Whitham modulation theory and exotic wave patterns of the good Jaulent–Miodek equation with step-like initial data

Abstract

The complete classification of solutions to the Riemann problem of the good Jaulent–Miodek equation is investigated by Whitham modulation theory. The one-phase and two-phase periodic wave solutions and the corresponding Whitham equations are derived based on the Lax pair of the good Jaulent–Miodek equation and the finite-gap integration method. In particular, the N-phase periodic wave solutions are proposed by algebro-geometric approach. Then the basic wave structures of rarefaction waves and DSWs are proposed analytically and graphically, which makes it possible to establish the classification of all the possible wave patterns evolving from initial discontinuities. The asymptotic results given by Whitham modulation theory are in excellent agreement with direct numerical simulations. Finally, a detailed description of the shallow-water dam break problem is demonstrated to find the possible physical significance of the wave patterns found in this work.

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].…”

Wave motions in discontinuous initial-value problem of the inviscid shallow water wave Jaulent-Miodek model

“…which has dispersion relation ω 2 = 4k 2 (4q 0 + k 2 ). It can be easy to see that the frequency ω is real and equation ( 5) is well-posed in the case of small amplitude solutions, thus we name the equation ( 5) as the good JM system, whose step-like initial problem has been explored in [25]. Fortunately, it is found that the equation ( 5) is also a completely integrable system with the Lax pair [25]…”

“…The formula (20) shows that the function µ is a real function. In view of the expression ( 16) of polynomial P (λ), it is observed that the exact solution µ of equation (25) only exists in the interval…”