Spiral Wave Solutions in Water Waves

TL;DR

This paper discovers and analyzes spiral wave solutions in linear and weakly nonlinear water wave equations, revealing their structure, persistence, and asymptotic behavior, thus introducing a new class of solutions in dispersive wave systems.

Contribution

It presents the first identification and analysis of spiral wave solutions in water wave equations, both linear and nonlinear, expanding the understanding of wave patterns in fluid dynamics.

Findings

Spiral waves evolve from initial conditions without external forcing.

Asymptotic analysis matches exact solutions, showing hyperbolic spiral structure.

Spiral waves persist under weak nonlinearity, indicating robustness.

Abstract

Spiral wave solutions are found in linear and weakly nonlinear irrotational water wave equations. These unsteady spiral waves evolve from suitable initial conditions; they are not induced by external forcing. In the linear case, a long-time asymptotic result is obtained via the method of stationary phase. The asymptotic approximation is found to be in good agreement with the exact solution and reveals hyperbolic spiral structure. Numerical simulations show that these spiral waves persist in the presence of weak nonlinearity. While spiral solutions are frequently found in excitable media governed by reaction-diffusion systems, they comprise a new class of interesting two space one time dimensional solutions in fundamental linear and nonlinear dispersive wave systems.