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

On the exact solutions of integrable models, there is a new classification way recently based on the property of associated spectral parameters [1]. Negatons, related to the negative spectral parameter, are usually expressed by hyperbolic functions, and positons are expressed by means of trigonometric functions related to the positive spectral parameters. The so-called complexiton, which is expressed by combinations of trigonometric and hyperbolic functions, is related to the complex spectral parameters. For many important integrable systems such as the Korteweg–de Vries (KdV) equation, there is no nonsingular positon. But negatons can be both singular and nonsingular. Especially, the nonsingular negatons are just the solitons which have been studied extensively. The known complexitons for (1 + 1)-dimensional integrable systems are singular. But some types of the analytical positons and complexitons in (2 + 1)and (3 + 1)-dimensions can be easily obtained because of the existence of arbitrary functions in their expressions of exact functions [2, 3]. For other integrable systems, the complexiton solutions have been presented by different approaches. For example, using the Casoratian technique, the authors of [4] constructed the complexiton solutions to the Toda lattice equation through the Casoratian formulation. 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. On the other hand, the coupled integrable systems, which have attracted more and more attention from the mathematicians and physicists, have also been studied extensively since the first coupled KdV system was put forward by Hirota and Satsuma. Since then, many other coupled KdV systems are constructed such as the Ito system, the Nutku–Og̃uz model, and so on. Recently, new positon, negaton, and complexiton solutions for the two types of the coupled KdV system are presented by means of the Darboux transformation [9, 10]. The analytical positon, negaton, and complexiton solutions for the coupled modified KdV (mKdV) system are given out directly in [11] from the zero seed solution by means of Darboux transformation. For different integrable systems, these new positon, negaton, and complexiton solutions are analytical or singular. It is known that some generalized KdV–mKdV system have solitary wave solutions and explicit solutions in terms of Jacobi elliptic functions [12, 13]. A special