gKP Equation Research Articles

Weak compactons of nonlinearly dispersive KdV and KP equations

AbstractA weak formulation is devised for the equation, which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.

Dynamics of quasi-periodic, bifurcation, sensitivity and three-wave solutions for (n + 1)-dimensional generalized Kadomtsev-Petviashvili equation.

This study endeavors to examine the dynamics of the generalized Kadomtsev-Petviashvili (gKP) equation in (n + 1) dimensions. Based on the comprehensive three-wave methodology and the Hirota's bilinear technique, the gKP equation is meticulously examined. By means of symbolic computation, a number of three-wave solutions are derived. Applying the Lie symmetry approach to the governing equation enables the determination of symmetry reduction, which aids in the reduction of the dimensionality of the said equation. Using symmetry reduction, we obtain the second order differential equation. By means of applying symmetry reduction, the second order differential equation is derived. The second order differential equation undergoes Galilean transformation to obtain a system of first order differential equations. The present study presents an analysis of bifurcation and sensitivity for a given dynamical system. Additionally, when an external force impacts the underlying dynamic system, its behavior resembles quasi-periodic phenomena. The presence of quasi-periodic patterns are identified using chaos detecting tools. These findings represent a novel contribution to the studied equation and significantly advance our understanding of dynamics in nonlinear wave models.

The positive multi-complexiton solution to a generalized Kadomtsev–Petviashvili equation

Recently, Lin et al. (Appl. Math. Lett. 78 (2018) 112–117) established the resonant multiple-wave solution of a generalized Kadomtsev–Petviashvili (gKP) equation with applications in applied sciences using the linear superposition technique. In the present paper, according to the results obtained by Lin et al. and adopting a systematic method established by Zhou and Manukure, the positive multi-complexiton solution to the generalized Kadomtsev–Petviashvili equation is constructed for the first time. Several simulations in three dimensions are conducted formally in order to examine the dynamic behavior of positive multi-complexiton solutions, particularly the positive single-, double-, and triple-complexion waves. The current research outcomes definitely will enrich studies related to the gKP equation and its exact solutions.

New conservation laws of the Boussinesq and generalized Kadomtsev–Petviashvili equations via homotopy operator

According to the tools of linear algebra and calculus of variations, the conservation laws of Boussinesq and generalized Kadomtsev–Petviashvili (gKP) equations are investigated using multipliers and scaling methods. Using the Euler–Lagrange operator, the determining equations are calculated to find the multipliers of Boussinesq equation and the actual density of gKP equation. Then, the flux and density pairs of Boussinesq equation and the corresponding flux of gKP equation are obtained through 2-dimensional homotopy operator.

Some Exact Solutions to Generalized Kadomtsev-Petviashvili Equation

Most of the papers have explored the interactions between solitons with a zero background, while reports about exact solutions for nonzero background are rare. Hence, this paper is aimed at exploring the breather, lump, and interaction solutions with a small perturbation to ( 2 + 1 )-dimensional generalized Kadomtsev-Petviashvili (gKP) equation. General high-order periodic breather solutions are obtained using Hirota’s bilinear method with a small perturbation. At the same time, combining the use of long wave limit methods and module resonance constraints, general lump solutions and mixed solutions to gKP equation are generated. Finally, the space-time structures of the breather solutions, lump solutions, and interaction solutions are investigated and discussed.

M-lump waves and their interaction with multi-soliton solutions for a generalized Kadomtsev–Petviashvili equation in (3+1)-dimensions

The aim of this work is a study of a generalized Kadomtsev–Petviashvili equation in (3+1)-dimensions that arise in fluid mechanics and plasma physics. We use a long-wave method to construct some new solutions for a gKP equation such as single-M-lump wave, double-M-lump wave, triple-M-lump wave, interaction phenomena of one-M-lump with one-soliton and two-soliton solutions, as well as an interaction between two-M-lump and one-soliton solutions are constructed. To our knowledge presented solutions are new and have never been described before. Moreover, obtained solutions are shown graphically to more understand its physical phenomena.

Residual symmetries, consistent-Riccati-expansion integrability, and interaction solutions of a new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation

With the aid of the Painlevé analysis, we obtain residual symmetries for a new (3+1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the (3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.

A trilinear analysis for lump-type wave, breather wave and BK-type wave solutions of a (3+1)-dimensional [formula omitted]-gKP equation

This paper investigates some novel exact analytical solutions, including lump-type wave, breather wave as well as BK-type wave solutions for a (3+1)-dimensional p¯-gKP equation. Firstly, the trilinear p¯-gKP equation is well established by the multivariate trilinear operators. Secondly, three types of wave solutions for the p¯-gKP equation are obtained by utilizing the trilinear method. A kind of new interaction solutions among the breather wave and the kink-type waves are also generated, i.e. the so called BK-type waves, which enriches the types of solutions of PDEs. Through the way of resetting different space constants, we adjust the coordinates of kink-type waves in order for them colliding with the breather wave, after that, the transformed kink-type waves gradually swallow the breather wave. Finally, the solitary wave solutions for the (3+1)-dimensional gKP equation are successfully constructed through using a perturbation method.

Modulational instability, multiple Exp-function method, SIVP, solitary and cross-kink solutions for the generalized KP equation

The multiple Exp-function method is employed for seeking the multiple soliton solutions to the generalized (3+1)-dimensional Kadomtsev-Petviashvili (gKP) equation, where contains one-wave, two-wave, and triple-wave solutions. The periodic wave including (exponential, $ \cosh $ hyperbolic, and $ \cos $ periodic), cross-kink containing (exponential, $ \sinh $ hyperbolic, and $ \sin $ periodic), and solitary containing (exponential, $ \tanh $ hyperbolic, and $ \tan $ periodic) wave solutions are obtained. In continuing, the modulation instability is engaged to discuss the stability of obtained solutions. Also, the semi-inverse variational principle is applied for the gKP equation with four major cases. The physical phenomena of these received multiple soliton solutions are analyzed and demonstrated in figures by choosing the specific parameters. By means of symbolic computation these analytical solutions and corresponding rogue waves are obtained with the help of Maple software. Via various three-dimensional, curve, and density charts, dynamical characteristics of these waves are exhibited.

Lie symmetries, optimal system and group-invariant solutions of the (3+1)-dimensional generalized KP equation

By applying the Lie symmetry method, abundant group-invariant solutions are constructed for a (3+1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation, which provides a water wave model for long waves of small amplitude with weakly non-linear restoring forces and frequency dispersion. Infinitesimal generators of symmetries, the commutation table of the symmetry Lie algebra, and an optimal system of one-dimensional Lie symmetry sub-algebras are presented. The governing gKP equation is reduced into several nonlinear ordinary differential equations utilizing thrice symmetry reductions. Consequently, abundant group invariant solutions are obtained in the shapes of different dynamical wave structures of solitons, multi-solitons, W-shaped solitons, doubly solitons, kink-type solitons, lump-type solitons interaction between parabolic waves and lump solitons, and annihilation multi-solitons profiles. The physical interpretation of the resulting soliton solutions is illustrated by three dimensional graphical through numerical simulation. The obtained group-invariant solutions involve many arbitrary functions, thereby exhibiting rich physical structures and including the existing solutions in the literature.

Multi-lump or lump-type solutions to the generalized KP equations in $$(N+1)$$-dimensions

We aim to explore lump or lump-type solutions to the generalized Kadomtsev–Petviashvili (gKP) equations in $$(N+1)$$-dimensions via the long wave limit technique. The construction procedure for presenting lump or lump-type solutions is improved. The key step is that all the involved parameters are extended to the complex field. We first furnish lump solutions from the corresponding soliton solutions to the $$(2+1)$$-dimensional gKP equation. In particular, a general class of multi-lump solutions of the $$(2+1)$$-dimensional gKP equation can be obtained. It is then shown that there exist lump-type solutions to the $$(N+1)$$-dimensional gKPI equations with $$N \ge 3$$ by means of the improved long wave limit technique.

Homoclinic breather waves, rogue waves and solitary waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation

PurposeThe purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.Design/methodology/approachThe authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.FindingsThe results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.Originality/valueThese results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Problem of stability of multidimensional solutions of the BK class equations in space plasma

The problem of stability of the multidimensional solutions of the BK class equations describing the nonlinear waves which are forming on the low-frequency branch of oscillations in plasma for cases when β≡4πnT/B2≪1 and β>1 is studied. In first case, for ω<ωB=eB/Mc,kλD≪1 the FMS waves are excited, and their dynamics under conditions kx2≫k⊥2, vx≪cA near the cone of θ=arctan(M/m)1/2, is described by the equation of the BK class known as the GKP equation for magnetic field h=B∼/B with due account of the high order dispersive correction defined by values of plasma parameters and angle θ=(B,k). In another case, the dynamics of the finite-amplitude Alfvén waves propagating near-to-parallel to B is described by the equation of the same class known as the 3-DNLS equation for h=(By+iBz)/2B|1-β|. To study the stability of multidimensional solutions in both cases the method of investigation of the Hamiltonian bounding with deformation conserving momentum by solving the variation problem is used. As a result, we have obtained the conditions of existence of the 2D and 3D soliton solutions in the BK system for cases of the GKP and 3-DNLS equation (i.e. for the FMS and Alfvén waves, respectively) in dependence on the equations’ coefficients, i.e. on the parameters of both plasma and wave.

Analytic study of solutions for a (3 + 1) -dimensional generalized KP equation

The (3 + 1) -dimensional generalized KP (gKP) equation is an important nonlinear partial differential equation in theoretical and mathematical physics which can be used to describe nonlinear wave motion. Through the Hirota bilinear method, one-solition, two-solition and N-solition solutions are derived via symbolic computation. Two classes of lump solutions, rationally localized in all directions in space, to the dimensionally reduced cases in (2 + 1)-dimensions, are constructed by using a direct method based on the Hirota bilinear form of the equation. It implies that we can derive the lump solutions of the reduced gKP equation from positive quadratic function solutions to the aforementioned bilinear equation. Meanwhile, we get interaction solutions between a lump and a kink of the gKP equation. The lump appears from a kink and is swallowed by it with the change of time. This work offers a possibility which can enrich the variety of the dynamical features of solutions for higher-dimensional nonlinear evolution equations.

Modeling ultrashort electromagnetic pulses with a generalized Kadomtsev–Petviashvili equation

In this paper we derive a properly scaled model for the nonlinear propagation of intense, ultrashort, mid-infrared electromagnetic pulses (10–100 femtoseconds) through an arbitrary dispersive medium. The derivation results in a generalized Kadomtsev–Petviashvili (gKP) equation. In contrast to envelope-based models such as the Nonlinear Schrödinger (NLS) equation, the gKP equation describes the dynamics of the field’s actual carrier wave. It is important to resolve these dynamics when modeling ultrashort pulses. We proceed by giving an original proof of sufficient conditions on the initial pulse for a singularity to form in the field after a finite propagation distance. The model is then numerically simulated in 2D using a spectral-solver with initial data and physical parameters highlighting our theoretical results.

Solitary waves, rogue waves and homoclinic breather waves for a (2 + 1)-dimensional generalized Kadomtsev–Petviashvili equation

We study a (2 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which characterizes the formation of patterns in liquid drops. By using Bell’s polynomials, an effective way is employed to succinctly construct the bilinear form of the gKP equation. Based on the resulting bilinear equation, we derive its solitary waves, rogue waves and homoclinic breather waves, respectively. Our results can help enrich the dynamical behavior of the KP-type equations.

Study of lump solutions to dimensionally reduced generalized KP equations

In this paper, a new (\(3+1\))-dimensional generalized KP (gKP) equation is presented, and two classes of lump solutions, rationally localized in all directions in the space, to the dimensionally reduced cases in (\(2\,+\,1\))-dimensions are derived as well. The proposed method in this work is based on a Hirota bilinear differential equation, which implies that we can build the lump solutions to the presented reduced gKP equation from positive quadratic function solutions to the aforementioned bilinear equation. Moreover, there are totally six free parameters in the resulting lump solutions, so that we can get the sufficient and necessary conditions guaranteeing analyticity and rational localization of the solutions by using these six free parameters. In the meantime, two special cases are plotted as illustrative examples and some contour plots with different determinant values are given to show that the corresponding lump solution tends to zero when the determinant approaches zero.