Nonlinear waves, modulations and rogue waves in the modular Korteweg–de​ Vries equation

Abstract

Effects of nonlinear dynamics of solitary waves and wave modulations within the modular (also known as quadratically cubic) Korteweg–de Vries equation are studied analytically and numerically. Large wave events can occur in the course of interaction between solitons of different signs. Stable and unstable (finite-time-lived) breathers can be generated in inelastic collisions of solitons and from perturbations of two polarities. A nonlinear evolution equation on long modulations of quasi-sinusoidal waves is derived, which is the modular or quadratically cubic nonlinear Schrödinger equation. Its solutions in the form of envelope solitons describe breathers of the modular Korteweg–de Vries equation. The instability conditions are obtained from the linear stability analysis of periodic wave perturbations. Rogue-wave-type solutions emerging due to the modulational instability in the modular Korteweg–de Vries equation are simulated numerically. They exhibit similar wave amplification, but develop faster than in the Benjamin–Feir instability described by the cubic nonlinear Schrödinger equation.