Nonlocal symmetry and interaction solutions for the new (3+1)-dimensional integrable Boussinesq equation

The nonlocal symmetry of the new (3+1)-dimensional Boussinesq equation is obtained with the truncated Painlev\'{e} method. The nonlocal symmetry can be localized to the Lie point symmetry for the prolonged system by introducing auxiliary dependent variables. The finite symmetry transformation related to the nonlocal symmetry of the integrable (3+1)-dimensional Boussinesq equation is studied. Meanwhile, the new (3+1)-dimensional Boussinesq equation is proved by the consistent tanh expansion method and many interaction solutions among solitons and other types of nonlinear excitations such as cnoidal periodic waves and resonant soliton solution are given.

Nonlinear excitations and dynamic features of dust ion-acoustic waves in a magnetized electron–positron–ion plasma

A wide class of nonlinear excitations and the dynamics of wave groups of finite amplitude ion-acoustic waves are investigated in multicomponent magnetized plasma system comprising warm ions, and superthermal electrons as well as positrons in presence of negatively charged impurities or dust particles. Employing the reductive perturbation technique (RPT), the Korteweg–de-Vries (KdV) equation, and extended KdV equation are derived. The presence of excess superthermal electrons as well as positrons and other plasma parameters are shown to influence the characteristics of both compressive and rarefactive solitons as well as double layers (DLs). Also, we extend our investigation by deriving the nonlinear Schrödinger equation from the extended KdV equation employing a suitable transformation to study the wave group dynamics for long waves. The analytical and numerical simulation results demonstrate that nonlinear wave predicts solitons, “table-top” solitons, DLs, bipolar structure, rogue waves, and breather structures. Moreover, implementing the concept of dynamical systems, phase portraits of nonlinear periodic, homoclinic trajectories, and supernonlinear periodic trajectories are presented through numerical simulation.

Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

Under investigation in this paper is the higher-order Broer-Kaup(HBK) system, which describes the bidirectional propagation of long waves in shallow water. Via the standard truncated Painlevé expansion method, the residual symmetry of this system is derived. By introducing an appropriate auxiliary-dependent variable, the residual symmetry is successfully localized to Lie point symmetries. Via solving the initial value problems, the finite symmetry transformations are presented. However, the solution which obtained from the residual symmetry is a special group invariant solutions. In order to find more general solution of HBK system, we further generalize the residual symmetry method to the consistent tanh expansion (CTE) method and prove that the HBK system is CTE solvable, then the resonant soliton solutions and interaction solutions among different nonlinear excitations are obtained by the CET method.