A new nonlinear wave equation: Darboux transformation and soliton solutions

In this paper, a new nonlinear wave equation associated with a 2 × 2 matrix spectral problem is proposed by means of the zero-curvature equation and the polynomial expansion of the spectral parameter. With the help of a gauge transformation between the corresponding Lax pair, a Darboux transformation of the nonlinear wave equation is obtained. As an application, by taking different seed solutions and using the Darboux transformation, one can get a variety of types of exact solutions for the nonlinear wave equation, like one-soliton solution, two-soliton solution, periodic solution, and Akhmediev breather solution.