Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

Abstract

The formation of rogue waves in shallow water is presented in this Rapid Communication by providing the three lowest-order exact rational solutions to the Korteweg-de Vries (KdV) equation. They have been obtained from the modified KdV equation by using the complex Miura transformation. It is found that the amplitude amplification factor of such waves formed in shallow water is much larger than the amplitude amplification factor of those occurring in deep water. These solutions clearly demonstrate a potential hazard for coastal areas. They can also provide a solid mathematical basis for the existence of abnormally large-amplitude waves in other branches of nonlinear physics such as optics, unidirectional crystal growth, and in quantum mechanics.

New Travelling Wave Solution-Based New Riccati Equation for Solving KdV and Modified KdV Equations

“…Determine the positive integer m in Eq. (5). It can be done by balancing the highest-order derivative term and the highest-order nonlinear term in (4).…”

“…A plenty of efficient methods have been implemented to derive solutions in various forms to the KdVE-and mKdv-type equations. For example, Ankiewicz et al in [5] provided three lowest order exact rational solutions to the Kdv-type equations. Jia et al [6] constructed a Darboux transformation for the nonlocal mKdV-type equations finding exact solutions in the form of soliton, kink and antikink solutions.…”

Localized Waves for the Coupled Mixed Derivative Nonlinear Schrödinger Equation in a Birefringent Optical Fiber

“…As is well known, most nonlinear partial differential equations (PDEs) in mathematics and physics are integrable, including the Hirota equation [1], the nonlinear Schrödinger equation [2,3], the Gerdjikov-Ivanov equation [4], the Korteweg-de Vries equation [5,6], the Sasa-Satsuma equation [7], and so on. These nonlinear equations play an important role in various fields of nonlinear science such as water waves, nonlinear optics, and Bose-Einstein condensates.…”

Variational theory for a kind of non-linear model for water waves

“…Nowadays, the study of shallow water waves is a very hot topic [1][2][3][4]. This paper, we mainly focus on the Whitham-Broer-Kaup (WBK) equation [5][6][7], reads:…”