Overview on the generalized Volterra System, solutions and integrability

In this review paper we present the recent results obtained, using the Hirota bilinear formalism, on the semidiscrete and discrete forms of generalized Volterra system with any number of coupled equations. The bilinear forms and the multisoliton solutions are presented for the differential-difference multicomponent Volterra system and for the fully discretized version. For some particular cases of the general differential-difference Volterra system, graphical representations of solitons are displayed.

Introduced by Hirota and Satsuma [1,2,3] in the seventies, the Volterra system gained interest in the last decades and was widely investigated from different points of view.We will focus on the results obtained via Hirota formalism, a well known method that is successfully used for building soliton solutions and integrable discretizations for solitonic equations.Multisoliton solutions, rational, plateau, white and dark soliton solutions were obtained in [4,5,6] for semidiscrete Volterra and, later, completely integrable full discretizations were constructed [7,8,9,10].In this paper we are making an overview on multicomponent Volterra system with any M coupled equations with branched dispersion relation.The results obtained using the Hirota bilinear formalism [11,12,13] are presented, first for the semidiscrete system and then for the fully discretized Volterra.
The paper is organized as follows: after a brief introduction, in Chapter 2 we present the Hirota bilinear formalism for building integrable discretisations, in Chapter 3 we focus on the general case (∀ M ) of semidiscrete Volterra system with its bilinear form and multisoliton solutions that prove the complete integrability of the general system.In Chapter 4, the discrete general Volterra system is presented, more explicitly the bilinear form, the multisoliton solutions and the recovered nonlinear form.In Chapter 5 we summarize our conclusions.

Hirota bilinear formalism for constructing integrable discretizations
The Hirota method is a very efficient tool for deriving integrable discretizations of bilinear systems.The method consists in the following three steps: Step 1. Considering a semidiscrete bilinear form of the analysed system, we replace the ordinary derivatives in We also must replace t with mδ in the rest of the equation.In the above relations f and g are the tau functions, t is the continuous time variable, m is the discretized time variable and δ is the discretisation step on the time axis; Step 2. We impose the gauge invariance to the bilinear equations built T (∆ m , ∆ n , ...)f •g = 0, more precisely the equations must remain invariant to the transformations: f → f e θn+σm , g → ge θn+σm , where θ and σ are real constants.
Step 3. We must compute the multisoliton solution and recover the nonlinear form of the analyzed system.

General semidiscrete Volterra system
The general differential-difference Volterra system with branched dispersion has the form: where c 0 , c 1 , c 2 are arbitrary constants, Q n is a diagonal matrix of complex functions q ν (n, t), with ν = 1, M , and E σ i are permutation matrices [9].In more details, system (1) can be written as: where q ν = q ν (n, t) and q ν = q ν (n + 1, t), q ν = q ν (n − 1, t) for any ν = 1, M .Introducing the substitutions [4]: where Particular cases that were investigated in literature are semidiscrete mKdV with non-zero boundary conditions, which coresponds to M = 1 case, and the coupled Volterra system obtained for M = 2 [4,7,8].
There are several ways of demonstrating the complete integrability of a system.Usually, we consider that a partial discrete equation is integrable if an infinite number of independent integrals in involution can be computed from the Lax pair.However we focus on the Hirota formulation, which is an alternative method.In his formulation the proof of integrability requires the existence of a general multisoliton solution, more explicitly a solution describing multiple collisions of an arbitrary number of solitons having arbitrary parameters and phases, and considering all branches of dispersion relations.
the Hirota bilinear form was constructed in [9]: where g ν = g ν (n, t), g ν−1 = g ν−1 (n − 1, t), g ν+1 = g ν+1 (n + 1, t) and The N-ss solution solution constructed in [9] gives the following expressions for the tau functions g ν and f ν : where: with M branches of dispersion for each soliton (k j wave number, j = 1, N ): The solitons are experiencing only a phase shift due to the interaction.The asymptotic analysis in [9] proves that the amplitudes, shapes and velocities of the 2-soliton solution remain invariant before and after the interactions, in other words, their interactions are elastic and the solitons are very stable.We illustrate in Figure 1 the 1-, 2-and 3-soliton solutions for M = 2 for some chosen parameters and suitable intervals of n and t.

Integrable discretization for general Volterra system
We present one integrable discretization for (3) obtained in [10] applying the three steps of Hirota bilinear formalism given in Chapter 2.
The fully discrete gauge invariant bilinear equations for general Volterra system with any M coupled equations is given by: where: The N -soliton solution for ( 8) is : where the coefficients are: with the M possible branches of dispersion for each of the N solitons: The multisoliton solution for fully discret equations has the same phase factors and interaction terms as in the differential-difference case [9].What differs form the semidiscrete case is the dispersion relation.The nonlinear form recovered in [10] for the integrable discretization of differential-difference Volterra system with M coupled equations is: where

Conclusions
In this review paper we presented the recent results on the coupled multicomponent diferentialdifference general Volterra system with branched dispersion.The Hirota bilinear form, the multisoliton solutions and some graphical representations are presented for 1-, 2-, 3-soliton solutions of the semidiscrete Volterra system.Since the existence of the multisoliton solution is a criterion of complete integrability in the Hirota sense, the results prove that multicomponent semidiscrete general Volterra system preserves integrability.
Another recent result included in this review is an integrable discretization of the multicomponent differential-difference coupled Volterra system.The complete discretization was obtained using the Hirota method, a very useful tool for constructing integrable discretizations.The method consists in discretizing the differential Hirota bilinear operator, preserving gauge invariance, building multi-soliton solutions and recovering the nonlinear form using some auxiliary functions.The discrete bilinear form, the multisoliton solutions and also the nonlinear form are presented for discrete Volterra.