New Exact Solutions and Localized Excitations in a (2+1)-Dimensional Soliton System

In the study of nonlinear physics, the search of new exact solutions of nonlinear evolution equations (NEEs) is one of the most important problems. Various methods for obtaining exact solutions to NEEs have been proposed, such as the Lie group method of infinitesimal transformations [1], the nonclassical Lie group method [2], the Clarkson and Kruskal direct method (CK) [3 – 5], the conditional similarity reduction method [6 – 10] and the mapping approach [11 – 16]. In the past, with the help of the improved mapping approach, we have derived some exact excitations of (2+1)-dimensional NEEs, such as (2+1)-dimensional Broer-Kaup-Kupershmidt system, (2+1)-dimensional Boiti-Leon-Pempinelli system, (2+1)-dimensional generalized Broer-Kaup system [17 – 21]. The thought of the mapping approach is based on the reduction theory. Now an important question is whether some simple mapping equations, such as the Riccati equation, can be gotten by the method of conditional similarity reduction, i.e., for a given NEE whether we can transform the NEE to some simple equations which we want to obtain. If yes, new exact excitations of the NEE can be derived based on the exact solutions of these simple equations. In this paper, we try to extend the conditional similarity reduction method in order to find the conditional similarity reduction equation and some new exact excitations of the (2+1)-dimensional dispersive long-water