Exact solutions for the shallow water equations in two spatial dimensions: A model for finite amplitude rogue waves

Abstract

Exact solutions for the shallow water equations in two spatial dimensions are obtained from a matrix formulation of the governing system. These fully nonlinear long waves exhibit linear velocity and quadratic free surface displacement fields in the spatial variables. No breaking phenomenon is observed. A special moving boundary can be defined where a fixed mass of fluid inside this circular region may display a rogue wave type behavior. The fluid first converges towards the center of a domain, and the amplitude of the free surface will attain a finite maximum. Subsequently, the fluid reverses paths and rushes away from the center. The kinetic and potential energy of the rogue wave can be computed analytically for all time. As the shallow water equations are employed, these rogue waves are fully nonlinear. There is no restriction on the amplitude, quite unlike the weakly nonlinear assumptions of the nonlinear Schrödinger and Korteweg—de Vries theories. Furthermore, the maximum is truly localized in two spatial dimensions horizontally and the wave is not a ‘long-crested’ one.

Keywords

• Shallow water equations
• Exact solutions
• Rogue waves
• Finite amplitude
• Compressible Euler equations
• Symmetry reduction