Resumen de Breather Transitions and Their Mechanisms of a (2+1)-Dimensional Sine-Gordon Equation and a Modified Boussinesq Equation in Nonlinear Dynamics

Qian Gao, Shou-Fu Tian, Ji-Chuan Liu, Yan-Qiang Wu

• Studying the transformation mechanisms of soliton solutions of some high-dimensional equations can aid in comprehending the physical phenomena of the relevant nonlinear wave interactions. Therefore, the transitions and mechanisms of nonlinear waves in the (2+1)-dimensional sine-Gordon and (2+1)-dimensional modified Boussinesq equations are investigated for the first time by means of characteristic line and phase shift analysis, and the dynamic behavior of various nonlinear transformed waves is analyzed. Firstly, we derive the N-soliton solution by applying the Hirota bilinear method, from which the breather solution is constructed by changing the parameters into a complex form in pairs. In addition, via the characteristic-line analysis, we present the mechanism for the transformation of the breather solution, which is composed of the nonlinear superposition of a solitary wave and periodic wave. When the characteristic lines of the two parts are parallel, we find that various transformed nonlinear wave structures can be obtained, such as M-type solitons, oscillating M-type solitons, multi-peak solitons, quasi-sine waves and so on. Finally, we demonstrate that the geometric properties of the characteristic lines vary with time essentially resulting in the time-varying properties of nonlinear waves, which have never been found in (1+1)-dimensional systems. Overall, the study of breath-wave transitions in the (2+1)-dimensional sine-Gordon equation and the modified Boussinesq equation provides valuable insights into the bahavior of nonlinear systems and wave propagation.