Study On State Transformation Of Nonlinear Wave Equations In High Dimensions

The construction of nonlinear wave solutions of nonlinear evolution equations is one of the most important aspects of nonlinear science.In this paper,the dynamics of nonlinear wave in(2+1)-dimensional Ito equation and(3+1)-dimensional BKP equation are studied by the nonlinear superposition principle,characteristic line and phase shift analysis.The details are as follows:Firstly,the state transformation of first-order nonlinear waves is studied based on the(2+1)-dimensional Ito equation.The N-soliton solution is obtained via the Hirota bilinear method,from which the breath-wave solution is derived by changing values of wave numbers into complex forms.Through the analysis of the characteristic line of the breather wave,the conversion condition is obtained.On this basis,we prove that breather waves and lumps can be transformed into various nonlinear waves,including the multi-peak soliton,M-shaped soliton,quasi-anti-dark soliton,three types of quasi-periodic waves and W-shaped soliton.The distribution phase diagram of the above-mentioned nonlinear waves is displayed on the(k1,l1)-plane,and the gradient relationship of the transformed nonlinear waves is revealed.Through nonlinear superposition mechanism,characteristic line and phase shift analysis,the formation mechanism of transformed nonlinear waves and their locality,oscillation,and time-varying characteristics are studied.Then,based on the high-order breath wave solutions,the interactions between those transformed nonlinear waves are investigated,such as the completely elastic mode,semi-elastic mode,inelastic mode,and collision-free mode.We reveal that the diversity of transformed nonlinear waves,time-varying property and shape-changed collision mainly appear as a result of the difference of phase shifts of the solitary wave and periodic wave components,and presented that the dynamics of the double shape-changed collisions.Finally,we explore the dynamical properties of transformed nonlinear waves for the(3+1)-dimensional B-type Kadomtsev-Petviashvili(BKP)equation.Different from the(1+1)-or(2+1)-dimensional case,three types of conversion conditions are analytically derived in different spatial coordinates,by which the breath waves can be converted into diverse TNWs,including the M-shaped kink soliton,kink soliton with multi peaks,(quasi-)kink soliton,and(quasi-)periodic wave,and the lump can be converted into the line rouge wave.The above aspects by introducing three techniques are further used to explicate the essence of oscillation,locality and shape-changed evolution of transformed nonlinear waves.