Modified Kadomtsev-Petviashvili Equation Research Articles

Oblique interactions of weakly nonlinear long waves in dispersive systems

Studies on the oblique interactions of weakly nonlinear long waves in dispersive systems are surveyed. We focus mainly our concentration on the two-dimensional interaction between solitary waves. Two-dimensional Benjamin–Ono (2DBO) equation, modified Kadomtsev–Petviashvili (MKP) equation and extended Kadomtsev–Petviashvili (EKP) equation as well as the Kadomtsev–Petviashvili (KP) equation are treated. It turns out that a large-amplitude wave can be generated due to the oblique interaction of two identical solitary waves in the 2DBO and the MKP equations as well as in the KP-II equation. Recent studies on exact solutions of the KP equation are also surveyed briefly.

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.

Commutativity of Pfaffianization and Bäcklund transformations: the KP equation

The Pfaffianization procedure is applied to the modified KP equation. As a result, we derive a coupled modified KP system, which is shown to constitute a bilinear Bäcklund transformation for the coupled KP (cKP) equation introduced by Hirota and Ohta. This means that commutativity of Pfaffianization and Bäcklund transformation is valid for the KP equation. As a by-product, we explicitly derive a coupled system from Bäcklund transformation formulae of the cKP equation, which solves an open problem proposed by Hirota. In addition to starting from the Bäcklund transformation, a Lax pair for the coupled KP equation is worked out. Finally, two kinds of Pfaffian solutions to the coupled modified KP equation are derived.

Two-dimensional Interaction of Solitary Waves in a Modified Kadomtsev–Petviashvili Equation

Two-dimensional interaction of two solitary waves is investigated numerically on the basis of a Modified Kadomtsev–Petviashvili equation. Two types of interaction are dealt with; one is the interac...

Soliton Resonances for the MKP-II

Using the second flow - the Derivative Reaction-Diffusion system, and the third one of the dissipative SL(2,R) Kaup-Newell hierarchy, we show that the product of two functions, satisfying those systems is a solution of the modified Kadomtsev-Petviashvili equation in 2+1 dimension with negative dispersion (MKP-II). We construct Hirota's bilinear representation for both flows and combine them together as the bilinear system for MKP-II. Using this bilinear form we find one and two soliton solutions for the MKP-II. For special values of parameters our solution shows resonance behaviour with creation of four virtual solitons. Our approach allows one to interpret the resonance soliton as a composite object of two dissipative solitons in 1+1 dimensions.

On therth dispersionless Toda hierarchy: factorization problem, additional symmetries and some solutions

For a family of Poisson algebras, parametrized by by an integer number r, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th dispersionless Toda hierarchy. Moreover, additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the r-th dispersionless KP, r-th dispersionless Dym and r-th dispersionless Toda equations. Some solutions are derived by examining the imposition of a time invariance to the potential r-th dispersionless Dym equation, for which a complete integral is presented and therefore an appropriate envelope leads to a general solution. Symmetries and Miura maps are applied to get new solutions and solutions of the r-th dispesionless modified KP equation.

Traveling wave solutions of a generalized modified Kadomtsev–Petviashvili equation

A method to generate traveling wave solutions of the generalized modified Kadomtsev–Petviashvili equation is reported and several physical solutions, including conditions of their existence, are presented.

Boundary Conditions for Multidimensional Integrable Equations

We suggest an efficient method for finding boundary conditions compatible with integrability for multidimensional integrable equations of Kadomtsev-Petviashvili type. It is observed in all known examples that imposing an integrable boundary condition at a point results in an additional involution for the t-operator of the Lax pair. The converse is also likely to be true: if constraints imposed on the coefficients of the t-operator of the L-A pair result in a broader group of involutions of the t-operator, then these constraints determine integrable boundary conditions. New examples of boundary conditions are found for the Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili 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.

Algebro-geometric solution of the 2+1 dimensional Burgers equation with a discrete variable

The quasiperiodic solution of the 2+1 dimensional Burgers equation with a discrete variable is obtained through three steps: (a) decomposition into a symplectic map plus two finite-dimensional Hamiltonian systems; (b) straightening out of both the discrete and the continuous flows in the Jacobian variety; (c) inversion into the original variables. Inner relation with the modified Kadomtsev–Petviashvili equation is presented. The explicit theta function solutions for these two 2+1 integrable models are given.

Decomposition of certain nonlinear evolution equations and their quasi-periodic solutions

A new 2+1-dimensional nonlinear evolution equation is proposed. With the help of the known 1+1-dimensional soliton equations, this new 2+1-dimensional evolution equation and the modified Kadomtsev–Petviashvili equation are separated into compatible Hamiltonian systems of ordinary differential equations. Using the generating function flow method, the involutivity and the functional independence of the integrals are proved. The Abel–Jacobi coordinates are introduced to straighten out the associated flows. The Riemann–Jacobi inversion problem is discussed, from which quasi-periodic solutions of the 1+1-dimensional soliton equations, the new 2+1-dimensional evolution equation and the modified Kadomtsev–Petviashvili equation are obtained.

Explicit solutions for some (2+1)-dimensional nonlinear evolution equations

Some (2+1)-dimensional nonlinear evolution equations, including the Kadomtsev-Petviashvili II (KPII) equation and the modified Kadomtsev-Petviashvili (KP) equation, are decomposed into two (1+1)-dimensional soliton systems in the coupled KdV hierarchy. With the help of the decomposition and the Darboux transformation, some explicit solutions of these (2+1)-dimensional nonlinear evolution equations such as new soliton solutions of the KPII equation and the modified KP equation are obtained.

Quasi-periodic solutions for some (2 + 1)-dimensional integrable models generated by the Jaulent-Miodek hierarchy

Some ( 2+1 )-dimensional integrable models, including the modified Kadomtsev–Petviashvili equation, generated by the Jaulent–Miodek hierarchy are investigated. With the help of the Jaulent–Miodek eigenvalue problem, these ( 2+1 )-dimensional integrable models are separated into compatible Hamiltonian systems of ordinary differential equations. Using the generating function flow method, the involutivity and the functional independence of the integrals are proved. The Abel–Jacobi coordinates are introduced, from which the quasi-periodic solutions for these ( 2+1 )-dimensional integrable models are derived by resorting to the Riemann theta functions. PACS number: 0230J

On the decomposition of the modified Kadomtsev–Petviashvili equation and explicit solutions

The (2+1)-dimensional modified Kadomtsev–Petviashvili equation is decomposed into two known (1+1)-dimensional soliton equations. A Darboux transformation of the two known (1+1)-dimensional soliton equations is derived with the help of a gauge transformation of the spectral problem. As an application, explicit solutions of the two (1+1)-dimensional soliton equations and two new explicit solutions of the (2+1)-dimensional modified Kadomtsev–Petviashvili equation are obtained.

On the dressing method in multidimension

The dressing method based on the system of linear inhomogeneous algebraic equations is proposed for construction the particular solutions of n-dimensional ( n is integer) nonlinear partial differential equations (PDE). The classical integrable systems are among them. We represent an example of (3+1)-dimensional generalization of modified Kadomtsev–Petviashvili equation (mKP) with particular solutions.

Wave propagation through a 2D lattice

Nonlinear wave propagation through a 2D lattice is investigated. Using reductive perturbation method, we show that this can be described by Kadomtsev–Petviashvili (KP) equation for quadratic nonlinearity and modified KP equation for cubic nonlinearity, respectively. With quadratic and cubic nonlinearities together, the system is governed by an integro-differential equation. We have also checked the integrability of these equations using singularity analysis and obtained solitary wave solutions.

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.

Coupled modified Kadomtsev–Petviashvili equations in dispersive elastic media

The present work considers two-dimensional non-linear wave propagation in an infinite, homogeneous micropolar elastic medium. The reductive perturbation method is directly applied to a Lagrangian whose Euler–Lagrange equations give the field equations for a geometrically non-linear micropolar elastic medium. It is shown that the behavior of non-linear waves in the long-wave approximation is governed by the two-coupled modified Kadomtsev–Petviashvili equations. Travelling wave solutions of the non-linear evolution equations are considered by means of a modified Hirota method.

A New Class of Soliton Solutions for the (Modified) Kadomtsev-Petviashvili Equation

We construct solutions of the Kadomtsev-Petviashvili equation and its counterpart, the modified Kadomtsev-Petviashvili equation, with an infinite number of solitons by a careful examination of the limits of N- soliton solutions as N → ∞. We give sufficient conditions to ensure that these limits exist and satisfy the (modified) Kadomtsev-Petviashvili equation.

Quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation

The (2 + 1)-dimensional modified Kadomtsev-Petviashvili equation is decomposed into systems of integrable ordinary differential equations resorting to the nonlinearization of Lax pairs. Abel-Jacobi coordinates are introduced to straighten the flows, from which quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation are obtained in terms of Riemann theta functions.