Darboux Transformation And Rogue Wave Solutions Of Nonlinear Wave Equation

The main line of this paper is to construct some primary Darboux transformation(DT) for nonlinear wave equation. In addition, primary DT is extended to general DT. With which we can generate rogue wave solutions for nonlinear wave equation and analyse the characterization of rogue wave via some figures. The main work is presented as follows:(1) The DT of the Heisenberg ferromagnet(HF) equations is derived via the Lax pair of the HF equations. Besides, a generalized DT for the HF equation is constructed by the Darboux matrix method. And, different rogue wave solutions are obtained when the real parameters a, ? and ? with different values. The HF equations have three potentials. Rogue waves of two potentials appear under the background of periodic wave, and the other one is regular rogue wave, which appears single and double peaks corresponding different values of the parameters.(2) Based on 33? matrix type Lax pair, we constructed a general DT of a generalized coupled nonlinear Schr?dinger equations in nonlinear optics. As application of the general DT, Nth-order rogue wave solutions of equations are obtained. In particular, the dynamics of the general first- and second-order rogue waves are discussed and illustrated through some figures.(3) Based on 22 ? matrix type Lax pair, a variable coefficient complex modified Korteweg-de Vries(vcc-mKdV) equation is considered. We introduced in detail infer process of DT of vcc-mKdV equation. And on this basis, we obtain a recursive formula of the(N-1) th-step generalized DT and the Nth-order rogue wave solutions for vcc-mKdV equation. We investigate the propagation of the first- to third-order rogue wave solutions of vcc-mKd V equation through the coefficients and parameters with different values.