Abstract
The semidiscrete complex modified Korteweg–de Vries equation (semidiscrete cmKdV), which is the second member of the semidiscrete nonlinear Schrődinger hierarchy (Ablowitz–Ladik hierarchy), is solved using the Hirota bilinear formalism. Starting from the focusing case of semidiscrete form of cmKdV, proposed by Ablowitz and Ladik, we construct the bilinear form and build the multi-soliton solutions. The complete integrability of semidiscrete cmKdV, focusing case, is proven and results are discussed.
Highlights
The complex modified Korteweg–de Vries equation has attracted a lot of attention due to the wide range of physical applications in fields like nematic optical fibers [1], few-cycle optical pulses [2] and due to the important results obtained such as conserved quantities, stability of solitary wave solutions, numerical simulations, symmetry constraints, Darboux transformation, and various solutions [3,4,5,6]
The semidiscrete complex modified Korteweg–de Vries equation, introduced by Ablowitz and Ladik in [7] has the following form: d dt where we denote by Φn the complex function Φ(n, t), where t is the time variable and n is the discrete space variable with step 1
From the asymptotic analysis of solutions, we find that the amplitudes, shapes and velocities of the 2-soliton solution remain invariant before and after the interactions, which means that their interactions are elastic and the solitons are very stable as the only changes are in their initial phases
Summary
The complex modified Korteweg–de Vries (cmKdV) equation has attracted a lot of attention due to the wide range of physical applications in fields like nematic optical fibers [1], few-cycle optical pulses [2] and due to the important results obtained such as conserved quantities, stability of solitary wave solutions, numerical simulations, symmetry constraints, Darboux transformation, and various solutions [3,4,5,6]. In this paper we are going to investigate de integrability of the first version of semidiscrete cmKdV [7], focusing case, and build the multi-soliton solution. There are several ways of investigating the integrability of a dynamical system such as: a direct computation of the conserved quantities [13], the computation of Lie symmetries [14,15,16], the existence of the Lax pair [17, 21], building the Hirota bilinear form and computing the multi-soliton solutions [18, 19].
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