New bright soliton solutions for Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equations and bidirectional propagation of water wave surface

The governing equations for fluid flows, i.e. Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) model equations represent a water wave model. These model equations describe the bidirectional propagating water wave surface. In this paper, an auto-Bäcklund transformation is being generated by utilizing truncated Painlevé expansion method for the considered equation. This paper determines the new bright soliton solutions for [Formula: see text] and [Formula: see text]-dimensional nonlinear KP-BBM equations. The simplified version of Hirota’s technique is utilized to infer new bright soliton solutions. The results are plotted graphically to understand the physical behavior of solutions.

New interaction solutions of the similarity reduction for the integrable (2+1)-dimensional Boussinesq equation

The nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation is studied by the standard truncated Painlevé expansion. This nonlocal symmetry can be localized to the Lie point symmetry of the prolonged system by introducing two auxiliary dependent variables. The corresponding finite symmetry transformation and similarity reduction related to the nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation are studied. The rational solution, the triangle solution, two solitoff-interaction solution and the soliton–cnoidal interaction solutions for the new [Formula: see text]-dimensional Boussinesq equation are presented analytically and graphically by selecting the proper arbitrary constants.

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.

Lump, mixed lump-soliton, and periodic lump solutions of a (2+1)-dimensional extended higher-order Broer–Kaup System

In this paper, a (2+1)-dimensional extended higher-order Broer–Kaup system is introduced and its bilinear form is presented from the truncated Painlevé expansion. By taking the auxiliary function as the ansatzs including quadratic, exponential, and trigonometric functions, lump, mixed lump-soliton, and periodic lump solutions are derived. The mixed lump-soliton solutions are classified into two cases: the first one describes the non-elastic collision between one lump and one line soliton, which exhibits fission and fusion phenomena. The second one depicts the interaction consisting of one lump and two line soliton, which generates a rogue wave excited from two resonant line solitons.

Symmetry Analysis, Bäcklund Transformations, and Interaction Solutions for an AB Modified KdV System

An AB modified KdV (AB-mKdV) system which can be used to describe two-place event is studied in this manuscript. Because the AB-mKdV system is considered as a special reduction of the famous AKNS system, the properties of the AKNS system are first revealed by using symmetry analysis. The nonlocal symmetries related to truncated Painlevé expansion, the finite transformation, and the symmetry reduction solutions of the AKNS system are presented. The corresponding Bäcklund transformations and the interaction solutions of the AB-mKdV system are constructed based on the special reduction. The results demonstrate that the AB-mKdV system possesses many kinds of interaction solutions, such as the interactions between kink and soliton and kink and cnoidal waves. The soliton can be changed from bright to dark during propagation.

Rational Solutions for the (2+1)-Dimensional Modified KdV-CBS Equation

In this paper, with the help of symbolic computation, three types of rational solutions for the (2+1)-dimensional modified KdV-Calogero-Bogoyavlenkskii-Schiff equation are derived. By means of the truncated Painlevé expansion, we show that the (2+1)-dimensional modified KdV-Calogero-Bogoyavlenkskii-Schiff equation can be written as a trilinear-linear equation, from which we get explicit representation for rational solutions of the (2+1)-dimensional modified KdV-Calogero-Bogoyavlenkskii-Schiff equation.

Residual Symmetries and Bäcklund Transformations of Strongly Coupled Boussinesq–Burgers System

In this article, we construct a new strongly coupled Boussinesq–Burgers system taking values in a commutative subalgebra Z 2 . A residual symmetry of the strongly coupled Boussinesq–Burgers system is achieved by a given truncated Painlevé expansion. The residue symmetry with respect to the singularity manifold is a nonlocal symmetry. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is obtained with the help of Lie’s first theorem. Further, the linear superposition of multiple residual symmetries is localized to a Lie point symmetry, and a N-th Bäcklund transformation is also obtained.

Construction of rational solutions for the (2+1)-dimensional Broer–Kaup system

Based on the quartic–linear form of the (2[Formula: see text]+[Formula: see text]1)-dimensional Broer–Kaup system which is derived from its truncated Painlevé expansion, three kinds of rational solutions are obtained through ansatz and symbolic computation with Maple. In general, these kinds of solutions obtained from quartic–linear equation are different from the ones which are generated via bilinear equation. Figures are presented to show the dynamical features of these solutions.

Resonant Soliton and Soliton-Cnoidal Wave Solutions for a (3+1)-Dimensional Korteweg-de Vries-Like Equation

The residual symmetry of a (3+1)-dimensional Korteweg-de Vries (KdV)-like equation is constructed using the truncated Painlevé expansion. Such residual symmetry can be localized and the (3+1)-dimensional KdV-like equation is extended into an enlarged system by introducing some new variables. By using Lie’s first theorem, the finite transformation is obtained for this localized residual symmetry. Further, the linear superposition of multiple residual symmetries is localized and the n-th Bäcklund transformation in the form of the determinants is constructed for this equation. For illustration more detail, the first three multiple wave solutions-the collisions of resonant solitons are depicted. Finally, with the aid of the link between the consistent tanh expansion (CTE) method and the truncated Painlevé expansion, the explicit soliton-cnoidal wave interaction solution containing three kinds of Jacobian elliptic functions for this equation is derived.