On an integrable system related to the relativistic Toda lattice – Bäcklund transformation and integrable discretization

Vinet, Luc; Yu, Guo‐Fu; Zhang, Yingnan

doi:10.1080/10236198.2015.1012074

Cited by 8 publications

(5 citation statements)

References 35 publications

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“…then we obtain the following coupled bilinear form where c 1 , c 2 are arbitrary constants. The bilinear Bäcklund transformation and soliton solutions (with c 1 = c 2 = 1) for the rToda lattice were given in [51]. According to theorem 2.2, the 3-periodic wave solution to the coupled bilinear system (4.23) and (4.24) has the form f m,n = θ (η; 0|τ ) with η j = ω j t + k j n + l j m + η 0 j , j = 1, 2, 3.…”

Section: (F) Relativistic Toda Latticementioning

confidence: 99%

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Numerical calculation ofN-periodic wave solutions to coupled KdV–Toda-type equations

Zhang

Sun

2021

Proc. R. Soc. A.

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In this paper, we study the N -periodic wave solutions of coupled Korteweg–de Vries (KdV)–Toda-type equations. We present a numerical process to calculate the N -periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N = 3, we give some detailed examples to show the N -periodic wave solutions to the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

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Section: (F) Relativistic Toda Latticementioning

confidence: 99%

Section: (F) Relativistic Toda Latticementioning

confidence: 99%

Numerical calculation ofN-periodic wave solutions to coupled KdV–Toda-type equations

Zhang

Sun

2021

Proc. R. Soc. A.

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“…The references provided are just a few representative examples, as numerous other recent works delve into this topic as well. For deeper investigation, additional progress and comprehensive research studies can be found in publications like [41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57], along with their respective reference sections. These works collectively contribute to the evolving understanding of integrable discretization methodologies.…”

Section: Introductionmentioning

confidence: 99%

Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies

Hu,

Yu,

Zhang

2024

Open Communications in Nonlinear Mathematical Physics

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In this paper, we give a procedure for discretizing recursion operators by utilizing unified bilinear forms within integrable hierarchies. To illustrate this approach, we present unified bilinear forms for both the AKNS hierarchy and the KdV hierarchy, derived from their respective recursion operators. Leveraging the inherent connection between soliton equations and their auto-B\"acklund transformations, we discretize the bilinear integrable hierarchies and derive discrete recursion operators. These discrete recursion operators exhibit convergence towards the original continuous forms when subjected to a standard limiting process.

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“…Here, we need to point out that the RTL − (α) system that we study here is different from the RTL equation in Refs. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. To our knowledge, some integrable properties, various exact solutions via the discrete generalized (m, 2N − m)-fold DT and asymptotic analysis for Equation (6) or Equation (1) have not been studied before.…”

Section: Introductionmentioning

confidence: 99%

A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions

2021

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This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m,2N−m)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.

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On an integrable system related to the relativistic Toda lattice – Bäcklund transformation and integrable discretization

Cited by 8 publications

References 35 publications

Numerical calculation ofN-periodic wave solutions to coupled KdV–Toda-type equations

Numerical calculation ofN-periodic wave solutions to coupled KdV–Toda-type equations

Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies

A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions

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