The Whitham Modulation Solution of the Complex Modified KdV Equation

This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg-de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever's algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler-Poisson-Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory.