An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions

Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.