Complexiton solutions to integrable equations

Wronskian solutions and integrability for a generalized variable-coefficient forced Korteweg–de Vries equation in fluids

“…Compared with the Hirota bilinear method [12,[38][39][40], the Wronskian technique is used to verify the N -soliton solutions by the direct substitution into the nonlinear evolution equation [41][42][43][44][45][46]. In this paper, we will construct and verify the Wronskian solutions for (1) directly.…”

“…Except for the soliton solutions, the Wronskian technique can also be applied to construct the rational solutions, positons, negatons, breathers, complexitons and interaction solutions of the continuous nonlinear evolution equations (NLEEs), such as the KdV equation and Boussinesq equation [44][45][46]. For the discrete NLEEs, the Casoratian technique, a discrete version of Wronskian technique, has been developed [51,52].…”

New Positon, Negaton, and Complexiton Solutions for a Coupled Korteweg–de Vries – Modified Korteweg–de Vries System

“…The new rational solutions, solitons, positons, negatons, and complexiton solutions for the KdV equation are given by the Wronskian formula with the help of its bilinear form [5]. Ma provided the complexiton solutions of the KdV equation and the Toda lattice equation through the Wronskian and Casoratian techniques [6]. The authors of [7,8] presented the positons, negatons, and complexitons and their interaction solutions for the Boussinesq equation through its Wronskian determinant.…”

A note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev–Petviashvili equation

“…The so-called complexiton, which is related to the complex spectral parameters, is expressed by combinations of trigonometric functions and hyperbolic-functions. Prof. Ma provided the complexiton solutions of the KdV equation and the Toda lattice equation through the Wronskian and Casoratian techniques [2]. The authors of [3] presented the positons, negatons and complexitons and their interaction solutions for the Boussinesq equation through its Wronskian determinant.…”