Generalised (2+1)-dimensional Super MKdV Hierarchy for Integrable Systems in Soliton Theory

Abstract

Much attention has been given to constructing Lie and Lie superalgebra for integrable systems in soliton theory, which often have significant scientific applications. However, this has mostly been confined to (1+1)-dimensional integrable systems, and there has been very little work on (2+1)-dimensional integrable systems. In this article, we construct a class of generalised Lie superalgebra that differs from more common Lie superalgebra to generate a (2+1)-dimensional super modified Korteweg-de Vries (mKdV) hierarchy, via a generalised Tu scheme based on the Lax pair method where the Hamiltonian structure derives from a generalised supertrace identity. We also obtain some solutions of the (2+1)-dimensional mKdV equation using the G′/G2 method.

Nonlinear integrable couplings of a generalized super Ablowitz‐Kaup‐Newell‐Segur hierarchy and its super bi‐Hamiltonian structures

“…Substituting Equation 24 into 23 and comparing the coefficient of −n−2 of both sides of Equation 23 yield…”

“…For example, the super Ablowitz-Kaup-Newell-Segur (AKNS) hirearchy, [2][3][4][5][6][7][8][9][10] super Dirac hierarchy, 4,[11][12][13] super Kaup-Newell (KN) hierarchy, [14][15][16] and others. [17][18][19][20][21][22][23][24][25][26] Among those, Hu 27 and Ma 4 have made a great contribution. Hu 27 proposed the super-trace identity at the first time, which is an effective tool to constructing super Hamiltonian structures of super integrable systems.…”

(2 + 1)‐Dimensional Coupled Model for Envelope Rossby Solitary Waves and Its Solutions as well as Chirp Effect

“…Many methods to solve the equations are proposed, such as traveling wave method [21], Darboux transformation method [22][23][24], Hirota method [25,26], homogeneous balance method [27], Jacobi elliptic function method [28], Symmetry method [29,30], Rational solutions [31][32][33]; meanwhile the features of equations are also discussed [34][35][36]. In this paper, we plan to adopt the trial function method and derive the exact solution of model.…”

Algebro-Geometric Solutions for a Discrete Integrable Equation

“…As we all know, the generation of integrable system, determination of exact solution, and the properties of the conservation laws are becoming more and more rich [1][2][3][4][5]; in particular, the discrete integrable systems have many applications in statistical physics, quantum physics, and mathematical physics [6][7][8][9][10][11]. It is worth discussing the properties of discrete integrable systems, such as Darboux transformations [12,13], Hamiltonian structures [14][15][16], exact solutions [17], and the transformed rational function method [18].…”