The Two-Component Breather Solution of the Hirota Equation

Abstract

A nonlinear wave equation which describes different nonlinear effects in various fields of research, was considered. In two particular cases, this equation was reduced to the Sine-Gordon equation and the Born-Infeld equation. Using the slowly varying envelope approximation and the generalized perturbative reduction method, the nonlinear wave equation was transformed to coupled nonlinear Schrödinger equations for auxiliary functions. An explicit analytical solution of a nonlinear wave equation in the form of a two-breather molecule was obtained. One breather oscillated with the sum, and the other with the difference of frequencies and wave numbers. The obtained solution coincides with the solutions of the two-breather molecule found in a number of well-known equations from different areas of physics. It is shown that in a particular case of the small amplitude waves, a solution in the form of a two-breather molecule for the nonlinear Klein-Gordon equation coincides with the vector 0π pulse of the self-induced transparency which is presented under less stringent conditions compared to the same solution of this equation obtained earlier.

Two-Component Nonlinear Wave of the Cubic Boussinesq-Type Equation

“…The dispersion relation and the connection between parameters Ω ± and Q ± are determined from Eqs. (10) and (15). The parameters of the nonlinear pulse from Eqs.…”

“…Substituting Eq. ( 8) into (7) we obtain the connection between the parameters ω and k in the form αω 2 + βk 2 − γkω − δk 4 + µk 3 ω − νω 2 k 2 = 0 (10) and the cubic Boussinesq-type equation for envelope function Ûl :…”

“…Among single-component solitary waves, soliton and breather are considered quite often. The two-component vector pulse is a bound state of two nonlinear wave packets with same velocities and different polarizations (for waveguide mode) or different frequencies and wave numbers with the same or various polarizations [1][2][3][4][5][6][7][8][9][10][11].…”

“…[14][15][16]. Later such vector two-component waves also were studied for waves of completely different nature and in various fields of physics such as plasma physics, hydrodynamics, field theory, metamaterials, etc [8][9][10].…”