, then we perform a graph-theoretic analysis of this issue and finally, we provide a general stability criterion.

We illustrate the results of each section with the same example: the so-called finite spectrum assignment (FSA) architecture which is used for the control of dead-time systems [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF]. This example -and the methods that we use to study it -are actually representative of a large class of controllers for delay systems which ranges from the primeval Smith predictor to the general observer-predictor structure [START_REF] Mirkin | Every stabilizing dead-time controller has an observer-predictor-based structure[END_REF]. A linear dead-time systems is governed by an ordinary differential equation ẋ(t) = Ex(t) + F u(t) but the only information available at time t is the delayed variable x(t -T ). If we apply the control u(t) = -Gy(t) where y(t) is the predicted value of x(t) based on x(t -T ) and the history of u defined by

y(t) = e T E x(t -T ) + ∫ 0 -T e -θE F u t (θ)dθ then the closed-loop dynamics is ẋ(t) = Ex(t) -F Gy(t) y(t) = e T E x(t -T ) - ∫ 0 -T e -θE F Gy t (θ)dθ
(3) Since this system is not overly simplistic (it combines effectively differential and algebraic equations, its delays are discrete and distributed), it is a good testbed for the theory exposed in this paper.

The Initial Value Problem

Delay Operators

We have described general delays as operators applied to continous functions: if C(X, Y ) denotes the space of continuous functions from X to Y , a delay operator with memory length r is a bounded linear operator from C([-r, 0], C j ) to C i for some integers i and j. But delays also have two alternate (and equivalent) representations -as measures and as convolution kernels -which we will use extensively in the next sections.

We start with a quick example of these representations: consider y(t) = x(t -1) where x(t) is a scalar variable. We will associate to this delayed expression three distinct representations m 1 , m 2 and m 3 such that by construction

y(t) = m 1 x t = ∫ 0 -1 x t (θ) dm 2 (θ) = ∫ 1 0 x(t -θ)dm 3 (θ). (4)
The first one m 1 is an operator applied to continuous scalar functions defined on [-1, 0]; here, equation (4) clearly mandates that m 1 χ = χ(-1). The representations m 2 and m 3 are measures; equation (4) holds if m 2 is δ -1 , the Dirac measure at t = -1 and m 3 is δ 1 , the Dirac measure at t = 1. Now if instead we consider the distributed delay y(t) = ∫ t t-1 x(θ)dθ, equation ( 4) is satisfied with

m 1 χ = ∫ 0 -1 χ(t) dt, m 2 = dt| [-1,0] and m 3 = dt| [0,1] .
In general, we deal with vector-valued variables; in this broader context, the term "measure" may refer to several things. We use the term scalar measure for a complex-valued and countably additive function defined for the bounded Borel subsets of the real line [START_REF] Fell | Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles[END_REF]. A matrix-valued measure (resp. vector-valued measure) is a countably additive function defined for the bounded Borel subsets of the real line whose values are complex matrices (resp. vectors) of a fixed size. Equivalently, it is as a matrix (resp. vector) whose elements are scalar measures.

Given a matrix-valued measure L, we define the integral of the vector function ϕ : R → C j with respect to L as the vector of C i whose k-th element is given by

∀ k ∈ {1, . . . , j}, [∫ dL ϕ ] k = j ∑ l=1 ∫ ϕ l dL kl .
Riesz's representation theorem provides a bijection between delay operators with memory length r and matrix-valued measures supported on [-r, 0]. We may therefore use the same symbol L to denote both objects, and this convention yields for any ϕ ∈ C([-r, 0], C j )

Lϕ = ∫ dL ϕ.
The nature of the argument of L (function or set) determines without ambiguity which representation of L is used in a given context. When the argument is a set, we will also drop the parentheses when it improves readability: typically, instead of L({0}), we will use the lighter notation L{0}.

Let L * refer to the measure obtained by symmetry of the measure L around t = 0, which is defined for any bounded Borel set B by L * (B) = L(-B). This construct obviously provides a new bijection between delay operators with memory length r and matrix-valued measures supported on [0, r], which is related to the representation of delay operators as convolutions.

We say that a measure µ is limited on the left if its support is included in [-r, +∞) for some r ∈ R, causal if its support is included in [0, +∞) and strictly causal if additionally µ{0} = 0. The convolution µ * ν of two scalar measures limited on the left µ and ν is the scalar measure defined for any bounded Borel set B by

(µ * ν)(B) = ∫ ν(B -x)dµ(x).
The convolution of two vector or matrix-valued measures limited on the left combines the scalar convolution and the linear algebra product: for example, the convolution of two measures L and M with values in C i×j and C j×k is the measure L * M , with values in C i×k , given by

(L * M )(B) α,β = j ∑ γ=1 (L α,γ * M γ,β )(B).
We may identify a (locally integrable) function f with the measure given by

f (B) = ∫ B f (t) dt.
This enables us to define the convolution of functions and measures, which may be identified with a function, and of two functions, which may be identified with an absolutely continous function. Additionally, we implicitly extend by 0 functions that are defined on a proper subset of real line -for example functions defined on [-r, 0] or [-r, +∞). With these conventions, if L is a delay operator with memory r and ϕ ∈ C([-r, +∞), C i ) then for almost every t > 0

Lϕ t = (L * * ϕ)(t).

Well-Posedness

Several concepts of solution exist for the problem formally described by ∀ t > 0 ẋ(t) = Ax t + By t y(t) = Cx t + Dy t , (x(0), x 0 , y 0 ) = (ϕ, χ, ψ).

(5)

A classic (or continuous) solution is a pair of continuous functions (x, y) from [-r, +∞) to C n+m such that ẋ exists, is continous on (0, +∞) and satisfies the system equations and initial conditions. In this setting, necessarily the functions χ and ψ are continous, ϕ = χ(0) and the consistency condition

ψ(0) = Cχ + Dψ holds.
The continuity assumption ensures that the right-hand side of the system equations is defined for any time t; if we relax this assumption, we may generalize the concept of solution to locally integrable solutions [START_REF] Salamon | Control and observation of neutral systems[END_REF]M. Delfour and Karrakchou 1987).

Let L be a delay operator with memory length r and L * the corresponding convolution kernel. For any continuous function z defined on [-r, +∞), we have Lz t = (L * * z)(t) for any t > 0 but the right-hand side of this equation is still properly defined -as a locally integrable function of t -if z is merely locally integrable, a strong incentive to rewrite the initial value problem as a convolution equation. Using the Heaviside step function e, defined by

e(t) = 1 if t ≥ 0, 0 otherwise,
we may also rewrite the (integral form of) the differential equation as a convolution equation. We end up with the following definition:

Definition -Solution of the Initial Value Problem. A pair of locally integrable functions (x, y), defined on [-r, +∞), with values in C n+m , is a (locally integrable) solution of the DDAE initial value problem ( 5) if

(x(0 + ), x 0 , y 0 ) = (ϕ, χ, ψ) and if there is a f ∈ C n such that [ x y ] (t) = [ e * A * e * B * C * D * ] * [ x y ] (t) + [ f 0 ] for a.e. t > 0.
This definition makes sense even if the initial function data is not continuous, does not meet the consistency condition, or if χ(0) ̸ = ϕ. It is however consistent with the concept of continuous solution when such solutions exist. The constant vector f is uniquely determined by the initial value: we have x(0

+ ) = ϕ if and only if f = ϕ -(e * A * * χ + e * B * * ψ)(0).
Now, because of its algebraic component, DDAE system (5) may have no solutions or multiple solutions; we may actually easily exhibit an algebro-differential system (with no delay) with this property. Consider

∀ t > 0, ẋ(t) = 0 y(t) = x(t) + y(t) (6)
It is defined formally by n = 1, m = 1, for example r = 1 (any nonnegative value is admissible) and for any χ 0), and Dψ = ψ(0). If x(0 + ) = 0, then x(t) = 0 for t > 0 is the unique solution of the first system equation; the second equation becomes y(t) = y(t), hence arbitrary values of y(t) for t > 0 satisfy it: multiple solutions exist. On the contrary, if x(0 + ) = 1, then necessarily x(t) = 1 for any t > 0; the second equation would become y(t) = 1 + y(t), which no function y(t) may satisfy: there are no solutions to this initial value problem.

∈ C 0 ([-r, 0], R n ) and ψ ∈ C 0 ([-r, 0], R m ), Aχ = 0, Bψ = 0, Cχ = χ(
A simple assumption that yields uniqueness of the solution is explicitness: a DDAE system is explicit if the right-hand side of its algebraic equation only depends strictly causally on the variable y, that is, if D{0} = 0. For example, system (6) is not explicit since y(t) = x(t) + y(t) and thus D{0} = 1; replace this equation with y(t) = x(t) + y(tϵ) for any ϵ > 0 and it becomes explicit.

However this assumption is too conservative for some practical use cases, including the kind of sound composition of input-output systems that is exposed in the next section. Therefore we state in this section a well-posedness theorem applicable under a more general assumption that ensures that the system is merely equivalent to an explicit system.

Let I p be the p × p identity matrix and X be the product space

X = C n × L 2 ([-r, 0], C n+m ). (7)
endowed with the norm

∥(ϕ, χ, ψ)∥ 2 X = |ϕ| 2 + ∫ 0 -r |χ(t)| 2 + |ψ(t)| 2 dt. ( 8
)
Theorem -Well-posedness. DDAE systems such that I m -D{0} is invertible are well-posed in the product space X: when (ϕ, χ, ψ) ∈ X, there is a unique solution (x, y) to the initial value problem (5), it belongs to L 2 loc ([-r, +∞), C n+m ), x is continuous and for any t > 0 there is some α > 0 such that

∥(x(t), x t , y t )∥ X ≤ α∥(ϕ, χ, ψ)∥ X . ( 9
)
It is classic that this result holds when the system is explicit [START_REF] Salamon | Control and observation of neutral systems[END_REF]; M. [START_REF] Delfour | State space theory of linear time invariant systems with delays in state, control, and observation variables. I, II[END_REF][START_REF] Boisgérault | Growth bound of delay-differential algebraic equations[END_REF]. Therefore, to prove the general case, we only need to demonstrate the following lemma:

Lemma -Equivalent Systems. Assume that the matrix J := I m -D{0} associated to system (5) is invertible and denote E and F the delay operators E := J -1 C and F := J -1 (D -D| {0} ). The original initial value problem and the one defined by

∀ t > 0 ẋ(t) = Ax t + By t y(t) = Ex t + F y t , (x(0), x 0 , y 0 ) = (ϕ, χ, ψ). ( 10
)
have the same solutions but the latter system is always explicit:

F {0} = 0.
Proof. We may decompose the convolution kernel of D into two components:

D * = (D| [-r,0] ) * = (D| {0} ) * + (D| [-r,0) ) * .
The first component is instantaneous: for any locally integrable function y

((D| {0} ) * * y)(t) = D{0}y(t);
the second component is strictly causal:

(D| [-r,0) ) * {0} = D| [-r,0) {0} = 0.
Consequently, a pair of locally integrable functions (x, y) : [-r, +∞) → R n+m is a solution of the DDAE system equations ( 5) if and only if they satisfy, almost everywhere for t > 0, the equations

x(t) = (A * * x)(t) + (B * * y)(t) + f y(t) -D{0}y(t) = (C * * x)(t) + ((D| [-r,0) ) * * y)(t)
By assumption the matrix J = I m -D{0} is invertible, thus the second equation is equivalent to 0) . The functions x and y are solutions of the original DDAE system if and only if they are solutions -with the same initial values -of the system whose delay operators are A, B, E and F . This new DDAE system is explicit by construction:

y(t) = (E * * x)(t) + (F * * y)(t) with E = J -1 C and F = J -1 D| [-r,
F {0} = J -1 D| [-r,0) {0} = 0. ■

The FSA System

The delay operators A, B and C of system (3) are defined by Ax t = Ex(t), By t = -F Gy(t) and Cx t = e T E x(t -T ). Since the Dirac measure δ τ at τ ∈ R satisfies δ τ φ = φ(τ ), we have A = Eδ 0 , B = -F Gδ 0 , C = e T E δ -T or equivalently

A * = Eδ 0 , B * = -F Gδ 0 , C * = e T E δ T . ( 11
)
The fourth delay operator D is defined by

Dy t = - ∫ T 0 e θE F Gy(t -θ)dθ = -(e θE F Gdθ| [0,T ] * y)(t), hence D * = -e θE F Gdθ| [0,T ] . ( 12
)
In particular

D * {0} = - ∫ {0}
e θE F Gdθ = 0, thus this DDAE system is explicit and therefore it is well-posed.

Graph-Theoretic Analysis

Block diagrams used in control theory provide an alternative to the system-ofequations approach for the modelling of dynamical systems: in this context, dynamical systems are specified as diagrams made of blocks and wires as in figure 1. Blocks are independent input-output systems with their own behaviors: each block specifies how to compute the values of its output variables given the values of its input variables (and optionally of some local initial state). In figure 1, each rectangle is a block and the symbol it holds defines its behavior. Wires connect these blocks to build larger systems: they specify how the input of each block is computed as a combination of the outputs of all blocks. This classic description -actually a graph representation in disguise -offers new indsights into the structural properties of dynamical systems. In section 3.1, we complete for delay systems the informal definition of block diagrams that we have given so far and expose the connexion with graph theory. Then, in section 3.2, we focus on causality to design sound composition rules of subsystems, that ensure the well-posedness of the initial-value problem for the global system.

Block Diagrams

For linear time-invariant systems, block diagrams can always be decomposed into elementary input-output systems with a scalar input u(t) and a scalar output y(t), defined for t > 0, and characterized by a scalar convolution kernel k * . Additionally, for delay systems, only two types of elementary subsystems are required: the integrator, whose kernel is the Heaviside step function e, and (scalar) finite-memory delay operators. By definition, the behavior of an elementary system is governed by

y(t) = (k * * u)(t) + (k * * ω)(t), t > 0;
where ω, a scalar function defined on [-r, 0] that represents a finite memory of past input values, determines the system initial state.

A block diagram embeds a collection of such systems into a (directed, labeled) graph structure: each vertice i refers to an output variable y i and each edge j → i is labeled with a scalar kernel k * = K * ij ; the global initial state is determined by the collection of functions ω ij for every edge j → i. The functional analytic representation of this system is

y i (t) = ∑ j (K * ij * y j )(t) + ∑ j (K * ij * ω ij )(t), t > 0. ( 13
)
For example, the block diagram depicted in figure 1 has two nodes -associated to variables y 1 and y 2 -and four edges -whose labels are K * 11 = 0, K * 12 = e, K * 21 = 2δ 0 and K * 22 = δ T . Then y 1 (t), defined for t > 0, satisfies

y 1 (t) = (0 * y 1 )(t) + (e * y 2 )(t) + (0 * ω 11 )(t) + (e * ω 12 )(t) = (e * y 2 )(t) + (e * ω 12 )(t)
which, since (e * ω 12 )(t) is a constant c for t > 0, is the integral form of the differential equation ẏ1 (t) = y 2 (t) with initial value y 1 (0) = c. On the other hand y 2 (t), also defined for t > 0, satisfies

y 2 (t) = (2δ 0 * y 1 )(t) + (δ T * y 2 )(t) + (2δ 0 * ω 21 )(t) + (δ T * ω 22 )(t)
and since (2δ 0 * ω 21 )(t) = (δ T * ω 22 )(t) = 0 for any t > r, eventually the equation y 2 (t) = 2y 1 (t) + y 2 (t -T ) holds and the behavior of the system is ruled by a DDAE system. This is not accidental, since the following result holds:

Theorem. Every DDAE system has a canonical block diagram representation.

Proof. Suppose that for some constant vector f

x(t) = (e * A * * x)(t) + (e * B * * y)(t) + f y(t) = (C * * x)(t) + (D * * y)(t) , t > 0 and (x(0 + ), x| [-r,0] , y| [-r,0] ) = (ϕ, χ, ψ).
The vector f is uniquely determined by the initial state. We may define the auxiliary variable w(t) for t > 0 by w(t) = (A * * x)(t) + (B * * y)(t), and rewrite the first equation of the DDAE system as

x(t) = (e * w)(t) + (e * υ)(t), t > 0 for any function υ : [-r, 0] → C n such that (e * υ)(0) = f.
It is now plain that the variable z(t) = (x(t), w(t), y(t)) defined for t > 0, the matrix kernel and the initial state

K * =   0 eI n 0 A * 0 B * C * 0 D *   , ω ij = (χ, υ, ψ) j (14) satisfy z i (t) = ∑ j (K * ij * z j )(t) + ∑ j (K * ij * ω ij ), t > 0. ( 15
)
which is the functional analytic representation of a block diagram. ■

Causality Analysis

We say that a scalar input-output system is causal (resp. strictly causal) if its convolution kernel is causal (resp. strictly causal). For example, since the kernel of the integrator is the function e, it is strictly causal:

e{0} = ∫ {0} e(t) dt = 0.
A (discrete or pure) delay of T > 0 seconds satisfies y(t) = u(t -T ); its kernel is k * = δ T and it is strictly causal. Similarly, a distributed delay with bounded delay r, defined by the equation

y(t) = ∫ 0 -r h(θ)u(t + θ)dθ
for some locally integrable function h : [-r, 0] → R, is also strictly causal. A gain y(t) = ku(t) is causal, but it is strictly causal only if k = 0.

Definition -Causality Loop.

A causality loop of a block diagram with kernel matrix K * of size n × n is a finite sequence of integers i 0 , i 1 , . . . , i j in {1, . . . , n} such that i 0 = i j and for any ℓ ∈ {0, . . . , j -1}, the system with kernel k * ℓ = K * i ℓ+1 i ℓ is not strictly causal. The block diagram of figure 1 has no causality loop, since the only element of its matrix K * which is not strictly causal is K * 21 = 2δ 0 which is not on the matrix diagonal. The following result is applicable in this case:

Theorem. If the block diagram representation ( 14)-(15) of DDAE system (5) has no causality loop then D{0} is nilpotent.

Proof. Let K * be the kernel matrix of the block diagram representation of DDAE system (5). In this proof, for any matrix-valued measure M , we denote M 0 the matrix M {0}. Let A[K] be the adjacency matrix of the block diagram directed graph, stripped off its strictly causal edges:

A[K] ij = 1 if [K 0 ] ij ̸ = 0 0 otherwise.
The graph-theoretic concept of loop corresponds to our definition of causality loop. Element (i, j) of the p-th power A[K] p of the adjacency matrix is the number of distinct paths with p edges between vertices j and i. If there is a loop of length p in the block diagram, there is also a loop whose length is an arbitrary multiple of p, hence the adjacency matrix is not nilpotent. Conversely, if the adjacency matrix is not nilpotent, there is a pair (i, j) such that A[K] n ij ̸ = 0; since any path between j and i goes through n + 1 vertices and the graph has only n distinct vertices, at least one of them is repeated: the path necessarily contains a loop.

Now, if A[K] p = 0 for some p ∈ N * then [K 0 ] p = 0. Indeed, since A[K] p ij = ∑ ℓ1,...,ℓp-1 A[K] iℓ1 . . . A[K] ℓp-1j
and all elements of the adjacency matrix are non-negative, if its p-th power is zero, there is for each term in the sum above at least one factor A[K] νµ which is zero and thus [K 0 ] p νµ = 0. Consequently, since

[K 0 ] p ij = ∑ ℓ1,...,ℓp-1 [K 0 ] iℓ1 . . . [K 0 ] ℓp-1j ,
the p-th power of K 0 is also zero. But for any integer p ≥ 2, we have

[K 0 ] p =   0 0 0 A 0 0 B 0 C 0 0 D 0   p =   0 0 0 B 0 [D 0 ] p-2 C 0 0 B 0 [D 0 ] p-1 [D 0 ] p-1 C 0 0 [D 0 ] p   .
Consequently, if the block diagram has no causality loop, A[K] is nilpotent and the matrix D 0 = D{0} is also nilpotent. ■

Note that the absence of causality loop does not necessarily provide explicitness: for example, the delay operator D of the DDAE system governed by y 1 (t) = y 2 (t) and y 2 (t) = y 1 (t -1) satisfies

D{0} = [ 0 1 0 0 ]
which is not zero. However, it is plain that its block-diagram representation has no causality loop: it contains only one subsystem which is not strictly causal and this subsystem connects variable y 1 to the other variable y 2 . This is sufficient to ensure its well-posedness:

Corollary. A DDAE system without causality loop is well-posed.

Proof. If DDAE system (5) has no causality loop, the matrix D{0} is nilpotent and [D{0}] m = 0. Consequently, the matrix I m -D{0} is invertible:

[I m -D{0}] -1 = I m + D{0} + • • • + [D{0}] m-1 .
The well-posedness theorem of the previous section provides the conclusion. ■

Integrator

Pure Delay

Distributed Delay

Figure 2: FSA closed-loop system block diagram. Each rectangular block characterizes an input-output system by its (function or measure) convolution kernel.

A triangle with symbol K represents a gain: a system whose output y(t) and input u(t) satisfy y(t) = Ku(t).

Block Diagram of the FSA System

In the block diagram representation of system (3), the auxiliary variable w satisfies w(t) = Ex(t) -F Gy(t), t > 0.

Since x(t) = e * w(t) + e * υ(t) is the only (system of) equation(s) with some variable w j in the right-hand side or some variable x i in the left hand-side, there is a unique outgoing edge of w j (with a nonzero label) which is also the unique incoming edge of the variable x j . This label is the function e, strictly causal, thus no causality loop may contain either of the components of the state variables w or x. Now the possible causality loops that could remain would contain only vertices which are components of the state variable y. But the edge between y j and y i is

H * ij where H * = (-e tE F Gdt)| [0,T ] .
Again, this kernel is a function and hence is strictly causal. Thus no such loop may exist and the initial-value problem is well-posed. This analysis may also be carried out graphically on figure 2. Once the integrator, the pure delay and the distributed delay -which are strictly causal subsystems -are removed from the diagram, it is plain that there is no loop and thus that the system has no causality loop.

Stability

Test in the Laplace Domain

There is a general and simple test in the Laplace domain for the exponential stability of a DDAE system. It can also be used to determine its growth bound, defined as the infimum of the σ + such that for any initial value, there is a κ > 0 such that the solution satisfies ∥(x(t), x t , y t )∥ X ≤ κe σ + t for any t > 0; exponential stability corresponds to a negative growth bound.

A few definitions are in order first: the Laplace transform LL of a compactly supported measure L is the function defined on C by

LL(s) = ∫ e -st dL(t).
The Laplace transform of L * satisfies LL * (s) = L(t → e st ). The characteristic matrix of the system is then defined as

∆(s) = [ sI n 0 0 I m ] -L [ A * B * C * D * ] (s) (16)
and its characteristic function as its determinant.

We may now state the criterion:

Theorem. The growth bound σ of a DDAE system without causality loop is:

σ = sup {ℜ(s) | s ∈ C, det ∆(s) = 0}. ( 17
)
The proof of this theorem is carried out in [START_REF] Boisgérault | Growth bound of delay-differential algebraic equations[END_REF] in the case of explicit systems. Here, we expose the proof in the general case.

Lemma. DDAE systems (5) and (10) have simultaneously singular characteristic matrices ∆(s) and ∆ † (s).

Proof. Let J = I m -D{0}, E = J -1 C and F = J -1 (D -D| {0} ); we have

LE * = J -1 LC * and LF * = J -1 (LD * -D{0})
and since

I m + J -1 D{0} = J -1 , I m -LF * = J -1 (I m -LD * ). Finally, ∆ † (s) = [ sI n 0 0 I m ] -L [ A * B * E * F * ] (s) = [ I n 0 0 J -1 ] ∆(s)
and the characteristic matrices are simultaneously singular. ■

The determinant and adjugate of the characteristic matrix ∆ † have a quasipolynomial structure:

det ∆ † (s) = n ∑ i=0 c i (s)s i , adj ∆ † (s) = n ∑ i=0 C i (s)s i (18)
where the c i (resp. C i ) are entire functions (resp. matrices of entire functions) bounded on any right-hand plane. It is plain that the leading coefficient of the characteristic function is given by c n (s) = det ∆ 0 (s) where ∆ 0 is the characteristic matrix of the system y(t) = F y t .

Lemma -Zero Clusters. Let H σ be the open right half-plane {s ∈ C | ℜ(s) > σ} and let Z η be the set of points whose distance to the zeros of det ∆ 0 is at most η.

For any σ ∈ R and ϵ > 0, there is a η > 0 such that:

i. any connected component Λ of the set Z η is bounded, ii. Λ is a subset of H σ-ϵ whenever Λ ∩ H σ ̸ = ∅, iii. | det ∆ 0 | has a positive lower bound on H σ-ϵ \ Z η .
The proof (from [START_REF] Boisgérault | Growth bound of delay-differential algebraic equations[END_REF]) is given in the appendix for the sake of completeness.

Lemma -Characteristic Function Zeros. Let σ ∈ R. If the function det ∆ 0 has an infinite number of zeros on H σ , the function det ∆ † has an infinite number of zeros on H σ-ϵ for any ϵ > 0.

Proof. Suppose that det ∆ 0 has an infinite number of zeros in H σ and let η > 0 be as in the zero clusters lemma. The zeros of det ∆ 0 are isolated, thus the collection of components Λ of Z η such that Λ ∩ H σ ̸ = ∅, which is locally finite by construction, is also infinite: for any compact set K, there is a Λ such that Λ ∩ K = ∅.

Since det ∆ † (s) has a quasi-polynomial structure with leading coefficient det ∆ 0 (s) s n and since det ∆ 0 (s) has a positive lower bound on

H σ-ϵ \ Z η , there is a compact set K such that | det ∆ † (s) -s n det ∆ 0 (s)| < |s n det ∆ 0 (s)| whenever s ∈ H σ-ϵ \ Z η and s ̸ ∈ K.
Rouché's theorem is applicable to any of the components Λ in the complement of K, each of which contains at least one zero of det ∆ 0 . Thus, det ∆ † has an infinite number of zeros in H σ-ϵ . ■

We can associate to the initial-value problem (5), or (10) which is equivalent, a one-parameter family of operators (exp(At)) t≥0 defined by

(x(t + ), x t , y t ) = exp(At)(ϕ, χ, ψ) for t ≥ 0, (ϕ, χ, ψ) ∈ X.
It is a strongly continous semigroup on the product space X; its infinitesimal generator A is defined by

A(ϕ, χ, ψ) = (Aχ + Bψ, χ, ψ)
on the domain

{(ϕ, χ, ψ) ∈ C n × W 1,2 ([-r, 0], C n+m ) | χ(0) = ϕ, ψ(0) = Eχ + F ψ}.
The resolvent operator (sI -A) -1 exists if and only if ∆ † (s) is non-singular and moreover, for any real number σ there are constants κ σ and λ σ such that

∥(sI -A) -1 ∥ ≤ κ σ ∥∆ † (s) -1 ∥ + λ σ (19)
if ℜs ≥ σ and det ∆ † (s) ̸ = 0 (see [START_REF] Salamon | Control and observation of neutral systems[END_REF]).

We may now prove the main result:

Proof. Let s(A) be the spectral bound of A. We show that for any σ > s(A), ∥∆ † (s) -1 ∥ is bounded on H σ . Thus, by inequality (19), ∥(sI -A) -1 ∥ is similarly bounded and the Gearhart-Prüss theorem proves the result.

The quasi-polynomial structure of the adjugate matrix provides for some κ ≥ 0

∥adj ∆ † (s)∥ ≤ κ(1 + |s| n ) on H σ .
∆ † that | det ∆ † (s)| ≥ κ ′ 2 (1 + |s| n )
on H σ except on some compact set K; by continuity of det ∆ † , this estimate still holds on all of H σ with a possibly smaller κ ′ . Finally, for any s

∈ H σ , ∥∆ † (s) -1 ∥ = ∥adj ∆ † (s)∥/| det ∆ † (s)| ≤ 2κ/κ ′ . ■

Stability of the FSA System

Since the kernels of FSA system (3) are given by ( 11) and ( 12), its characteristic matrix satisfies

∆(s) = [ sI n -E F G -e (sIn-E)T I n + P (s)F G ] where P (s) is the analytic extension to C of the meromorphic function s → [sI n -E] -1 (I n -e (sIn-E)T ).
A straightforward computation provides

∆(s) [ I n 0 I n I n ] = [ sI n -E + F G F G P (s)(sI n -E + F G) I n + P (s)F G ]
and the right-hand side of this equation can be factored into

[ I n 0 P (s) I n ] [ I n F G 0 I n ] [ sI n -E + F G 0 0 I n ] .
Thus, the characteristic function satisfies

det ∆(s) = det(sI n -E + F G)
and it roots are the eigenvalues of E -F G. When the delay-free system ẋ(t) = Ex(t) + F u(t) is controllable, finite-dimensional control theory provides a gain matrix G that assigns these eigenvalues to n arbitrary locations in the complex plane, and thus we may achieve a "finite spectrum assignment" of the closed-loop system. In particular, it's possible to select a gain matrix G which exponentially stabilizes system (3). We may now proceed to the proof:

Proof (Zero Clusters Lemma). The function z → det ∆ 0 (iz) meets the assumptions of theorem VIII in [START_REF] Levinson | Gap and density theorems[END_REF]) and is of exponential type mr. Thus, the number of distinct zeros N (ρ) of det ∆ 0 whose modulus is less than ρ is such that lim sup ρ→+∞ N (ρ)/ρ ≤ 2mr/π. If the component of Z η that contains a zero s is unbounded, there are at least n + 1 zeros in the closed disk centered at s of radius 2ηn and thus lim sup ρ→+∞ N (ρ)/ρ ≥ 1/2η. Consequently, if η < π/4mr, every connected component of Z η is bounded.

The proofs of statements ii. and iii. use a similar argument. In each case we consider a sequence s n of complex numbers of bounded real part and the functions f n (s) = det ∆ 0 (s + iℑs n ). Since inequality (20) holds, these functions are locally uniformly bounded, hence there is a subsequence of s n which converges to some real number x and, by Montel's theorem, a corresponding subsequence of f n that converges locally uniformly to an entire function f ∞ . By (21), f ∞ is not identically zero, thus by Hurwitz's theorem, if m is the multiplicity of x if f ∞ (x) = 0, or 0 otherwise, for any sufficiently small δ > 0, det ∆ 0 has exactly m zeros in the open disk B(s n , δ) for an infinite number of values of n.

To prove ii., we assume the existence of a sequence Λ n of connected components of Z ηn where η n → 0 and such that Λ n ∩ H σ ̸ = ∅ but Λ n ̸ ⊂ H σ-ε . The Λ n are eventually bounded and there is a tuple of zeros in Λ n with a first element such that ℜs + η n > σ, a last element such that ℜsη n ≤ σϵ and a distance between consecutive points that is at most 2η n . Let s n to be the first element of this tuple; the real part of this sequence is bounded by (21). But for any δ > 0, the number of zeros in B(s n , δ) converges to +∞, a contradiction with the previous paragraph.

To prove iii., we assume the existence of a sequence s n in H σ-ϵ \ Z η such that det ∆ 0 (s n ) → 0 when n → +∞; the real part of this sequence is bounded and

Figure 1 :

 1 Figure 1: Block diagram of a simple delay system.

  measure µ. Let Σ m be the set of permutations of {1, ..., m} anddet * M = ∑ σ∈Σm sgn(σ)M 1,σ(1) * . . . * M m,σ(m) .As ∆ 0 (s) = I m -LF * (s), det ∆ 0 = Lµ where µ = det * (δ 0 I m -F * ). The complex measure µ is a sum of convolution products of m complex measures supported on [0, r], hence it is supported on [0, mr]. Consequently, det ∆ 0 is an entire function that satisfies the inequality| det ∆ 0 (s)| ≤ |µ|([0, mr]) max(1, exp(-ℜ(s)mr)).(20)Since F {0} = 0, we also have µ{0} = 1, which yields lim ℜs→+∞ det ∆ 0 (s) = 1.(21)

  Since the resolvent operator is defined on H σ-ϵ for any ϵ > 0 such that s(A) < σϵ, det ∆ † has no zero in H σ-ϵ . Thus, by the characteristic function zeros lemma, det ∆ 0 has at most a finite number of zeros on H σ-ϵ/2 and thus on H σ . By the zero clusters lemma, on H σ and away from these zeros, | det ∆ 0 | has a positive lower bound κ ′ . It follows from the quasipolynomial structure of det

Appendix

We provide in this appendix a proof of the zero clusters lemma. First we derive some properties of det ∆ 0 by expressing it as the Laplace transform of a complex we may select s n such that ℜs n → x. Now, f ∞ (x) = lim n→+∞ f n (ℜs n ) = 0, thus for some δ < η and for some value of n, there is at least one zero of det ∆ 0 in B(s n , δ), which is a contradiction. ■


P Gaillard
email: pierre.gaillard@u-bourgogne.fr

From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N -2 parameters, the

Keywords: numbers : 33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd

From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N

-2 parameters

Introduction

We consider the Kadomtsev-Petviashvili equation (KP) which can be written in the form (4u t -6uu x + u xxx ) x -3u yy = 0, (1) 1 subscripts x, y and t denoting partial derivatives. Kadomtsev and Petviashvili [1] first proposed that equation in 1970. That equation is considered as a model, for example, for surface and internal water waves [START_REF] Ablowitz | On the evolution of packets of water waves[END_REF], and in nonlinear optics [START_REF] Pelinovsky | Self-focusing of plane dark solitons in nonlinear defocusing media[END_REF]. In 1974 Dryuma showed how the KP equation could be written in Lax form [START_REF] Dryuma | On analytical solutions of the two-dimensional Kortewegde Vries equation[END_REF]. The inverse scattering transform (IST)for the KPI equation is due to Manakov [START_REF] Manakov | The inverse scattering transform for the time-dependent Schrodinger equation and Kadomtsev-Petviashvili equation[END_REF].

The first rational solutions were found in 1977 by Manakov,Zakharov,Bordag and Matveev [6].

From the end of the seventies, a lot of methods have been carried out to solve that equation. Dubrovin constructed for the first time in 1981 [START_REF]Dubrovin Theta functions and non-linear equations[END_REF] the solutions to KPI given in terms of Riemann theta functions in the frame of algebraic geometry.

Various researches were conducted and more general rational solutions of the KPI equation were obtained. We can mention the following studies of Krichever in 1978 [START_REF] Krichever | Rational solutions of the Kadomtcev-Petviashvili equation and integrable systems of n particules on a line[END_REF][START_REF] Krichever | Holomorphic bundles over Riemann surfaces and the KPI equation[END_REF], Satsuma and Ablowitz in 1979 [START_REF] Satsuma | Two-dimensional lumps in nonlinear dispersive systems[END_REF], Matveev in 1979 [11], Freeman and Nimmo in 1983 [START_REF] Freeman | Nimmo Rational solutions of the KdV equation in wronskian form[END_REF][START_REF] Freeman | The use of Bäcklund transformations in obtaining N-soliton solutions in wronskian form[END_REF], Pelinovsky and Stepanyants in 1993 [START_REF] Pelinovsky | New multisolitons of the Kadomtsev-Petviashvili equation[END_REF], Pelinovsky in 1994 [START_REF] Pelinovsky | Rational solutions of the Kadomtsev-Petviashvili hierarchy and the dynamics of their poles. I. New form of a general rational solution[END_REF], Ablowitz and Villarroel [START_REF] Ablowitz | Solutions to the time dependent schrödinger and the Kadomtsev-Petviashvili equations[END_REF][START_REF] Villarroel | On the discrete spectrum of the nonstationary Schrdinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation[END_REF] in 1997-1999, Biondini andKodama [18, 19, 20] in 2003-2007. In particular in 1999, the inverse scattering method has been applied by Prinari to obtain solutions to the KPI and KPII equations [START_REF] Prinari | Inverse scattering theory for the KP equations[END_REF].

In the following, we recall the results of the author about the representations of solutions to the KPI equation. We have expressed the solutions in terms of Fredholm determinants of order 2N depending on 2N -1 parameters. We have also given another representation in terms of wronskians of order 2N with 2N -1 parameters. These representations allow to obtain an infinite hierarchy of solutions to the KPI equation, depending on 2N -1 real parameters . We have used these results to build rational solutions to the KPI equation, making a parameter to 0 tend to 0. Here we construct rational solutions of order N depending on 2N -2 parameters without the presence of a limit. That gives a new method to build rational solutions. We prove that these families depending on 2N -2 parameters for the N -th order can be written as a ratio of two polynomials of x, y and t of degree 2N (N + 1). That provides an effective method to build an infinite hierarchy of rational solutions of order N depending on 2N -2 real parameters. We present here only the explicit rational solutions of order 4, depending on 6 real parameters, and the representations of their modulus in the plane of the coordinates (x, y) according to the real parameters a We need to define some notations. First one defines real numbers λ j such that -1 < λ ν < 1, ν = 1, . . . , 2N depending on a parameter ǫ which will be intended to tend towards 0; they can be written as

1 , b 1 , a 2 , b 2 , a 3 , b 3 and time t.
λ j = 1 -2ǫ 2 j 2 , λ N +j = -λ j , 1 ≤ j ≤ N, (2)
The terms κ ν , δ ν , γ ν , τ ν and x r,ν are functions of λ ν , 1 ≤ ν ≤ 2N ; they are defined by the formulas :

κ j = 2 1 -λ 2 j , δ j = κ j λ j , γ j = 1-λj 1+λj ,; x r,j = (r -1) ln γj -i γj +i , r = 1, 3, τ j = -12iλ 2 j 1 -λ 2 j -4i(1 -λ 2 j ) 1 -λ 2 j , κ N +j = κ j , δ N +j = -δ j , γ N +j = γ -1 j , x r,N +j = -x r,j , , τ N +j = τ j j = 1, . . . , N.
(3) e ν 1 ≤ ν ≤ 2N are defined in the following way :

e j = 2i 1/2 M -1 k=1 a k (je) 2 k+1 -i 1/2 M -1 k=1 b K (je) 2 k+1 , e N +j = 2i 1/2 M -1 k=1 a k (je) 2 k+1 + i 1/2 M -1 k=1 b k (je) 2 k+1 , 1 ≤ j ≤ N, a k , b k ∈ R, 1 ≤ k ≤ N -1. (4)
ǫ ν , 1 ≤ ν ≤ 2N are real numbers defined by :

ǫ j = 1, ǫ N +j = 0 1 ≤ j ≤ N. (5)
Let I be the unit matrix and D r = (d jk ) 1≤j,k≤2N the matrix defined by :

d νµ = (-1) ǫν η =µ γ η + γ ν γ η -γ µ exp(iκ ν x -2δ ν y + τ ν t + x r,ν + e ν ). (6)
Then we recall the following result1 :

Theorem 2.1 The function v defined by v(x, y, t) = -2 |n(x, y, t)| 2 d(x, y, t) 2 (7)
where

n(x, y, t) = det(I + D 3 (x, y, t)), (8)
d(x, y, t) = det(I + D 1 (x, y, t)), (9)
and D r = (d jk ) 1≤j,k≤2N the matrix

d νµ = (-1) ǫν η =µ γ η + γ ν γ η -γ µ exp(iκ ν x -2δ ν y + τ ν t + x r,ν + e ν ). ( 10
)
is a solution to the KPI equation ( 1), dependent on 2N -

1 parameters a k , b k , 1 ≤ k ≤ N -1 and ǫ.
We recall a second result on the solutions to KPI equation obtained recently by the author in terms of wronskians. We need to define the following notations : [START_REF] Matveev | Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation depending on functional parameters[END_REF] with the arguments

φ r,ν = sin Θ r,ν , 1 ≤ ν ≤ N, φ r,ν = cos Θ r,ν , N + 1 ≤ ν ≤ 2N, r = 1, 3,
Θ r,ν = κν x 2 + iδ ν y -i xr,ν 2 -i τν 2 t + γ ν w -i eν 2 , 1 ≤ ν ≤ 2N. ( 12
)
We denote W r (w) the wronskian of the functions φ r,1 , . . . , φ r,2N defined by

W r (w) = det[(∂ µ-1 w φ r,ν ) ν, µ∈[1,...,2N ] ]. (13)
We consider the matrix D r = (d νµ ) ν, µ∈ [1,...,2N ] defined in [START_REF] Satsuma | Two-dimensional lumps in nonlinear dispersive systems[END_REF]. Then we have the following statement2

φ r,ν (w) = sin( κν x 2 + iδ ν y -i xr,ν 2 -i τν 2 t + γ ν w -i eν 2 ), 1 ≤ ν ≤ N, φ r,ν (w) = cos( κν x 2 + iδ ν y -i xr,ν 2 -i τν 2 t + γ ν w -i eν 2 ), N + 1 ≤ ν ≤ 2N, r = 1, 3,
κ ν , δ ν , x r,ν , γ ν , e ν being defined in(3), ( 2) and ( 4).

From those two preceding results, we construct rational solutions to the KPI equation as a quotient of two determinants. We use the following notations :

X ν = κ ν x 2 + iδ ν y -i x 3,ν 2 -i τ ν 2 t -i e ν 2 , Y ν = κ ν x 2 + iδ ν y -i x 1,ν 2 -i τ ν 2 t -i e ν 2 ,
for 1 ≤ ν ≤ 2N , with κ ν , δ ν , x r,ν defined in [START_REF] Pelinovsky | Self-focusing of plane dark solitons in nonlinear defocusing media[END_REF] and parameters e ν defined by [START_REF] Dryuma | On analytical solutions of the two-dimensional Kortewegde Vries equation[END_REF]. We define the following functions :

ϕ 4j+1,k = γ 4j-1 k sin X k , ϕ 4j+2,k = γ 4j k cos X k , ϕ 4j+3,k = -γ 4j+1 k sin X k , ϕ 4j+4,k = -γ 4j+2 k cos X k , (14)
for 1 ≤ k ≤ N , and

ϕ 4j+1,N +k = γ 2N -4j-2 k cos X N +k , ϕ 4j+2,N +k = -γ 2N -4j-3 k sin X N +k , ϕ 4j+3,N +k = -γ 2N -4j-4 k cos X N +k , ϕ 4j+4,N +k = γ 2N -4j-5 k sin X N +k , (15)
for 1 ≤ k ≤ N . We define the functions ψ j,k for 1 ≤ j ≤ 2N , 1 ≤ k ≤ 2N in the same way, the term X k is only replaced by Y k .

ψ 4j+1,k = γ 4j-1 k sin Y k , ψ 4j+2,k = γ 4j k cos Y k , ψ 4j+3,k = -γ 4j+1 k sin Y k , ψ 4j+4,k = -γ 4j+2 k cos Y k , (16)
for 1 ≤ k ≤ N , and

ψ 4j+1,N +k = γ 2N -4j-2 k cos Y N +k , ψ 4j+2,N +k = -γ 2N -4j-3 k sin Y N +k , ψ 4j+3,N +k = -γ 2N -4j-4 k cos Y N +k , ψ 4j+4,N +k = γ 2N -4j-5 k sin Y N +k , (17)
for 1 ≤ k ≤ N .

The following ratio q(x, t)

:= W 3 (0) W 1 (0)
can be written as

q(x, t) = ∆ 3 ∆ 1 = det(ϕ j,k ) j, k∈[1,2N ] det(ψ j,k ) j, k∈[1,2N ] . ( 18
)
The terms λ j depending on ǫ are defined by λ j = 1 -2jǫ 2 . All the functions ϕ j,k and ψ j,k and their derivatives depend on ǫ. They can all be prolonged by continuity when ǫ = 0. We use the following expansions

ϕ j,k (x, y, t, ǫ) = N -1 l=0 1 (2l)! ϕ j,1 [l]k 2l ǫ 2l + O(ǫ 2N ), ϕ j,1 [l] = ∂ 2l ϕ j,1 ∂ǫ 2l (x, y, t, 0), ϕ j,1 [0] = ϕ j,1 (x, y, t, 0), 1 ≤ j ≤ 2N, 1 ≤ k ≤ N, 1 ≤ l ≤ N -1, ϕ j,N +k (x, y, t, ǫ) = N -1 l=0 1 (2l)! ϕ j,N +1 [l]k 2l ǫ 2l +O(ǫ 2N ), ϕ j,N +1 [l] = ∂ 2l ϕ j,N +1 ∂ǫ 2l (x, y, t, 0), ϕ j,N +1 [0] = ϕ j,N +1 (x, y, t, 0), 1 ≤ j ≤ 2N, 1 ≤ k ≤ N, 1 ≤ l ≤ N -1.
We have the same expansions for the functions ψ j,k .

ψ j,k (x, y, t, ǫ) = N -1 l=0 1 (2l)! ψ j,1 [l]k 2l ǫ 2l + O(ǫ 2N ), ψ j,1 [l] = ∂ 2l ψ j,1 ∂ǫ 2l (x, y, t, 0), ψ j,1 [0] = ψ j,1 (x, y, t, 0), 1 ≤ j ≤ 2N, 1 ≤ k ≤ N, 1 ≤ l ≤ N -1, ψ j,N +k (x, t, ǫ) = N -1 l=0 1 (2l)! ψ j,N +1 [l]k 2l ǫ 2l +O(ǫ 2N ), ψ j,N +1 [l] = ∂ 2l ψ j,N +1 ∂ǫ 2l (x, y, t, 0), ψ j,N +1 [0] = ψ j,N +1 (x, t, 0), 1 ≤ j ≤ 2N, 1 ≤ k ≤ N, N + 1 ≤ k ≤ 2N..
Then we get the following result :

Theorem 2.3 The function v defined by v(x, y, t) = -2 | det((n jk) j,k∈[1,2N ] )| 2 det((d jk) j,k∈[1,2N ] ) 2 (19)
is a rational solution to the KPI equation ( 1).

(4u t -6uu x + u xxx ) x -3u yy = 0,
where

n j1 = ϕ j,1 (x, y, t, 0), 1 ≤ j ≤ 2N n jk = ∂ 2k-2 ϕj,1 ∂ǫ 2k-2 (x, y, t, 0), n jN +1 = ϕ j,N +1 (x, y, t, 0), 1 ≤ j ≤ 2N n jN +k = ∂ 2k-2 ϕj,N+1 ∂ǫ 2k-2
(x, y, t, 0),

d j1 = ψ j,1 (x, y, t, 0), 1 ≤ j ≤ 2N d jk = ∂ 2k-2 ψj,1 ∂ǫ 2k-2 (x, y, t, 0), d jN +1 = ψ j,N +1 (x, y, t, 0), 1 ≤ j ≤ 2N d jN +k = ∂ 2k-2 ψj,N+1 ∂ǫ 2k-2 (x, y, t, 0), 2 ≤ k ≤ N, 1 ≤ j ≤ 2N (20)
The functions ϕ and ψ are defined in ( 14),( 15), ( 16), [START_REF] Villarroel | On the discrete spectrum of the nonstationary Schrdinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation[END_REF].

Remark 2.1 In general solutions of KPI have the form

u(X, Y, T ) = -2 ∂ 2 ∂X 2 ln F (X, Y, T ) (21)
for some function F (X, Y, T ), due to the formulation of solutions through the inverse scattering method.

It is clear that the formulation given in this article is different from those given by ( 21). If it were the case, then necessarily, F should be of degree 20 in X,and an elementary calculation proves that the expression of u would be the quotient of a polynomial of degree 38 in X on a polynomial of degree 40 in X in the case of order 4 of the article, what is not the case.

Proof : In each column k (and N +k) of the determinants appearing in q(x, t), we successively eliminate the powers of ǫ strictly inferior to 2(k -1); then each common term in the numerator and denominator is factorized and simplified; in the end, we take the limit when ǫ goes to 0. First, the components j of the columns 1 and N +1 are respectively equal by definition to

ϕ j1 [0] + 0(ǫ) for C 1 , ϕ jN +1 [0] + 0(ǫ) for C N +1 of ∆ 3 , and ψ j1 [0] + 0(ǫ) for C ′ 1 , ψ jN +1 [0] + 0(ǫ) for C ′ N +1 of ∆ 1 .
At the the first step of the reduction, we replace the columns C k by C k -C 1 and C N +k by C N +k -C N +1 for 2 ≤ k ≤ N , for ∆ 3 ; the same changes for ∆ 1 are made. Each component j of the column C k of ∆ 3 can be rewritten as

N -1 l=1 1 (2l)! ϕ j,1 [l](k 2l -1)ǫ 2l and the column C N +k replaced by N -1 l=1 1 (2l)! ϕ j,N +1 [l](k 2l - 1)ǫ 2l for 2 ≤ k ≤ N .
For ∆ 1 , we make the same reductions, each component j of the column C ′ k can be rewritten as

N -1 l=1 1 (2l)! ψ j,1 [l](k 2l -1)ǫ 2l and the column C ′ N +k replaced by N -1 l=1 1 (2l)! ψ j,N +1 [l](k 2l -1)ǫ 2l for 2 ≤ k ≤ N . The term k 2 -1
2 ǫ 2 for 2 ≤ k ≤ N can be factorized in ∆ 3 and ∆ 1 in each column k and N + k , and so those common terms can be simplified in the numerator and denominator. If we restrict the developments at order 1 in columns 2 and N +2, we respectively get ϕ j1 [START_REF] Kadomtsev | On the stability of solitary waves in weakly dispersing media[END_REF]

+ 0(ǫ) for the component j of C 2 , ϕ jN +1 [1] + 0(ǫ) for the component j of C N +2 of ∆ 3 , and ψ j1 [1] + 0(ǫ) for the component j of C ′ 2 , ψ jN +1 [1] + 0(ǫ) for the component j of C ′ N +2 of ∆ 1 . We can continue this algorithm up to the columns C N , C 2N of ∆ 3 and C ′ N , C ′ 2N of ∆ 1 .
Then we take the limit when ǫ tends to 0, and q(x, y, t) can be replaced by Q(x, y, t) defined by :

Q(x, y, t) := ϕ 1,1 [0] . . . ϕ 1,1 [N -1] ϕ 1,N +1 [0] . . . ϕ 1,N +1 [N -1] ϕ 2,1 [0] . . . ϕ 2,1 [N -1] ϕ 2,N +1 [0] . . . ϕ 2,N +1 [N -1] . . . . . . . . . . . . . . . . . . ϕ 2N,1 [0] . . . ϕ 2N,1 [N -1] ϕ 2N,N +1 [0] . . . ϕ 2N,N +1 [N -1] 2      ψ 1,1 [0] . . . ψ 1,1 [N -1] ψ 1,N +1 [0] . . . ψ 1,N +1 [N -1] ψ 2,1 [0] . . . ψ 2,1 [N -1] ψ 2,N +1 [0] . . . ψ 2,N +1 [N -1] . . . . . . . . . . . . . . . . . . ψ 2N,1 [0] . . . ψ 2N,1 [N -1] ψ 2N,N +1 [0] . . . ψ 2N,N +1 [N -1]      2 (22)
So the solution to the KPI equation takes the form :

v(x, y, t) = -2Q(x, y, t)
and we get the result. 2

2.2

The structure of the families of rational solutions of order N depending on 2N -2 parameters

Here, we give a theorem which states, in this representation, the structure of the rational solutions to the KPI equation. In this section we use the notations defined in the previous sections. The functions ϕ and ψ are defined in ( 14), ( 15), ( 16), [START_REF] Villarroel | On the discrete spectrum of the nonstationary Schrdinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation[END_REF].

Theorem 2.4 The function v defined by

v(x, y, t) = -2 det((n jk) j,k∈[1,2N ] ) 2 (det((d jk) j,k∈[1,2N ] )) 2 (23)
is a rational solution to the KPI equation ( 1) quotient of two polynomials n(x, y, t) and d(x, y, t) depending on 2N -2 real parameters a j and b j , 1 ≤ j ≤ N -1.

n and d are polynomials of degrees 2N (N + 1) in x, y and t.

The terms n jk and d jk are defined by ( 20) and the functions ϕ and ψ are defined in ( 14), ( 15), ( 16), [START_REF] Villarroel | On the discrete spectrum of the nonstationary Schrdinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation[END_REF].

Proof : From the previous result ( 22), we have to analyze the functions ϕ k,1 , ψ k,1 and ϕ k,N +1 , ψ k,N +1 . The functions ϕ k,j and ψ k,j differ only by the term of the argument x 3,k , so only the study of functions ϕ k,j will be carried out. Then we study functions ψ k,j and it can be easily deduced from the analysis of ϕ k,j .

We study the expansions of those functions in ǫ. We denote (l kj ) k,j∈ [1,2N ] the matrix defined by

l kj = ∂ 2j-2 ∂ǫ 2j-2 ϕ k1 , l k,j+N = ∂ 2j-2 ∂ǫ 2j-2 ϕ k,1+N , 1 ≤ j ≤ N, 1 ≤ k ≤ 2N, ∂ 0
∂x 0 ϕ meaning ϕ. Each coefficient of the matrix (l kj ) k,j∈ [1,2N ] must be evaluated, the power of x, y and t in the coefficient of ǫ 2(m-1) for the column m ∈ [1, 2N ]. We remark that with those notations, the matrix (l kj ) k,j∈ [1,2N ] evaluated in ǫ = 0 is exactly (n kj ) k,j∈ [1,2N ] defined in [START_REF] Biondini | Line Soliton Interactions of the Kadomtsev-Petviashvili Equation[END_REF]. There are four cases to study depending on the parity of k.

1. Case l k1 for k odd, k = 2s + 1.

l k1 = (-1) s sin(2ǫ(1-ǫ 2 ) 1 2 x+4iǫ(1-ǫ 2 ) 1 2 (1-2ǫ 2 )y-(12ǫ(1-ǫ 2 ) 1 2 (1-2ǫ 2 ) 2 +16ǫ 3 (1-ǫ 2 ) 3 2 )t -i ln 1 + iǫ(1 -ǫ 2 ) -1 2 1 -iǫ(1 -ǫ 2 ) -1 2 -e 1 ) × ǫ k-2 (1 -ǫ 2 ) -k-2 2 = (-1) s sin ǫ( p l=0 c 2l ǫ 2l x+2i p l=0 c 2l ǫ 2l (1-2ǫ 2 )y+ p l=0 h 2l ǫ 2l t+2 p l=0 (-1) l ǫ 2l (1 -ǫ 2 ) -2l+1 2 (2l + 1) - N -1 l=1 ãl ǫ 2l + i N -1 l=1 bl ǫ 2l + O(ǫ p+1 )) × ǫ k-2 ( r l=1 g 2l ǫ 2l + O(ǫ r+1 )) = (-1) s sin ǫ( p l=0 (c 2l x+d 2l y+h 2l t+f 2l +O(ǫ p+1 ))ǫ 2l )×ǫ k-2 ( r l=1 g 2l ǫ 2l +O(ǫ r+1 )) = q l=0 (-1) l+s ǫ 2l (2l + 1)! ( p n=0 (c 2n x+d 2n y+h 2l t+f 2n +O(ǫ p+1 ))ǫ 2n ) 2l+1 ×ǫ k-1 ( r l=1 g 2l ǫ 2l +O(ǫ r+1 )) = q l=0
(-1) l+s ǫ 2l (2l + 1)! (

p n=0 P n (x, y, t)ǫ 2n ) 2l+1 × ǫ k-1 r l=1 g 2l ǫ 2l + O(ǫ t )
where P n (x, y, t) is a polynomial of order 1 in x, y and t.

l k,1 = q l=0 ǫ 2l α0+...+αp=2l+1
β α0,...,αp P 0 (x, y, t) α0 . . . P p (x, y, t) αp ǫ 2(α1+2α2+pαp) ×ǫ 2s r l=1

g 2l ǫ 2l +O(ǫ t ) = q l=0 ǫ 2l α0+...+αp=2l+1 Q α0,...,αp (x, y, t)ǫ 2(α1+2α2+pαp) × ǫ 2s r l=1 g 2l ǫ 2l + O(ǫ t ),
where Q α0,...,αp (x, y, t) is a polynomial of order 2l + 1 in x, y and t.

Terms in ǫ 0 are obtained for l = 0 in the two summations with α 0 = 1.

For the column m, we search the terms in ǫ 2m-2 with the maximal power in x, y and t. It is obtained for 2l + k -1 = 2m -2, which gives l = ms -1.

We get the following result

Proposition 2.1 deg(n 2s+1,m ) = 2(m -s) -1 for s ≤ m -1, n 2s+1,m = 0 for s ≥ m. (24)
2. Case l k1 for k even, k = 2s.

l k1 = (-1) s+1 cos(2ǫ(1-ǫ 2 ) 1 2 x+4iǫ(1-ǫ 2 ) 1 2 (1-2ǫ 2 )y-(12ǫ(1-ǫ 2 ) 1 2 (1-2ǫ 2 ) 2 +16ǫ 3 (1-ǫ 2 ) 3 2 )t -i ln 1 + iǫ(1 -ǫ 2 ) -1 2 1 -iǫ(1 -ǫ 2 ) -1 2 -e 1 ) × ǫ k-2 (1 -ǫ 2 ) -k-2 2 = (-1) s+1 cos ǫ( p l=0 c 2l ǫ 2l x+2i p l=0 c 2l ǫ 2l (1-2ǫ 2 )y+ p l=0 h 2l ǫ 2l t+2 p l=0 (-1) l ǫ 2l (1 -ǫ 2 ) -2l+1 2 (2l + 1) - N -1 l=1 ãl ǫ 2l + i N -1 l=1 bl ǫ 2l + O(ǫ p+1 )) × ǫ k-2 ( r l=1 g 2l ǫ 2l + O(ǫ r+1 )) = (-1) s+1 cos ǫ( p l=0 (c 2l x+d 2l y+h 2y t+f 2l +O(ǫ p+1 ))ǫ 2l )×ǫ k-2 ( r l=1 g 2l ǫ 2l +O(ǫ r+1 )) = q l=0 (-1) l+d+1 ǫ 2l (2l)! ( p n=0 (c 2n x+d 2n y+h 2l t+f 2n +O(ǫ p+1 ))ǫ 2n ) 2l ×ǫ k-2 ( r l=1 g 2l ǫ 2l +O(ǫ r+1 )) = q l=0 (-1) l+s+1 ǫ 2l (2l)! ( p n=0 P n (x, y, t)ǫ 2n ) 2l × ǫ 2s-2 r l=1 g 2l ǫ 2l + O(ǫ t )
where P n (x, y, t) is a polynomial of order 1 in x, y and t.

l k,1 = q l=0 ǫ 2l α0+...+αp=2l
β α0,...,αp P 0 (x, y, t) α0 . . . P p (x, y, t) αp ǫ 2(α1+2α2+pαp) ×ǫ 2s-2 r l=1

g 2l ǫ 2l +O(ǫ t ) = q l=0 ǫ 2l α0+...+αp=2l Q α0,...,αp (x, y, t)ǫ 2(α1+2α2+pαp) × ǫ 2s-2 r l=1 g 2l ǫ 2l + O(ǫ t ),
where Q α0,...,αp (x, y, t) is a polynomial of order 2l in x, y and t.

Terms in ǫ 0 are obtained for l = 0 in the two summations with α 0 = 1.

For the column m, we search the terms in ǫ 2m-2 with the maximal power in x and t. It is obtained for 2l + k -2 = 2m -2, which gives l = ms. So we have :

Proposition 2.2 deg(n 2s,m ) = 2(m -s) for s ≤ m, n 2s,m = 0 for s > m. ( 25
) 3. Case l k M 2 +1 for k odd, k = 2s + 1. l k M 2 +1 = (-1) s cos(2ǫ(1-ǫ 2 ) 1 2 x-4iǫ(1-ǫ 2 ) 1 2 (1-2ǫ 2 )y+i ln 1 + iǫ(1 -ǫ 2 ) -1 2 1 -iǫ(1 -ǫ 2 ) -1 2 -e M 2 +1 -(12ǫ(1 -ǫ 2 ) 1 2 (1 -2ǫ 2 ) 2 + 16ǫ 3 (1 -ǫ 2 ) 3 2 )t) × ǫ M -k-1 (1 -ǫ 2 ) -M -k-1 2 = (-1) s (cos ǫ( p l=0 c 2l ǫ 2l x-2i p l=0 c 2l ǫ 2l (1-2ǫ 2 )y+ p l=0 h 2l ǫ 2l t-2 p l=0 (-1) l ǫ 2l (1 -ǫ 2 ) -2l+1 2 (2l + 1) - N -1 l=1 ãl ǫ 2l + i N -1 l=1 bl ǫ 2l + O(ǫ p+1 )) × ǫ M -k-1 ( r l=1 g 2l ǫ 2l + O(ǫ r+1 )) = (-1) s (cos ǫ( p l=0 (c 2l x+d 2l y+h 2l t+f 2l )ǫ 2l +O(ǫ p+1 ))×ǫ M -k-1 ( r l=1 g 2l ǫ 2l +O(ǫ r+1 )) = q l=0 (-1) l+s ǫ 2l (2l)! ( p n=0 (c 2n x+d 2n y+h 2l t+f 2n +O(ǫ p+1 ))ǫ 2n ) 2l ×ǫ M -k-1 ( r l=1 g 2l ǫ 2l +O(ǫ r+1 )) = q l=0 (-1) l+s ǫ 2l (2l)! ( p n=0 P n (x, y, t)ǫ 2n + O(ǫ p+1 )) 2l × ǫ M -2s-2 ( r l=1 g 2l ǫ 2l + O(ǫ r+1 ))
where P n (x, y, t) is a polynomial of order 1 in x and t.

l k, M 2 +1 = q l=0 ǫ 2l α0+...+αp=2l β α0,...,αp P 0 (x, y, t) α0 . . . P p (x, y, t) αp ǫ 2(α1+2α2+pαp) × ǫ M -2s-2 r l=1 g 2l ǫ 2l + O(ǫ t ) = q l=0 ǫ 2l α0+...+αp=2l Q α0,...,αp (x, y, t)ǫ 2(α1+2α2+pαp) ×ǫ M -2s-2 r l=1 g 2l ǫ 2l +O(ǫ t ),
where Q α0,...,αp (x, y, t) is a polynomial of order 2l in x, y and t.

The terms in ǫ 0 (column M 2 + 1) are obtained for l = 0 in the two summations with α 0 = 1. For the column M 2 +m, we search the terms in ǫ 2m-2 with the maximal power in x and t. It is obtained for 2l + 2(Ns -1) = 2m -2, which gives l = m + s -N . So we obtain the following result

Proposition 2.3 deg(n 2s+1,m+ M 2 ) = 2m + 2s -M for s ≥ M 2 -m, n 2s+1,m = 0 for s < M 2 -m. (26) 4. Case l k,1+ M 2 for k even, k = 2s. l k M 2 +1 = (-1) s sin(2ǫ(1-ǫ 2 ) 1 2 x-4iǫ(1-ǫ 2 ) 1 2 (1-2ǫ 2 )y+i ln 1 + iǫ(1 -ǫ 2 ) -1 2 1 -iǫ(1 -ǫ 2 ) -1 2 -e M 2 +1 -(12ǫ(1 -ǫ 2 ) 1 2 (1 -2ǫ 2 ) 2 + 16ǫ 3 (1 -ǫ 2 ) 3 2 )t) × ǫ M -k-1 (1 -ǫ 2 ) -M -k-1 2 = (-1) s sin ǫ( p l=0 c 2l ǫ 2l x-2i p l=0 c 2l ǫ 2l (1-2ǫ 2 )y+ p l=0 h 2l t-2 p l=0 (-1) l ǫ 2l (1 -ǫ 2 ) -2l+1 2 (2l + 1) - N -1 l=1 ãl ǫ 2l + i N -1 l=1 bl ǫ 2l + O(ǫ p+1 )) × ǫ M -k-1 ( r l=1 g 2l ǫ 2l + O(ǫ r+1 )) = (-1) s sin ǫ( p l=0 (c 2l x+d 2l y+h 2l t+f 2l )ǫ 2l +O(ǫ p+1 ))×ǫ M -k-1 ( r l=1 g 2l ǫ 2l +O(ǫ r+1 )) = q l=0 (-1) l+s ǫ 2l (2l + 1)! ( p n=0 (c 2n x+d 2n y+h 2l t+f 2n +O(ǫ p+1 ))ǫ 2n ) 2l+1 ×ǫ M -k ( r l=1 g 2l ǫ 2l +O(ǫ r+1 )) = q l=0 (-1) l+s ǫ 2l (2l + 1)! ( p n=0 P n (x, y, t)ǫ 2n + O(ǫ p+1 )) 2l+1 × ǫ M -2s ( r l=1 g 2l ǫ 2l + O(ǫ r+1 ))
where P n (x, y, t) is a polynomial of order 1 in x, y and t.

l k,1 = q l=0 ǫ 2l α0+...+αp=2l+1 β α0,...,αp P 0 (x, y, t) α0 . . . P p (x, y, t) αp ǫ 2(α1+2α2+pαp) × ǫ M -2s r l=1 g 2l ǫ 2l + O(ǫ t ) = q l=0 ǫ 2l α0+...+αp=2l+1 Q α0,...,αp (x, y, t)ǫ 2(α1+2α2+pαp) ×ǫ M -2s r l=1 g 2l ǫ 2l +O(ǫ t ),
where Q α0,...,αp (x, y, t) is a polynomial of order 2l + 1 in x,y and t.

The terms in ǫ 0 are obtained for l = 0 in the two summations with α 0 = 1.

For the column M 2 + m, we search the terms in ǫ 2m-2 with the maximal power in x and t. It is obtained for 2l + Mk = 2m -2, which gives l = m + s -N -1. We get the following result Proposition 2.4

deg(n 2s,m+ M 2 ) = 2m + 2s -M -1 for s ≥ M 2 + 1 -M, n 2s,m+ M 2 = 0 for s < M 2 + 1 -m. (27)
These results can be rewritten in the following way Proposition 2.5

deg(n j,k ) = 2k -j for j ≤ 2k, n j,k = 0 for j > 2k, deg(n j,k ) = 2k + j -2M -1 for j ≥ 2M + 1 -2k, n j,k = 0 for j < 2M + 1 -2k. ( 28
)
Now we can evaluate the degree of the determinant of the matrix (n kj ) k,j∈ [1,2N ] .

From the previous analysis, x, y and t have necessarily the same power in each n kj . The maximal power in x, y and t, is successively taken in each column. It is realized by the following product Applying the result given in [START_REF] Gaillard | Families of quasi-rational solutions of the NLS equation and multi-rogue waves[END_REF] we get

deg(det(n kj ) k,j∈[1,2N ] ) = N j=1 deg(n j,j ) + N j=1 deg(n N +j,2N +1-j ) = N j=1 2j -j + N j=1 2(M + 1 -j) -2M -1 + M 2 + j = N j=1 j + N j=1 N + 1 -j = N (N + 1).
We have the same argument for the determinant det(d kj ) k,j∈ [1,2N ] , we have deg(det

(d kj ) k,j∈[1,2N ] ) = N (N + 1).
Thus the quotient det((n kj) j,k∈ [1,2N ] ) det((d kj) j,k∈ [1,2N ] ) defines a quotient of two polynomials, each of them of degree N (N + 1). As the expression of the solution v is given by n(x,y,t)n * (x,y,t) d(x,y,t) 2

, it gives the result on the degrees of the polynomials |n| 2 and d 2 . So we obtain det((n kj) j,k∈ [1,2N ] )

2 det((d kj) j,k∈[1,2N ] )
2 is a quotient of two polynomials of degree 2N (N + 1). 23 Explicit expression of rational solutions of order 4 depending on 6 parameters

In the following, we explicitly construct rational solutions to the KPI equation of order 4 depending on 6 parameters.

Because of the length of the expression, it cannot be given in that paper. We only give the expression without parameters in the appendix.

We give patterns of the modulus of the solutions in the plane (x, y) of coordinates in function of the parameters a 1 , a 2 , a

Conclusion

From the previous representations of the solutions to the KPI equation given by the author in terms of Fredholm determinants of order 2N depending on 2N -1 real parameters and in terms of wronskians of order 2N depending on 2N -1 real parameters, we succeed in obtaining rational solutions to the KPI equation depending on 2N -2 real parameters. These solutions can be expressed in terms of a ratio of two polynomials of degree 2N (N + 1) in x, y and t. That gives a new approach to find explicit solutions for higher orders and try to describe the structure of those rational solutions.

In the (x, y) plane of coordinates, different structures appear. For a given t, when one parameter grows and the other ones are equal to 0 we obtain triangles or rings; for a 1 = 0 or b 1 = 0 and the other parameters equal to zero, we obtain a triangle with 10 peaks; for a 2 = 0 or b 2 = 0, and other parameters equal to zero, we obtain two concentric rings of 5 peaks on each of them; in the last case, when a 3 = 0 or b 3 = 0, and the other parameters equal to zero, we obtain one ring with 7 peaks. It will be relevant to go on this study for higher orders to try to understand the structure of those rational solutions.

Appendix : Because of the length of the complete expression, we only give in this appendix the explicit expression of the rational solution of order 4 to KPI equation without parameters. They can be written as

v

Theorem 2 . 2 2 (W 1

 2221 The function v defined by v(x, y, t) = -2 |W 3 (φ 3,1 , . . . , φ 3,2N )(0)| (φ 1,1 , . . . , φ 1,2N )(0)) 2 is a solution to the KPI equation depending on 2N -1 real parameters a k , b k 1 ≤ k ≤ N -1 and ǫ, with φ r ν defined by

  3 , b 1 , b 2 , b 3 , and time t. When at least one parameter is not equal to 0, we observe the presence of ten peaks. The maximum of modulus of those solutions is checked equal in this case N = 4 to 2(2N + 1) 2 = 2 × 9 2 = 162.

Figure 1 .

 1 Figure 1. Solution of order 4 to KPI, on the left for t = 0; in the center for t = 0, 01; on the right for t = 0, 1; all the parameters to equal to 0.

Figure 2 .

 2 Figure 2. Solution of order 4 to KPI, on the left for t = 0, 2; in the center for t = 10; on the right for t = 50; all the parameters to equal to 0.

Figure 3 .

 3 Figure 3. Solution of order 4 to KPI for t = 0, on the left for a 1 = 10 3 ; in the center for b 1 = 10 3 ; on the right for a 2 = 10 6 ; all the other parameters to equal to 0.

Figure 4 .

 4 Figure 4. Solution of order 4 to KPI for t = 0, on the left for b 2 = 10 6 ; in the center for a 3 = 10 9 ; on the right for b 3 = 10 9 ; all the other parameters to equal to 0.

20 k=0

 20 f k (Y, T )X k , H 4 (X, Y, T ) = 20 k=0 h k (Y, T )X k , Q 4 (X, Y, T ) = 20 k=0 q k (Y, T )X k . f 20 = 1, f19= -60 T , f 18 = 1710 T 2 + 10 Y 2 -30, f 17 = -30780 T 3 + (-540 Y 2 + 2340)T , f 16 = 392445 T 4 + 45 Y 4 + (13770 Y 2 -78030)T 2 -1350 Y 2 -675, f 15 = -3767472 T 5 + (-220320 Y 2 + 1542240)T 3 + (-2160 Y 4 + 67680 Y 2 + 23760)T , f 14 = 28256040 T 6 + 120 Y 6 + (2478600 Y 2 -20655000)T 4 -9000 Y 4 + (48600 Y 4 -1587600 Y 2 -210600)T 2 +16200 Y 2 -19800, f 13 = -169536240 T 7 +(-20820240 Y 2 +201262320)T 5 + (-680400 Y 4 +23133600 Y 2 -3402000)T 3 +(-5040 Y 6 +378000 Y 4 -680400 Y 2 +831600)T , f 12 = 826489170 T 8 + 210 Y 8 + (135331560 Y 2 -1488647160)T 6 -27720 Y 6 + (6633900 Y 4 -234397800 Y 2 + 112776300)T 4 + 224700 Y 4 + (98280 Y 6 -7371000 Y 4 +14175000 Y 2 -18937800)T 2 +163800 Y 2 -330750, f 11 = -3305956680 T 9 +(-695990880 Y 2 + 8583887520)T 7 +(-47764080 Y 4 +1751349600 Y 2 -1512529200)T 5 +(-1179360 Y 6 +88452000 Y 4 -191872800 Y 2 + 297410400)T 3 + (-7560 Y 8 + 977760 Y 6 -7182000 Y 4 -4989600 Y 2 + 12209400)T , f 10 = 10909657044 T 10 + q 20 = 1, q 19 = -60 T , q 18 = 1710 T 2 + 10 Y 2 + 10, q 17 = -30780 T 3 + (-540 Y 2 + 180)T , q 16 = 392445 T 4 + 45 Y 4 + (13770 Y 2 -22950)T 2 -270 Y 2 + 405, q 15 = -3767472 T 5 + (-220320 Y 2 + 660960)T 3 + (-2160 Y 4 + 15840 Y 2 -16560)T , q 14 = 28256040 T 6 + 120 Y 6 + (2478600 Y 2 -10740600)T 4 -1800 Y 4 + (48600 Y 4 -421200 Y 2 + 437400)T 2 + 1800 Y 2 + 16200, q 13 = -169536240 T 7 + (-20820240 Y 2 + 117981360)T 5 + (-680400 Y 4 + 6804000 Y 2 -8845200)T 3 + (-5040 Y 6 + 75600 Y 4 -75600 Y 2 -680400)T , q 12 = 826489170 T 8 + 210 Y 8 + (135331560 Y 2 -947320920)T 6 -4200 Y 6 + (6633900 Y 4 -75184200 Y 2 + 130466700)T 4 + 6300 Y 4 + (98280 Y 6 -1474200 Y 4 +2381400 Y 2 +14175000)T 2 +113400 Y 2 +425250, q 11 = -3305956680 T 9 +(-695990880 Y 2 +

2

  Rational solutions to KPI equation of order N depending on 2N -2 parameters 2.1 Families of rational solutions of order N depending on 2N -2 parameters

  4 (x, y, t) = -2 |n 4 (x, y, t)| 2 (d 4 (x, y, t)) 2

	with	n 4 (x, y, t) d 4 (x, y, t)	=	F 4 (2x, 4y, 4t) -iH 4 (2x, 4y, 4t) Q 4 (2x, 4y, 4t)
	with
		F 4 (X, Y, T ) =

The proof of this result has been given by the author[START_REF] Gaillard | Rational solutions to the KPI equation and multi rogue waves[END_REF][START_REF] Gaillard | Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves[END_REF]

The proof of this result has been given[START_REF] Gaillard | Rational solutions to the KPI equation and multi rogue waves[END_REF][START_REF] Gaillard | Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves[END_REF].


Science, technology and society: the social representations approach Nikos Kalampalikis* Martin W. Bauer** Thémis Apostolidis*** The present special issue explores the role that science and technology play in modern common sense and daily practices. The modern techno-sciences introduce new systems of production, regulation, opinion, knowledge and know-how. They propose new dieting rules at the crossroads of private and public health, research and nutrition. Techno-science makes available new technologies of reproduction; in changing ecological practices climate science impacts our lifestyles. Dieting habits, gender relations, bio-medical ethics, our relationship to the natural world, the wish for a child, projection, transmission and identification within the family are re-defined, acquire legal acceptance, upset language, and disturb traditions and religious codes, foster political debates and mass fears [START_REF] Kalampalikis | Giving or giving back: New psychosocial insights from sperm donors in France[END_REF]. We witness the opening of a new era of societal representations. These novelties afford new practices and forge new thinking that invite social sciences -in particular social psychology -to question their theoretical models and approaches with respect to new political, legal, psychological and societal realities. The aim of this special issue is to present empirical research that illustrates the contributions of a way of doing social psychology to approaching these challenges outside the simple expectations of fostering acceptance of innovations. Social psychology has more to offer 'than a ride on the back of technocratic dreams' [START_REF] Bauer | Le fait accompli et son influence sociale[END_REF].

The reader will notice in this collection of papers the authors base their research on the social representations approach as theoretical, methodological, and above all open and reflective paradigm. This is no coincidence, but demonstrates the pertinence and vivacity of this paradigm and its heuristic value for the investigation of changing science-society relations in different sociocultural contexts. This adds to an emerging interdisciplinary research field of public understanding of science (PUS) with evident long-term fertility and renewing creativity [START_REF] Suerdem | PUS in turbulent times II -A shifting vocabulary that brokers interdisciplinary knowledge[END_REF]. This also goes back to Moscovici's original purpose in proposing the concept of 'social representa-tions' some 50 years ago, which he recently formulated as follows: "I must admit that originally I did not aim at introducing into social psychology a concept derivative of Durkheim or Lévy-Bruhl, nor trying to differentiate it in order to adapt it later to the social Zeitgeist. To the contrary, it is the problem of the transformation of science while spreading and the emergence of a post-scientific common sense, the one of our social psychology, which have lead me there" (Moscovici, 2013, p. 108, our translation). In this original impetus, the social representation approach is an early anti-thesis to the still going diffusion paradigm, which assumes that new ideas and techniques are ready-made and fixed as they spread, the only variables being the speed and reach of diffusion. The evidence continues to show that diffusion is more wishful thinking than reality; it occurs in the very special case of a somnambulant public.

In this issue, our purpose is to investigate this circulation of new scientific insights in the production of knowledge and social practices and in their direct repercussion on reality and daily life. Life incorporates more and more scientific knowledge and technique in the design of objects directly linked with our projects (with permanence and variation), in our relationship with nature and culture through environmental stakes and scientific controver-sies, in the new parenthood technically assisted, in our relationship to fashion, in the definition of professions and the representa-tions of how science works. Social communication also plays a key source of data for this research work. Papers also analyze mass media materials as rhetorical resources or the interlocutory process of dynamic expression of ideas. This formal and informal communication constitutes the force and the framework of the modern mentality which characterizes our thinking societies.

Our special issue is divided into three thematic sections: environment, bioethics and science and daily life. The first three papers address environmental issues. Peta Callaghan and Martha Augoustinos (University of Adelaide) investigate the controversy over climate change in Australia. Their reflections shed light on the complexity and pertinence of preoccupations through the repercussions these controversies have within the scientific sphere. They analyze the protagonists' harsh rhetoric and polemic that is manifest in the Australian press. Their analyses evidence the effects of the style of communication and the type of knowledge used by both parties as rhetorical resources in order to forge the social representation of a changing climate in both the sciences and in public.

Sabine Caillaud (Paris-Descartes University) and Uwe Flick (Freie Universität Berlin) compare new environmental practices in two European socio-cultural contexts: France and Germany. Their design involves triangulation methodology of non-experts in each context. They show how the relationship with nature and culture affects daily practices and environmental knowledge differently. Acquaintance with the causes of climate change, risk perception and the anchoring in categories, moral or political, illustrate the underlying tensions in these two socio-culturally different contexts.

Carla Mouro (ISCTE -University Institute of Lisbon), Paula Castro (ISCTE), Nicole Kronberger (University of Linz) and Patricia Duarte (ISCTE) examine the 'clean energy' frame in European countries as a way of linking environmental matters, technolog-ical innovations and legal regulations. Their analysis makes use of the 2010 Eurobarometer (life sciences and public understanding of science). The survey was carried out in 32 European countries.

The analysis evidences the significance of a country's timing of entry into the European Union as a factor for forming consent on different forms of energy (solar versus nuclear).

Two papers focus on bioethical challenges. Sophie Zadeh, Tabitha Freeman and Susan Golombok (University of Cambridge) discuss a rarely explored question, the experience of sperm donation, by talking to single mothers, a rare and sensitive group with access to new reproductive technology. With a qualitative approach the authors reveal the forms of parentage identity negotiations that these new technique of reproduction afford for single women. Identity negotiation is a dynamically constructed process in balancing ambivalence between personal choice and traditional family pattern. Caroline Léchot (University of Lausanne) and Vlad-Petre Glâveanu (Aalborg University) address the issue of late pregnancy in the Swiss press. They show the weight of culture as morality and medical ethics of reproductive technologies burdening the effort of building a normative commons.

Finally, three more papers explore relationship of science and daily life. Nelly Courvoisier (University of Lausanne-Unil), Eva Green (Unil), Mira Canciu (Unil) and Alain Clémence (Unil) conducted two field studies involving participants before and after a guided tour of the ATLAS particle detector at the European Organization of Nuclear Research (CERN). The results show how science becomes part of daily conversations on the visit experience in function of previous attitude toward science (more or less positive or nega-tive) and the extent of debriefing and comprehension.

Lucile Salesses (Aix-Marseille University-AMU) and Deborah Romain (AMU) examine the ethical fashion. Questioning both fashion and ethical consumption historically, they focus on the representations of a new composite object and take a critical look at its relative autonomy.

Chloé Eyssartier (CETE de l'Ouest/Center of technical studies of the equipment of the West) and Fabrice Hamelin (Ifsttar -French Institute of science and technology for transports, development and networks) discuss the impact of a technological device, auto-matic radars for reporting traffic offence in France, on the social representation of the professions involved in the implementa-tion, professional drivers and police constables. Their results evidence a gradual transformation of the profession of drivers and a more radical institutional reorganization of the police force in function of this new technology.

Our thematically focused special issue of the journal brings together eight texts and presents a panorama of research of twenty-one social psychologists from eight different (Australia, Austria, Denmark, France, Germany, Great Britain, Portugal, Switzerland). This demonstrates the richness, diversity and vivacity of a social psychology that is open to cultural and societal challenges, and testifies the necessity to study social cognition as knowledge-in-context using mixed methodologies. Furthermore, these developments illustrate the heuristic value of the social representations paradigm in the study of objects in engagement with issues raised by scientific and technological changes and its impact on the social life of contemporary societies [START_REF] Alessio | Représentations sociales et embryon humain : une étude comparative Brésil/France[END_REF].