New type of source extension for a two-dimensional special lattice equation and determinant solutions

Hong-Yan Wang,Guo-Qing Zhu

doi:10.1186/s13662-021-03219-w

Abstract

We present a new type of two-dimensional special lattice equations with self-consistent sources using the source generation procedure. Then we obtain the Grammy-type and Casorati-type determinant solutions of the coupled system. Further, we present the one-soliton and two-soliton solutions.

Highlights

1 Introduction Soliton equations with self-consistent sources (SESCSs) are integrable coupled generalizations of the soliton equations. These coupled systems are usually relevant to problems in certain areas of physics, such as hydrodynamics, solid-state physics, and plasma physics [1,2,3,4]
We present the Casoratian determinant solution to the bilinear special lattice ESCS in Eqs. (22)–(28), which can be expressed in the following Pfaffian form: fn = det φi(n + j – 1) 1≤i,j≤N
5 Discussion and conclusion In this study, we applied source generation procedure (SGP) to the bilinear form of the two-dimensional special lattice equation and presented a new type of special lattice ESCS given by Eqs. (49)–(52)

Summary

Introduction

Soliton equations with self-consistent sources (SESCSs) are integrable coupled generalizations of the soliton equations. (22)–(28) constitute the bilinear forms of the two-dimensional special lattice ESCS, and the functions in Eqs. Functions in (53)–(58) constitute the Casoratian determinant solutions of the bilinear ESCS in Eqs. Example 2 We set N = 2 in expressions (15)–(18), and the functions φi(n) and ψi(–n) possess the following structures: φi(n) = pni e(p2i +p–i 1)y+pit+p2i z pni eξi , i = 1, 2, ψi(–n) = qi–ne–(qi2+qi–1)y–qit–qi2z qi–neηi , i = 1, 2, where pi and qi are real constants. In this case the function fn is a second-order determinant, and C1(t) included in fn is chosen as e2β(t)/(p1 – q1).

Discussion and conclusion

14. Gegenhasi