Abstract
The Painlevé integrability of the higher-order Boussinesq equation is proved by using the standard Weiss-Tabor-Carnevale (WTC) method. The multisoliton solutions of the higher-order Boussinesq equation are obtained by introducing dependent variable transformation. The soliton molecule and asymmetric soliton of the higher-order Boussinesq equation can be constructed by the velocity resonance mechanism. Lump solution can be derived by solving the bilinear form of the higher-order Boussinesq equation. By some detailed calculations, the lump wave of the higher-order Boussinesq equation is just the bright form. These types of the localized excitations are exhibited by selecting suitable parameters.
Highlights
The soliton molecule as a boundary state is comprised by a balance of repulsive and attractive forces between solitons caused by the nonlinear and dispersive effects [1]
The velocity resonance mechanism is developed to some integrable systems, the (2+1)-dimensional fifth-order Korteweg-de Vries (KdV) equation [11], the complex modified KdV equation [12], the (3+1)-dimensional Boiti-LeonManna-Pempinelli equation [13], and so on [14,15,16]
In addition to the soliton molecule, lump solutions are a kind of rational function solutions which have become a hot field in nonlinear systems [18,19,20,21,22]
Summary
The soliton molecule as a boundary state is comprised by a balance of repulsive and attractive forces between solitons caused by the nonlinear and dispersive effects [1]. Our objective is to explore a higher-order Boussinesq equation which is considered as the combination between the fourth-order and sixth-order Boussinesq forms. We get lots of interesting results for the higher-order Boussinesq equation, such as the multisoliton, the soliton molecule, and lump solution. The soliton molecule of the higher-order Boussinesq equation is not valid for just the fourth-order Boussinesq equation or the sixth-order Boussinesq equation. The N-soliton solutions for the fourth-order [33] and the sixth-order Boussinesq equations [34] were obtained by the Hirota bilinear method. The soliton molecule of the higher-order Boussinesq equation is constructed by a new resonance condition.
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