Mechanisms Of Bright And Dark Transition States For The (2+1)-Dimensional Boussinesq Equation In Shallow Water

Under investigation in this work is the(2+1)-dimensional Boussinesq equation,which describes the dynamic behavior of the shallow water wave.We apply the Hirato bilinear method and complex parameters to present the solitary-wave solution and the first-order breath-wave solution.With the analysis of the nonlinear superposition mechanism and characteristic lines,the transition condition is given analytically.Based on this condition,a series of transformed waves are obtained,including the(quasi-)bell-shaped solitary waves,(oscillating)M-and W-shaped solitons,multi-peak solitons and(quasi-)periodic waves,which show different localization and oscillation.Some important features for these transformed waves are investigated in detail,such as the formation mechanisms,existence conditions,characteristic lines,localizations and oscillations and superposition mechanisms.More importantly,we study the time-varying properties by analyzing the characteristic lines and superposition mechanisms.What is more,by applying the improved transition condition,the seconder-order breath wave is transformed.Four kinds of collision modes are presented for these transformed waves by analyzing the wave components and characteristic lines,including the one-off,completely elastic,semi-elastic and inelastic collisions.The analysis of their nonlinear superposition mechanism and characteristic lines reveals the nature of different collisions,and explains that the deformation after collision is due to the phase change of the characteristic lines of the two wave sources.In particular,we show several dark transformed waves and analyzed the related properties of dark transformed waves.Finally,we take the interaction between a dark bell-type solitary wave and a dark multi-peak soliton as an example to study the collision characteristics between two dark transformed waves.The properties of the dark transformed wave are similar to those of the bright transformed wave,except for the feature of dark structure.