Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in 2+1 dimensions

TL;DR

This paper constructs explicit one- and two-periodic wave solutions for certain (2+1)-dimensional Hirota bilinear equations using Riemann theta functions, expanding the analytical tools for nonlinear wave equations.

Contribution

It introduces a method to derive explicit periodic solutions to Hirota bilinear equations in 2+1 dimensions using Riemann theta functions, with applications to specific nonlinear equations.

Findings

Explicit one- and two-periodic solutions for the equations.

Solutions involve arbitrary purely imaginary Riemann matrices.

Application to two nonlinear equations demonstrating wave propagation.

Abstract

Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the particular dependence of the equations on independent variables guarantees the existence of one-periodic and two-periodic wave solutions involving an arbitrary purely imaginary Riemann matrix. The resulting theory is applied to two nonlinear equations possessing Hirota bilinear forms: $u_t+u_{xxy}-3uu_y-3u_xv=0$ and $u_t+u_{xxxxy}-(5u_{xx}v+10u_{xy}u-15u^2v)_x=0$ where $v_x=u_y$, thereby yielding their one-periodic and two-periodic wave solutions describing one dimensional propagation of waves.