Whitham modulation theory of the defocusing AB system and its application

Whitham modulation theory and periodic solutions for the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

“…Kamchatnov extended this so that the method could be applied to a series of equations for the AKNS system [35][36][37][38][39][40][41]. In recent years, it is found that the periodic solutions derived by the finite-gap integration method can be further studied by the Whitham modulation theory [42][43][44][45][46][47]. The Whitham modulation theory is the basis for the development of dispersive hydrodynamics, and it is often used to analyze the slow evoluti on of nonlinear dispersive waves [48][49][50].…”

Exotic wave patterns in Riemann problem of the high‐order Jaulent–Miodek equation: Whitham modulation theory

“…The main purpose of this paper is to search for the full classification of solutions to the high‐order JM equation () by the finite‐gap integration method and Whitham modulation theory 8,14 . The finite‐gap integration method was presented by Flaschka, McLaughlin, and Forest 8 to study the slow modulations of N ‐phase trains for the KdV equation, which permits one to predict wave patterns arising from given initial values, and has been used to study the generalized nonlinear Schrödinger equations with self‐steepening nonlinearity, 45,46 the Camassa–Holm equation with small dispersion, 47 the polarization waves in a two‐component Bose–Einstein condensate, 48 the cmKdV equation, 21,49 the generalized Chen–Lee–Liu equation, 50 and the defocusing AB system 51 with initial discontinuity. Here we extend the method to determine the zero‐genus, one‐genus, and two‐genus solutions of the high‐order JM equation ().…”

“…The left oscillating pattern corresponds to {DSW-I}, the periodic wave solution is (47). While the right pattern corresponds to {DSW-VIII}, the periodic wave solution is (51). We now give the solutions for each region and boundary velocities (see Figure 15).…”

The long-time asymptotic behaviors of the solutions for the coupled dispersive AB system with weighted Sobolev initial data

“…Those results were generalized by introducing the generalized Darboux transformation to seek the high-order rogue wave solutions [8,9]. In addition, the modulation periodic wave solutions along with the corresponding Whitham equations were also derived via Whitham modulation theory [10]. Following a long development involving many researchers, for the problem where the long-time asymptotic of the solutions of the AB system as t → ∞, the nonlinear steepset descent method has been introduced in [11].…”