A Hierarchy of Lattice Soliton Equations Associated with a New Discrete Eigenvalue Problem and Darboux Transformations

Abstract

By considering a new discrete isospectral eigenvalue problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.

N-Fold Darboux transformation of the discrete Ragnisco–Tu system

“…There are some methods to construct solutions such as the inverse scattering transform method [13], the bilinear transformation method of Hirota [14], the Bäcklund and Darboux transformation techniques [15,16], the Fokas unified approach [17], the long-time asymptotics approach [18], and so on. Among them, the Darboux transformation is the most effective technique to find explicit solutions of the integrable differential-difference equations [10,11,[17][18][19][20][21][22][23][24][25][26]. This method based on Lax pairs has been proven to be one of the most fruitful algorithmic procedures to get explicit solutions of nonlinear evolution equations.…”

Hybrid Solutions of (3 + 1)‐Dimensional Jimbo‐Miwa Equation

“…In soliton theory, the study of integrability to nonlinear evolution is always a hot topic of interest, which can be regarded as a key step of their exact solvability. Many areas of integrable systems are researched, such as Painlevé analysis [1], Hamiltonian structure [2][3][4][5], Lax pair [6][7][8], Bäcklund transformation (BT) [9][10][11][12], infinite conservation laws [13][14][15][16], and bilinear integrability [17][18][19][20]. Based on the bilinear methods, we have got many kinds of solutions, such as lump solutions [21][22][23][24] and Pfaffian solution [25].…”

The Rational Solutions and the Interactions of the N-Soliton Solutions for Boiti-Leon-Manna-Pempinelli-Like Equation

“…Especially some key problems in engineering, science and modern physics are ultimately dependent on the specific solution of the nonlinear equations. So the solution of the nonlinear equations [1] [2] [3] [4] [5] occupies very important position no matter on theory research or practical application. But the diverse nonlinear equations of mathematical physics was descried with Hirota bilinear equations [6] [7] [8] and generalized bilinear equations [9] [11] [12] [13], such as the KdV equations [10] [11] [14] [15], the BLMP equations [16] [17] [18], the NLS equation, the Boussinesq equation [19], the KP equations [9] [20] [21] and so on.…”