mKP Equation Research Articles

Nonlinear ion-acoustic waves in weak magnetic fields with vortex-like electron distribution

The two-dimensional dynamics of nonlinear ion-acoustic waves in a weakly magnetized plasma comprising cold ions and trapped as well as free electrons is considered. It is shown that owing to departure from the Boltzmann electron distribution to a vortex-like one, the dynamics of small but finite-amplitude ion-acoustic waves is governed by a new nonlinear equation which is valid for both unmagnetized and magnetized plasmas. For exactly vanishing magnetic fields the modified Kadomtsev-Petviashvili (mKP) equation is recovered. For weak magnetic fields, however, the dynamics is mainly different from the mKP equation, depending on the amplitude. By increasing the magnetic field, the new equation becomes similar (but not identical) to the modified Zakharov-Kuznetsov (mZK) equation, which is fulfilled for very strong magnetic fields. The plane periodic and solitary wave solutions of this equation are obtained using the appropriate scalings.

Exact solutions of the Wick-type stochastic mKP equation

Abstract Variable coefficient and Wick-type stochastic mKP equations are researched, and their exact soliton solutions are showed by using the homogeneous balance and Hermite transform in the white noise space.

Algebro-Geometric Solution to the Modified Kadomtsev–Petviashvili Equation

The modified Kadomtsev–Petviashvili (mKP) equation is split into two soliton equations in the modified Jaulent–Miodek (mJM) hierarchy, and further into integrable finite-dimensional Hamiltonian systems (FDHSs) using the nonlinearization of Lax pairs. The Abel–Jacobi coordinate is introduced to linearize the mKP flow such that its solution is reduced as a linear superposition, expressed in the Abel–Jacobi variable. Finally, the algebro-geometric solution of the mKP equation is given via the Jacobi inversion.

Relation among nonlinear evolution equations and their reductions

Abstract The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.

A new constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources

The mKP equation with self-consistent sources (mKPESCS) is treated in the framework of the constrained mKP hierarchy. We introduce a new constrained mKP hierarchy which may be viewed as the stationary hierarchy of the mKP hierarchy with self-consistent sources. This offers a natural way to obtain the Lax representation for the mKPESCS. Based on the conjugate Lax pairs, we construct the generalized Darboux transformation with arbitrary functions in time t for the mKPESCS which, in contrast with the Darboux transformation for the mKP equation, provides a nonauto-Bäcklund transformation between two mKPESCSs with different degrees. The formula for n -times repeated generalized Darboux transformation is proposed and enables us to find the rational solutions (including the lump solutions), soliton solutions and the solutions of breather type of the mKPESCS.

The kink-soliton and antikink-soliton in quasi-one-dimensional nonlinear monoatomic lattice

The quasi-one-dimensional nonlinear monoatomic lattice is analyzed. The kink-soliton and antikink-soliton are presented. When the interaction of the lattice is strong in the x-direction and weak in the y-direction, the two-dimensional (2D) lattice changes to a quasi-one-dimensional lattice. Taking nearest-neighbor interaction into account, the vibration equation can be transformed into the KPI, KPII and MKP equation. Considering the cubic nonlinear potential of the vibration in the lattice, the kink-soliton solution is presented. Considering the quartic nonlinear potential and the cubic interaction potential, the kink-soliton and antikink-soliton solutions are presented.

On the decomposition of the modified Kadometsev–Petviashvili equation and the connection between the KN equation and a completely integrable finite-dimensional Hamiltonian system

The (2+1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation is decomposed into the known (1+1)-dimensional Kaup–Newell (KN) equation. By using the nonlinearization of the Lax pair, a classically integrable Hamiltonian system in the Liouville sense and the involutive solution of the mKP equation (1.1) are obtained from the first two nontrivial KN equations.

Some Special Types of Multisoliton Solutions of the Modified Kadomtsev-Petviashvili Equation**The project supported by Natural Science Foundation of Zhejiang Province of China under Grant No. 100039

Using the standard truncated Painlevé analysis approach, we have obtained some new special types of multisoliton solutions of a (2+1)-dimensional integrable model, the modified Kadomtsev-Petviashvili (mKP) equation.

Jacobi elliptic function solutions of nonlinear wave equations via the new sinh-Gordon equation expansion method

In this paper, based on the well-known sinh-Gordon equation, a new sinh-Gordon equation expansion method is developed. This method transforms the problem of solving nonlinear partial differential equations into the problem of solving the corresponding systems of algebraic equations. With the aid of symbolic computation, the procedure can be carried out by computer. Many nonlinear wave equations in mathematical physics are chosen to illustrate the method such as the KdV-mKdV equation, (2+1)-dimensional coupled Davey–Stewartson equation, the new integrable Davey–Stewartson-type equation, the modified Boussinesq equation, (2+1)-dimensional mKP equation and (2+1)-dimensional generalized KdV equation. As a consequence, many new doubly-periodic (Jacobian elliptic function) solutions are obtained. When the modulus m → 1 or 0, the corresponding solitary wave solutions and singly-periodic solutions are also found. This approach can also be applied to solve other nonlinear differential equations.

Solitary Wave Interaction in a Cold Collisionless Plasma

The interaction between two solitary waves in a cold collisionless plasma having different velocities has been studied using the multiscale expansion method. The propagation of nonlinear waves through a cold collisionless plasma is described by mKdV and KdV equations in (1+1) dimensions and mKP and KP equations in (2+1) dimensions for different orders of a small perturbation parameter under the same time and space scale transformations. The interacting system of equations are derived for mKdV and KdV equations as well as for mKP and KP equations. We have also analysed the interacting system of equations in (1+1) dimensions.

Algebro-geometric solutions of (2+1)-dimensional coupled modified Kadomtsev–Petviashvili equations

New (2+1)-dimensional integrable coupled modified Kadomtsev–Petviashvili (mKP) equations are proposed with the help of known (1+1)-dimensional soliton equations. The (2+1)-dimensional coupled mKP equations are decomposed into systems of solvable ordinary differential equations. The Abel–Jacobi coordinates are introduced to straighten the flows, from which new algebro-geometric solutions of the (2+1)-dimensional mKP equation and algebro-geometric solutions of the (2+1)-dimensional coupled mKP equations are obtained in terms of the Riemann theta functions.

Kadomtsev-Petviashvili solitons in inhomogeneous plasmas with finite temperature drifting ions

Two-dimensional ion acoustic solitons travelling in a spatially inhomogeneous plasma with finite temperature drifting ions are studied. The modified Kadomtsev-Petviashvili (mKP) equation governing the soliton propagation is obtained, and is solved for a constant density gradient. The amplitude and the energy of the soliton increase and the soliton width decreases as it travels into the decreasing plasma density.

Characteristic aspects of the stability of Kadomtsev-Petviashvili solitary waves in plasmas

By employing the reductive perturbation technique (RPT), the solitary waves in an ideal model of multidimensional space-plasmas are studied by the Kadomtsev-Petviashvili (K-P) equation. It has been shown that the presence of multitemperature electrons exhibits the compressive and rarefactive solitary waves bifurcated by the critical densities appear at the zeros of the nonlinear coefficient in the K-P equation. Because of the zeros, singularities in soliton propagation appear and lead to the derivation, with a modified RPT, of a modified K-P (mK-P) equation, which in fact exhibits not only the compressive and rarefactive solitary waves but also shows the formation of shocklike waves. The analyses on soliton propagation are also derived by the Sagdeev potential, confirming all those solitary waves observed by the implementation of the K-P equation. The main emphasis has been given to highlighting the stability of the soliton, and, even though they have singularities as well as shocklike waves, it was found that the solitary waves are stable, independent of their nature.

Delta -dressing and exact solutions for the (2+1)-dimensional Harry Dym equation

Exact implicit solutions with functional parameters of the (2+1)-dimensional Harry Dym (HD) equation are constructed by the delta -dressing method. The interrelation between the HD and modified Kadomtsev-Petviashvili (mKP) equations is established within the framework of the delta -dressing method. Knowledge of the mKP eigenfunctions allows the solutions of the (2+1)-dimensional HD equation to be represented in a simple parametric form.

Lax pair for the general KP equation with explicit x, y and t dependence

With the help of a commutativity condition, we construct the general KP equation with explicit x, y and t dependence, the Lax pair for the equation and the general MKP equation with explicit x, y and t dependence are obtained.

Lax pair, Bäcklund transformation, solitary wave solution and finite conservation laws of the general KP equation and MKP equation with variable coefficients

In this paper, we obtain the Lax pair, Bäcklund transformation, solitary wave solution and infinite conservation laws of the general KP equation and MKP equation with variable coefficients.

Inverse Spectral Transform for the Modified Kadomtsev‐Petviashvili Equation

The 2 + 1‐modified Kadomtsev‐Petviashvili (mKP) equation is studied by the inverse‐spectral‐transform method. The initial‐value problems for the mKP‐1 and mKP‐11 equations are solved by the nonlocal Riemann‐Hilbert and techniques for initial data decaying sufficiently rapidly at infinity. The lump solutions for the mKP‐I equation are found explicitly. Wide classes of the exact solutions for the mKP equation—namely, the rational solutions, including the plane lumps for the mKP‐I equation; solutions with functional parameters; the plane solitons; and breathers—are constructed by the use of the method based on the nonlocal . The Miura transformation between the mKP and KP equations is discussed.

Rational KP and mKP-solutions in Wronskian form

Rational solutions of the Kadomtsev-Petviashvili ( KP) and modified Kadomtsev-Petviashvili ( mKP) equations in Wronskian form are constructed for the first time. The relationship between the Wronskian technique and Hirota's bilinear formalism is discussed. It is shown that each pair of KP solutions giving rise to a mKP solution is connected via the KP-Bäcklund transformation.

On the interrelation between the solutions of the mKP and KP equations via the Miura transformation

It is shown that the classes of exact solutions with functional parameters and rational solutions of the modified Kadomtsev-Petviashvili (mKP) equation and the Kadomtsev-Petviashvili (KP) equation are connected by the (2+1)-dimensional Miura transformation. The correspondence between more particular classes of solutions of the mKP and the KP equations via the Miura transformation is established.