Rogue waves and breathers of the derivative Yajima-Oikawa long wave-short wave equations on theta-function backgrounds

Abstract

Based on the Baker-Akhiezer functions on Riemann surfaces, a general method is proposed to construct rogue-wave solutions and breather solutions of integrable nonlinear evolution equations associated with higher-order matrix spectral problems on theta-function periodic backgrounds. The derivative Yajima-Oikawa long wave-short wave equation associated with a 3×3 matrix Lax pair is chosen as an illustrative example of our method. First, a theta-function seed solution for the derivative Yajima-Oikawa long wave-short wave equation is derived by using the Weierstrass functions. Then, the general solution of the 3×3 matrix spectral problem and its derivatives with respect to the spectral parameter are obtained with the help of the Baker-Akhiezer functions. Finally, by means of Darboux transformations, explicit exact solutions such as rogue waves and breathers on theta-function backgrounds for the derivative Yajima-Oikawa long wave-short wave equation are constructed. Taking the advantage of the Jacobi θ-functions, these new solutions are numerically calculated and illustrated. In addition, by observing the explicit solutions, it is found that two rogue waves of different shapes may appear in one solution, which is a new phenomenon different from the solutions on constant backgrounds.