mKP Equation Research Articles

Soliton interaction in the modified Kadomtsev–Petviashvili-(II) equation

ABSTRACTWe study soliton interaction in the Modified Kadomtsev–Petviashvili-(II) equation (MKP-(II)) using the totally non-negative Grassmannian. One constructs the multi-kink soliton of MKP equation using the -function and the Binet–Cauchy formula, and then investigates the interaction between kink solitons and line solitons. Especially, Y-type kink-soliton resonance, O-type kink soliton and P-type kink soliton of X-shape are investigated. Their amplitudes of interaction are computed after choosing appropriate phases.

Self-Consistent Sources Extensions of Modified Differential-Difference KP Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11601247 and 11605096, the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant Nos. 2016MS0115 and 2015MS0116 and the Innovation Fund Programme of Inner Mongolia University No. 20161115

In this paper, we investigate a modified differential-difference KP equation which is shown to have a continuum limit into the mKP equation. It is also shown that the solution of the modified differential-difference KP equation is related to the solution of the differential-difference KP equation through a Miura transformation. We first present the Grammian solution to the modified differential-difference KP equation, and then produce a coupled modified differential-difference KP system by applying the source generation procedure. The explicit N-soliton solution of the resulting coupled modified differential-difference system is expressed in compact forms by using the Grammian determinant and Casorati determinant. We also construct and solve another form of the self-consistent sources extension of the modified differential-difference KP equation, which constitutes a Bäcklund transformation for the differential-difference KP equation with self-consistent sources.

Darboux transformation and nonautonomous solitons for a modified Kadomtsev–Petviashvili equation with variable coefficients

We study a modified Kadomtsev–Petviashvili (mKP) equation with variable coefficients, which has important applications in fluid dynamics, ferromagnetics and plasma. The Lax pair, Hirota bilinear form and Darboux transformation (DT) are obtained through the two-singular manifold method. The N-soliton solution with Grammian type in a compact determinant representation is derived from the N-fold DT. A complete classification of the mKP-type solitons is given, namely, the bright and dark solitons, as well as resonant bright and dark solitons. Dynamics of the nonautonomous solitons with parabolic, periodic and kink shapes under different kinds of variable-coefficient oscillations are shown. Interactions between the nonautonomous two bright solitons and two resonant bright solitons are discussed through the asymptotic analysis method.

Soliton interactions and Bäcklund transformation for a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili equation in fluid dynamics

In this paper, we investigate a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili (mKP) equation in fluid dynamics. With the binary Bell-polynomial and an auxiliary function, bilinear forms for the equation are constructed. Based on the bilinear forms, multi-soliton solutions and Bell-polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton interactions are presented. Based on the graphic analysis, Parametric conditions for the existence of the shock waves, elevation solitons and depression solitons are given, and it is shown that under the condition of keeping the wave vectors invariable, the change of [Formula: see text] and [Formula: see text] can lead to the change of the solitonic velocities, but the shape of each soliton remains unchanged, where [Formula: see text] and [Formula: see text] are the variable coefficients in the equation. Oblique elastic interactions can exist between the (i) two shock waves, (ii) two elevation solitons, and (iii) elevation and depression solitons. However, oblique interactions between (i) shock waves and elevation solitons, (ii) shock waves and depression solitons are inelastic.

Dynamics of the positron acoustic waves in electron–positron–ion magnetoplasmas

Dynamics of the positron acoustic waves in electron–positron–ion (e–p–i) magnetoplasmas with \(\kappa \)-distributed hot electrons and positrons is investigated in the frameworks of the Kadomtsev–Petviashili (KP) and modified Kadomtsev–Petviashili (mKP) equations. Employing the reductive perturbation technique, the KP and mKP equations are derived. Using the bifurcation theory of planar dynamical systems, the positron acoustic solitary wave solutions, the kink and anti-kink wave solutions are obtained. Considering an external periodic perturbation in the electron–positron–ion magnetoplasmas, the perturbed KP and mKP equations are studied via some qualitative and quantitative approaches. To corroborate in the fact that the perturbed KP and mKP equations can indeed give rise to the quasiperiodic and chaotic motions, the phase plane plots, time series plots, and the Poincare section are used. The quasiperiodic and developed chaos can be observed for the perturbed positron acoustic waves. The frequency (\(\omega \)) of the external periodic perturbation plays the role of the switching parameter in chaotic motions of the perturbed positron acoustic waves through quasiperiodic route to chaos. This work can be useful to understand the dynamics of nonlinear electromagnetic perturbations in space and laboratory plasmas consisting of \(\kappa \)-distributed hot electrons and positrons.

Effects of trapped electrons on the oblique propagation of ion acoustic solitary waves in electron-positron-ion plasmas

The characteristics of the nonlinear oblique propagation of ion acoustic solitary waves in unmagnetized plasmas consisting of Boltzmann positrons, trapped electrons and ions are investigated. The modified Kadomtsev-Petviashivili (mKP) equation is derived employing the reductive perturbation technique. The parametric effects on phase velocity, Sagdeev potential, amplitude and width of solitons, and electrostatic ion acoustic solitary structures are graphically presented with the relevant physical explanations. This study may be useful for the better understanding of physical phenomena concerned in plasmas in which the effects of trapped electrons control the dynamics of wave.

Dust Acoustic Compressive Waves in a Warm Dusty Plasma Having Non-Thermal Ions and Non-Isothermal Electrons

In this article an investigation is presented on the properties of dust acoustic (DA) compressive solitary wave propagation in an adiabatic dusty plasma, including the effect of non-thermal positive and negative ions and non-isothermal electrons. The reductive perturbation method has been employed to derive the lower degree modified Kadomtsev-Petviashivili (mK-P), 3D Schamel-Korteweg-de-Vries equation or modified Kadomtsev-Petviashivili (mK-P) equations for dust acoustic solitary waves in a homogeneous, unmagnetized and collisionless plasma whose constituents are non-isothermal electrons, singly charged positive and negative non-thermal ions and massive charged dust particles. The stationary analytical solutions of the lower degree mK-P and mK-P equations are numerically analyzed, where the effect of various dusty plasma constituents on DA solitary wave propagation is taken into account. It is observed that both the ions in dusty plasma play a key role in the formation of DA compressive solitary waves, and also the ion concentration and non-isothermal electrons control the transformation of the compressive potentials of the waves.

Two (2+1)-dimensional expanding dynamical systems associated to the mKP hierarchy

An isospectral problem with a parameter ϵ is presented, for which a modified KdV (mKdV) hierarchy is easily derived from the Tu scheme. Then two different expanding integrable models are obtained, respectively. The main purpose for this focuses on deriving a (2+1)-dimensional mKdV hierarchy (called the mKP hierarchy) and the corresponding different (2+1)-dimensional expanding models (including the linear and nonlinear), whose Hamiltonian structures are obtained by employing an identity developed by us in the paper. As the reduced consequences, the linear and nonlinear (2+1)-dimensional expanding models of the mKP equation are generated.

Modified Kadomtsev–Petviashvili Equation in (3+1) Dimensions: Multiple Front-Wave Solutions

A modified Kadomtsev–Petviashvili (mKP) equation in (3+1) dimensions is presented. We reveal multiple front-waves solutions for this higher-dimensional developed equation, and multiple singular front-wave solutions as well. The constraints on the coefficients of the spatial variables, that assure the existence of these multiple front-wave solutions are investigated. We also show that this equation fails the Painlevé test, and we conclude that it is not integrable in the sense of possessing the Painlevé property, although it gives multiple front-wave solutions.

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.

Algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations via Hamiltonian approach

The algebraic-geometrical solutions of three (2 + 1)-dimensional equations (including mKP equation and coupled mKP equation) are discussed by Hamiltonian approach. First, the Poisson structure on CN × RN is introduced to give a Hamiltonian system associated with the derivative nonlinear Schrödinger (DNLS) hierarchy. The Hamiltonian system is proved to be Liouville integrable, accordingly the solutions of three (2 + 1)-dimensional nonlinear equations can be solved by three compatible Hamiltonian flows. Second, the canonical separated variables and Hamilton–Jacobi theory is used to definite action-angle variables for Hamiltonian flows. At last, by Riemann–Jacobi inversion, the algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations are obtained. Besides, the algebraic-geometrical solutions of the first two DNLS equations are also given.

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.

Generalized Cauchy matrix approach for lattice KP-type equations

In this paper we generalize Cauchy matrix approach to study the lattice KP-type equations, including lattice potential KP equation, lattice potential mKP equation, lattice Schwarzian KP equation and lattice KP-type NQC equation. By solving the condition equation set, we give explicit formulae for all possible r, ts and M. Consequently, we get more solutions than usual multi-soliton solutions for these lattice KP-type equations.

BIFURCATIONS OF EXACT TRAVELING WAVE SOLUTIONS FOR THE (2 + 1)-DIMENSIONAL mKP EQUATION

For the (2 + 1)-dimensional mKP equation, what is the dynamical behavior of its traveling wave solutions and how does it depend on the parameters of the systems? This paper will answer these questions by using the methods of dynamical systems. Thirteen exact explicit parametric representations of the traveling wave solutions are given. Bifurcations of phase portraits of the corresponding singular traveling wave system are shown.

A new extended dispersionless mKP hierarchy and hodograph solution

In this article, a new extended dispersionless mKP hierarchy (exdmKPH) is constructed to obtain two types of dispersionless mKP equations with self-consistent sources (dmKPSCS) and their associated conservation equations. Two reductions of this hierarchy are used to get two types of the corresponding dispersionless mKdV equations with self-consistent sources (dmKdVSCS). A hodograph solution for the first type of dmKdVSCS and Bäcklund transformation between the extended dispersionless KP hierarchy (exdKPH) and exdmKPH are also given.

Multi-front waves for extended form of modified Kadomtsev-Petviashvili equation

An extended form of the modified Kadomtsev-Petviashvili (mKP) equation is investigated. The simplified form of the Hirota bilinear method established by Hereman and Nuseir is employed. Multi-front wave solutions are formally derived to the extended mKP equation and the mKP equation. The results show that the extension terms do not kill the integrability of the mKP equation.

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.

Soliton Resonances of the Nonisospectral Modified Kadomtsev-Petviashvili Equation

Many equations possess soliton resonances phenomenon, this paper studies the soliton resonances of the nonisospectral modified Kadomtsev-Petviashvili (mKP) equation by asymptotic analysis.

The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons

Based on the eigenfunction symmetry constraint of the q-deformed modified KP hierarchy, a q-deformed mKP hierarchy with self-consistent sources (q-mKPHSCSs) is constructed. The q-mKPHSCSs contain two types of q-deformed mKP equation with self-consistent sources. By the combination of the dressing method and the method of variation of constants, a generalized dressing approach is proposed to solve the q-deformed KP hierarchy with self-consistent sources (q-KPHSCSs). Using the gauge transformation between the q-KPHSCSs and the q-mKPHSCSs, the q-deformed Wronskian solutions for the q-KPHSCSs and the q-mKPHSCSs are obtained. The one-soliton solutions for the q-deformed KP (mKP) equation with a source are given explicitly.

From nonlinear Schrödinger hierarchy to some (2+1)-dimensional nonlinear pseudodifferential equations

The Poisson structure on CN×RN is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schrödinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev–Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev–Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel–Jacobi coordinates, and Riemann–Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.