General high-order localized waves to the Bogoyavlenskii–Kadomtsev–Petviashvili equation

Nonlinear Dynamics - In this work, the Bogoyavlenskii–Kadomtsev–Petviashvili equation is investigated. By means of the Hirota bilinear system and Pfaffian, we demonstrate that the...     

Various exact analytical solutions of a variable-coefficient Kadomtsev–Petviashvili equation

Nonlinear Dynamics - Under investigation is a generalized (3 + 1)-dimensional variable- coefficient Kadomtsev– Petviashvili equation in fluid mechanics. Various exact analytical solutions are...     

A new type of multiple-lump and interaction solution of the Kadomtsev–Petviashvili I equation

Nonlinear Dynamics - In this paper, the solution in the form of Grammian of the Kadomtsev–Petviashvili I equation is employed to investigate a new type of multiple-lump solution. The bound...     

Interactions among different types of nonlinear waves described by the Kadomtsev–Petviashvili equation

In nonlinear science, the interactions among solitons are well studied because the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlevé waves. In this paper, taking the Kadomtsev–Petviashvili (KP) equation as an illustration model, a new method is established to find interactions among different types of nonlinear waves. The nonlocal symmetries related to the Darboux transformation (DT) of the KP equation is localized after embedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions. It is shown that the essential and unique role of the DT is to add an additional soliton on a Boussinesq-type wave or a KdV-type wave, which are two basic reductions of the KP equation.     

Novel characteristics of lump and lump–soliton interaction solutions to the generalized variable-coefficient Kadomtsev–Petviashvili equation

Nonlinear Dynamics - With the inhomogeneities of media taken into account, a generalized variable-coefficient Kadomtsev–Petviashvili (vcKP) equation is proposed to model nonlinear waves in...     

Some group-invariant solutions of potential Kadomtsev–Petviashvili equation by using Lie symmetry approach

Nonlinear Dynamics - A variety of closed-form solutions such as multiple-front wave, kink wave, waves interaction, curve-shaped multisoliton, parabolic and stationary wave solutions have been...     

Resonance and deflection of multi-soliton to the (2+1)-dimensional Kadomtsev–Petviashvili equation

In this work, a modified three-soliton method with a perturbation parameter is proposed, and it is applied to the (2+1)-dimensional Kadomtsev–Petviashvili equation (KP), and new breather multi-soliton solutions are obtained. The dependence of new mechanical structures on the perturbed parameter for multi-soliton including resonance and deflection for KP equation are investigated and exhibited.     

Bilinear form and soliton interactions for the modified Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics

In this paper, we investigate the modified Kadomtsev–Petviashvili (mKP) equation for the nonlinear waves in fluid dynamics and plasma physics. By virtue of the rational transformation and auxiliary function, new bilinear form for the mKP equation is constructed, which is different from those in previous literatures. Based on the bilinear form, one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. Propagation and interactions of shock and solitary waves are investigated analytically and graphically. Parametric conditions for the existence of the shock, elevation solitary, and depression solitary waves are given. From the two-soliton solutions, we find that the (i) parallel elastic interactions can exist between the (a) shock and solitary waves, and (b) two elevation/depression solitary waves; (ii) oblique elastic interactions can exist between the (a) shock and solitary waves, and (b) two solitary waves; (iii) oblique inelastic interactions can exist between the (a) two shock waves, (b) two elevation/depression solitary waves, and (c) shock and solitary waves.     

Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation

Nonlinear Dynamics - In this work, a non-isospectral and variable-coefficient Kadomtsev–Petviashvili equation is considered using Hirota’s bilinear form and a direct assumption with...     

Abundant lump and lump–kink solutions for the new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

Nonlinear Dynamics - By utilizing the Hirota’s bilinear form and symbolic computation, abundant lump solutions and lump–kink solutions of the new (3 + 1)-dimensional...     

A direct Bäcklund transformation for a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation

In this paper, a Kadomtsev–Petviashvili–Boussinesq-like equation in (3+1)-dimensions is firstly introduced by using the combination of the Hirota bilinear Kadomtsev–Petviashvili equation and Boussinesq equation in terms of function f. And then a direct bilinear Bäcklund transformation of this new model is constructed, which consists of seven bilinear equations and ten arbitrary parameters. Based on this constructed bilinear Bäcklund transformation, some classes of exponential and rational traveling wave solutions with arbitrary wave numbers are presented.     

Geometrical patterns of time variable Kadomtsev–Petviashvili (I) equation that models dynamics of waves in thin films with high surface tension

Lump solutions are a prominent option for numerous models of nonlinear evolution. The intention of this research is to explore the variable coefficients Kadomtsev–Petviashvili equation. We auspiciously provide multiple soliton and M-lump solutions to this equation. Additionally, the presented results are also supplied with collision phenomena. Owing of its essential role, we employ appropriate parameter values to emphasis the physical characteristics of the provided results using 3D and contour charts. The outcomes of this work convey the physical characteristics of lump and lump interactions that occur in many dynamical regimes.

     

Novel evolutionary behaviors of localized wave solutions and bilinear auto-Bäcklund transformations for the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation

Water waves are common phenomena in nature, which have attracted extensive attention of researchers. In the present paper, we first deduce five kinds of bilinear auto-Bäcklund transformations of the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation starting from the specially exchange identities of the Hirota bilinear operators; then, we construct the N-soliton solutions and several new structures of the localized wave solutions which are studied by using the long wave limit method and the complex conjugate condition technique. In addition, the propagation orbit, velocity and extremum of the first-order lump solution on (x, y)-plane are studied in detail, and seven mixed solutions are summarized. Finally, the dynamical behaviors and physical properties of different localized wave solutions are illustrated and analyzed. It is hoped that the obtained results can provide a feasibility analysis for water wave dynamics.

     

A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions

Searching for higher-dimensional integrable models is one of the most significant and challenging issues in nonlinear mathematical physics. This paper aims to extend the classic lower-dimensional integrable models to arbitrary spatial dimension. We investigate the celebrated Kadomtsev–Petviashvili (KP) equation and propose its (n+1)-dimensional integrable extension. Based on the singularity manifold analysis and binary Bell polynomial method, it is found that the (n+1)-dimensional generalized KP equation has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, Bäcklund transformation as well as infinite conservation laws, and thus the (n+1)-dimensional generalized KP equation is proven to be completely integrable. Moreover, various types of localized solutions can be constructed starting from the N-soliton solutions. The abundant interactions including overtaking solitons, head-on solitons, one-order lump, two-order lump, breather, breather-soliton mixed solutions are analyzed by some graphs.

     

The mixed interaction of localized,breather, exploding and solitary wave for the (3+1)-dimensional Kadomtsev–Petviashvili equation in fluid dynamics

Nonlinear Dynamics - The (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation with weak nonlinearity, dispersion and perturbation can denote the development of the long waves and the...     

Studies on the breather solutions for the $$\mathbf{(2+1)}$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids and plasmas

Nonlinear Dynamics - In this paper, we study the $$(2 + 1)$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation, which has certain applications in fluids and plasmas. Via the...     

Localized excitation and folded solitary wave for an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

Nonlinear Dynamics - In this work, we employ the multi-linear variable separation approach to derive variable separation solution for a new extended (3+1)-dimensional B-type...