高维非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了修正的Kadomtsev-Petviashvili方程, 得到了它的近似孤立波解, 该解与精确解符合得非常好.结果表明，同伦分析法在求解高维非线性演化方程的孤立波解时, 仍然是一种行之有效的方法. 关键词： 同伦分析法 修正的Kadomtsev-Petviashvili方程 孤立波解     

The D'Alembert type waves and the soliton molecules in a (2+1)-dimensional Kadomtsev-Petviashvili with its hierarchy equation

For a one (2+1)-dimensional combined Kadomtsev-Petviashvili with its hierarchy equation, the missing D'Alembert type solution is derived first through the traveling wave transformation which contains several special kink molecule structures. Further, after introducing the Bäcklund transformation and an auxiliary variable, the N-soliton solution which contains some soliton molecules for this equation, is presented through its Hirota bilinear form. The concrete molecules including line solitons, breathers and a lump as well as several interactions of their hybrid are shown with the aid of special conditions and parameters. All these dynamical features are demonstrated through the different figures.     

On reduction of some classes of partial differential equations to equations with fewer variables and exact solutions

We establish a connection between the fundamental solutions to some classes of linear nonstationary partial differential equations and the fundamental solutions to other nonstationary equations with fewer variables. In particular, reduction enables us to obtain exact formulas for the fundamental solutions of some spatial nonstationary equations of mathematical physics (for example, the Kadomtsev-Petviashvili equation, the Kelvin-Voigt equation, etc.) from the available fundamental solutions to one-dimensional stationary equations.     

Soliton Resonances for the MKP-II

Using the second flow (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 dimensions with negative dispersion (MKP-II). We construct Hirota’s bilinear representations for both flows and combine them as the bilinear system for the MKP-II. Using this bilinear form, we find one- and two-soliton solutions for the MKP-II. For special values of the parameters, our solution shows resonance behavior with the creation of four virtual solitons. Our approach allows interpreting the resonance soliton as a composite object of two dissipative solitons in 1+1 dimensions.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 133–142, July, 2005.     

Backlund Transformation and Exact Solution to Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the homogeneous balance principle, we derive a Backlund transformation (BT) to (3+1)-dimensionaI Kadomtsev-Petviashvili (K-P) equation with variable coefficients if the variable coefficients are linearly dependent. Based on the BT, the exact solution of the (3+1)-dimensional K-P equation is given. By the same method, we derive a BT and the solution to (2+1)-dimensional K-P equation. The variable coefficients can change the amplitude of solitary wave, but cannot change the form of solitary wave.     

New exact solutions of (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation using a combination of lie symmetry and singular manifold methods

A new combination of Lie symmetry and Singular Manifold methods has been employed to study (3 + 1)-dimensional generalized Kadomtsev-Petviashvili (KP). Infinite-dimensional space of Lie vectors has been established. Single and dual linear combinations of Lie vectors are used after appropriate calculations of the arbitrary functions to reduce the equation to an ordinary differential equation (ODE). The resulting ODE is then analytically solved through the singular manifold method which resulted in a Bäcklund truncated series with seminal analysis leading to a Schwarzian differential equation in the Eigenfunction φ (η). Solving this differential equation leads to new analytical solutions.     

Lumps and their interaction solutions of a (2+1)-dimensional generalized potential Kadomtsev-Petviashvili equation

A (2+1)-dimensional generalized potential Kadomtsev-Petviashvili (gpKP) equation which possesses a Hirota bilinear form is constructed. The lump waves are derived by using a positive quadratic function solution. By combining an exponential function with a quadratic function, an interaction solution between a lump and a one-kink soliton is obtained. Furthermore, an interaction solution between a lump and a two-kink soliton is presented by mixing two exponential functions with a quadratic function. This type of lump wave just appears to a line $k_2x+k_3y+k_4t+k_5 \sim 0$. We call this kind of lump wave is a special rogue wave. Some visual figures are depicted to explain the propagation phenomena of these interaction solutions.     

Asymptotic behavior of the periodic wave solution for the (3+1)-dimensional Kadomtsev-Petviashvili equation

In this paper, the one- and two-periodic wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation are presented by means of the Hirota’s bilinear method and the Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.     

Painlevé analysis and new analytic solutions for variable-coefficient Kadomtsev-Petviashvili equation with symbolic computation

The variable-coefficient Kadomtsev-Petviashvili (KP) equation is hereby under investigation. Painlevé analysis is given out, and an auto-Bäcklund transformation is presented via the truncated Painlevé expansion. Based on the auto-Bäcklund transformation, new analytic solutions are given, including the soliton-like and periodic solutions. It is also reduced to a (1+1)-dimensional partial differential equation via classical Lie group method and the Painlevé I equation by CK direct method.     

Wronskian and Grammian Determinant Solutions for a Variable-Coefficient Kadomtsev-Petviashvili Equation

In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an exact solution of this equation through the Wronskian technique. In addition, we testify that this equation can be reduced to a Jacobi identity by considering its solution as a Grammian determinant by means of Pfaffian derivative formulae.