Modified Kadomtsev Petviashvili Research Articles

Interaction solutions for mKP equation with nonlocal symmetry reductions and CTE method

The nonlocal symmetries for the modified Kadomtsev–Petviashvili (mKP) equation are obtained with the truncated Painlevé method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing auxiliary dependent variables. The finite symmetry transformations and similarity reductions related with the nonlocal symmetries are computed. The multi-solitary wave solution and interaction solutions among a soliton and cnoidal waves of the mKP equation are presented. In the meantime, the consistent tanh expansion method is applied to the mKP equation. The explicit interaction solutions among a soliton and other types of nonlinear waves such as cnoidal periodic waves and multiple resonant soliton solutions are given.

Dust ion acoustic travelling waves in the framework of a modified Kadomtsev-Petviashvili equation in a magnetized dusty plasma with superthermal electrons

For the critical values of the parameters q and V, the work (Samanta et al. in Phys. Plasma 20:022111, 2013b) is unable to describe the nonlinear wave features in magnetized dusty plasma with superthermal electrons. To describe the nonlinear wave features for critical values of the parameters q and V, we extend the work (Samanta et al. in Phys. Plasma 20:022111, 2013b). To extend the work, we derive the modified Kadomtsev-Petviashvili (MKP) equation for dust ion acoustic waves in a magnetized dusty plasma with q-nonextensive velocity distributed electrons by considering higher order coefficients of ϵ. By applying the bifurcation theory of planar dynamical systems to this MKP equation, the existence of solitary wave solutions of both types rarefactive and compressive, periodic travelling wave solutions and kink and anti-kink wave solutions is proved. Three exact solutions of these above waves are determined. The present study could be helpful for understanding the nonlinear travelling waves propagating in mercury, solar wind, Saturn and in magnetosphere of the Earth.

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.

Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

We study here the Lie symmetries, conservation laws, reductions, and new exact solutions of (2+1) dimensional Zakharov-Kuznetsov (ZK), Gardner Kadomtsev-Petviashvili (GKP), and Modified Kadomtsev-Petviashvili (MKP) equations. The multiplier approach yields three conservation laws for ZK equation. We find the Lie symmetries associated with the conserved vectors, and three different cases arise. The generalized double reduction theorem is then applied to reduce the third-order ZK equation to a second-order ordinary differential equation (ODE) and implicit solutions are established. We use the Sine-Cosine method for the reduced second-order ODE to obtain new explicit solutions of ZK equation. The Lie symmetries, conservation laws, reductions, and exact solutions via generalized double reduction theorem are computed for the GKP and MKP equations. Moreover, for the GKP equation, some new explicit solutions are constructed by applying the first integral method to the reduced equations.

RECURSION OPERATORS FOR KP, mKP AND HARRY DYM HIERARCHIES

In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev–Petviashvili (KP), modified Kadomtsev–Petviashvili (mKP) and Harry Dym under n-reduction. This shows a new inherent relationship between them. To illustrate our construction, the recursion operator are calculated explicitly for 2-reduction and 3-reduction.

DECOMPOSITION OF THE MODIFIED KADOMTSEV–PETVIASHVILI EQUATION AND ITS FINITE BAND SOLUTION

The modified Kadomtsev–Petviashvili (mKP) equation is revisited from two 1 + 1-dimensional integrable equations whose compatible solutions yield a special solution of the mKP equation in view of a transformation. By employing the finite-order expansion of Lax matrix, the mKP equation is reduced to three solvable ordinary differential equations (ODEs). The associated flows induced by the mKP equation are linearized under the Abel–Jacobi coordinates on a Riemann surface. Finally, a finite band solution expressed by Riemann-theta functions for the mKP equation is obtained through the Jacobi inversion.

Darboux transformation and multi-soliton solutions for some (2+1)-dimensional nonlinear equations

Multi-soliton solutions of some (2+1)-dimensional nonlinear equations, including the modified Kadomtsev–Petviashvili equation, are obtained with the help of the Darboux transformation method and the decomposition technique.

Soliton solutions of Burgers' equation and the modified Kadomtsev–Petviashvili equation

From the Levi spectral problem, we obtain the (1+1)-dimensional Burgers' equation and the (2+1)-dimensional modified Kadomtsev–Petviashvili equation. Through the Miura transformation, the modified Kadomtsev–Petviashvili equation is transformed into the Kadomtsev–Petviashvili equation. Then by constructing some new Darboux transformations, we obtain new soliton solutions of Burgers' equation and find solitons fusion. By solving two (1+1)-dimensional soliton equations, new soliton solutions of the modified Kadomtsev–Petviashvili equation are obtained. And then explicit solutions of the Kadomtsev–Petviashvili equation are also obtained through the Miura transformation.

SOLITON RESONANCES FOR THE MODIFIED KADOMTSEV–PETVIASHVILI EQUATIONS IN UNIFORM AND NON-UNIFORM MEDIA

The phenomena of soliton resonance can occur in (2+1)-dimensional modified Kadomtsev–Petviashvili equation (mKP). In this paper, we study the multisoliton solutions in Hirota's forms for isospectral and non-isospectral mKP. We take two-soliton solutions as examples to investigate soliton resonances occurring in uniform and non-uniform media according to different values of the phase shift parameter.

Neumann type integrable reduction for nonlinear evolution equations in 1+1 and 2+1 dimensions

A family of new Neumann type systems is given in view of the nonlinearization technique, realizing the variable separation of the modified Jaulent–Miodek hierarchy and a new coupled modified Kadomtsev–Petviashvili equation on the symplectic submanifold TSN−1. By two Casimir functions and a special solution of the Lenard eigenvalue equation, we deduce the Lax–Moser matrix of the Neumann type systems that yields integrals of motion and the constrained Hamiltonians whose vector fields are tangent to TSN−1. Based on the Dirac–Poisson bracket and a Lax equation on TSN−1, a new systematic way is proposed to prove the Liouville integrability of a family of Neumann type systems synchronously. The Dirac–Poisson bracket and the generating function are used to reveal the explicit relation between infinite dimensional integrable systems and Neumann type systems, and then we point out that compatible solutions of Neumann type systems yield the finite parametric solutions of 1+1 and 2+1 dimensional integrable nonlinear evolution equations and the finite-gap solutions of the Novikov equation.

The non-isospectral modified Kadomtsev–Petviashvili equation with self-consistent sources and its coupled system

A new algebraic method called ‘source generation procedure’ is applied to construct non-isospectral soliton equations with self-consistent sources. As results, the non-isospectral modified Kadomtsev–Petviashvili equation with self-consistent sources (mKPESCS) and its Gramm-type determinant solutions are obtained by using the source generation procedure. Furthermore, a new coupled system of the non-isospectral mKPESCS and its Pfaffian solutions are constructed.

Inelastic interactions of the multiple-front waves for the modified Kadomtsev–Petviashvili equation in fluid dynamics, plasma physics and electrodynamics

Hereby investigated is the modified Kadomtsev–Petviashvili equation which can be used to describe the wave phenomena in fluid dynamics, plasma physics and electrodynamics. By virtue of the Cole–Hopf transformation and perturbation expansion method, we obtain the single- and multiple-front waves of such equation. Based on the structures of those (2 + 1)-dimensional solutions, inelastic interactions among the multiple-front waves are discussed which might provide us with the useful information on the dynamics of the relevant physical fields.

Binary Darboux Transformation for the Modified Kadomtsev–Petviashvili Equation

Via the elementary Darboux transformation (DT) of the modified Kadomtsev–Petviashvili (mKP) equation, a binary Darboux transformation (BDT) of the mKP equation is constructed.

Darboux Transformation and Grammian Solutions for Nonisospectral Modified Kadomtsev–Petviashvili Equation with Symbolic Computation

In the present paper, under investigation is a nonisospectral modified Kadomtsev–Petviashvili equation, which is shown to have two Painlevé branches through the Painlevé analysis. With symbolic computation, two Lax pairs for such an equation are derived by applying the generalized singular manifold method. Furthermore, based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the N-th-iterated potential transformation formula in the form of Grammian is also presented.

Regular soliton solutions and singular soliton solutions for the modified Kadomtsev–Petviashvili equations

In this paper, we study the positive and the negative modified Kadomtsev–Petviashvili (mKP) equations. The mKP equations are completely integrable equations that admit N-soliton solutions and an infinite number of conserved densities. Multiple-soliton solutions are obtained for the positive model, whereas multiple–singular soliton solutions are established for the negative model. The analysis depends mainly on the Hirota’s bilinear method.

Grammian solutions and Pfaffianization of the non-isospectral modified Kadomtsev–Petviashvili equation

The Grammian determinant solutions of the non-isospectral modified Kadomtsev–Petviashvili (mKP) equation are presented. Moreover, a new non-isospectral coupled system is constructed by using the Pfaffianization procedure. Furthermore, Gramm-type Pfaffian solutions of the non-isospectral coupled system are obtained.

Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev–Petviashvili equation

Recently, the (2+1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation was decomposed into two known (1+1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants’ representations of the two known (1+1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1+1)-dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2+1)-dimensional mKP equation are presented by using the decomposition. The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.

Darboux transformations for the isospectral and nonisospectral mKP equation

Darboux transformations for the isospectral and nonisospectral modified Kadomtsev–Petviashvili (mKP) equations are investigated. In the isospectral case it is an auto-Darboux transformation; however, in the nonisospectral case it is not auto-Darboux tranformation.

(2+1)-Dimensional Electron Acoustic Solitary Waves inan Unmagnetized Collisionless Plasma with Vortex-Like HotElectrons

In this paper, (2+1)-dimensional electron acoustic waves (EAW) inan unmagnetized collisionless plasma have been studied by thelinearized method and the reductive perturbation technique,respectively. The dispersion relation and a modifiedKadomtsev–Petviashvili (KP) equation have been obtained for theEAW in the plasma considering a cold electron fluid and avortex-like hot electrons. It is found from some numerical resultsthat the parameter β (the ratio of the free hot electrontemperature to the hot trapped electron temperature) effects on theamplitude and the width of the electron acoustic solitary waves(EASW). It can be indicated that the free hot electron temperatureand the hot trapped electron temperature have very importanteffect on the characters of the propagation for the EASW.

A modified Zabolotskaya–Khokhlov equation for systems having small quadratic nonlinearity

We examine three-dimensional, weakly diffracting, weakly nonlinear waves propagating in a uniform medium. A method of multiple scales is used to derive an extension of the Zabolotskaya–Khokhlov equation which is valid for systems having a quadratic nonlinearity parameter which is either zero or small. The effects of relaxation, dissipation, and dispersion are also considered resulting in general forms of the modified Kadomtsev–Petviashvili equation and the modified Khokhlov–Zabolotskaya–Kuznetsov equation. Further results include the delineation of the simplifications required when the system is such that waves propagate isotropically, i.e., where the shock and characteristic speeds are independent of the direction of propagation. We also illustrate our results with physical examples corresponding to the nonlinear acoustics of pressurized gases with a uniform wind and obliquely propagating Alfvén waves.