On the Inverse Scattering of the Time-Dependent Schrödinger Equation and the Associated Kadomtsev-Petviashvili (I) Equation

The Kadomtsev-Petviashvili equation, a two-spatial-dimensional analogue of the Korteweg-deVries equation, arises in physical situations in two different forms depending on a certain sign appearing in the evolution equation. Here we investigate one of the two cases. The initial-value problem, associated with initial data decaying sufficiently rapidly at infinity, is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation of a nonlocal Riemann-Hilbert problem in terms of scattering data expressible in closed form in terms of given initial data. The lump solutions, algebraically decaying solitons, are given a definite spectral characterization. Pure lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t. Many of the above results are also relevant to the problem of inverse scattering for the so-called time-dependent Schrödinger equation.     

On the Derivation of the Modified Kadomtsev-Petviashvili Equation

It is well known that the Kadomtsev-Petviashvili (KP) equation is the two-dimensional analogue of the Korteweg—de Vries (KdV) equation. We reconsider the derivation of the KP equation, modified to include the effects of rotation, in order to determine the nature of the initial conditions. The motivation for this is that if the solutions of the modified KP equation are assumed to be locally confined, then they satisfy a certain constraint, which appears to restrict considerably the class of allowed initial conditions. The outcome of the analysis presented here is that in general it is not permissible to assume that solutions of the modified KP equation are locally confined, and hence the constraint cannot be applied. The reason for this is the radiation of Poincaré waves, which appear behind the main part of the solution described by the modified KP equation.     

On the Inverse Scattering Transform for the Kadomtsev-Petviashvili Equation

The initial value problem of the Kadomtsev-Petviashvili equation for one choice of sign in the equation has been recently investigated in the literature. Here we consider the other choice of sign. We introduce suitable eigenfunctions which though bounded are not analytic in the spectral parameter. This, in contrast to the known case, prevents us from formulating the inverse problem as a nonlocal Riemann-Hilbert boundary value problem. Nevertheless a suitable formulation is given and a formal solution is constructed via a linear integral equation.     

Kadomtsev-Petviashvili方程的精确解

齐次平衡法是把非线性偏微分方程转换成带约束条件的线性偏微分方程的一种很好的方法 .本文在齐次平衡法的基础上具体讨论了KP方程的精确解 ,包括孤波解 ,一般的行波解 ,有理函数解和一种新类型的解 .     

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.     

The Rotation-Modified Kadomtsev-Petviashvili Equation: An Analytical and Numerical Study

The rotation-modified Kadomtsev-Petviashvili equation, derived by Grimshaw in 1985, is studied both analytically and numerically to determine the structure of solutions which are initially localized. It is shown that solitary-like waves can be found, whose wavefronts are curved in a direction transverse to the propagation direction, which remain unsteady, and which are always accompanied by trailing Poincaré waves. These effects are more noticeable as the effects of rotation are increased.     

扩展形式的修正的Kadomtsev-Petviashvili方程的多重峰波

研究修正的Kadomtsev-Petviashvili(mKP)方程的一个扩展形式.使用由Hereman和Nuseir提出的、一个可以信赖的、Hirota双线性法的简化形式.由该方程(这里称为mKP方程)直接导出多重峰波解.研究还表明,扩展项并不会破坏mKP方程的可积性.     

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

By using homogeneous balance principle,we derive a Baecklund trans-formation(BT) to (3 1)-dimensional 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.     

Stationary Solutions for a Generalized Kadomtsev-Petviashvili Equation in Bounded Domain

In this work, we are mainly concerned with the existence of stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain of Rn. We utilize variational method and critical point theory to establish our main results.     

The Cauchy Problem for the Kadomtsev-Petviashvili (KPII) Equation in Three Space Dimensions

We prove the existence and uniqueness of local solutions in time for the Kadomtsev-Petviashvili equation (KPII) in ?³ for initial data in a suitable anisotropic Sobolev space of functions which have essentially 1⁺ derivative in the variable x and 0⁺ derivatives in the variables y and z.     

2+1-维变系数广义Kadomtsev-Petviashvili方程的相似约化

借助于MATHEMATICA软件,将直接约化法推广并应用到2+1-维变系数广义Kadomtsev-Petviashvili(VCGKP)方程,获得了VCGKP方程的若干相似约化,其中包括PainleveⅠ型、PainleveⅡ型和PainleveⅣ型的约化.     

The Recursion Operator of the Kadomtsev-Petviashvili Equation and the Squared Eigenfunctions of the Schrödinger Operator

The recursion operator of the Kadomtsev-Petviashvili equation is algorithmically derived. This recursion operator is the two-spatial-dimensional analogue of the Lenard operator of the Korteweg-deVries equation. It is also the “squared” eigenfunction operator of the time-dependent Schrödinger operator. The existence of the recursion operator suggests that the Kadomtsev-Petviashvili equation is a hi-Hamiltonian system.     

On the low regularity of the fifth order Kadomtsev-Petviashvili I equation

We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as , while α∈R. We introduce an interpolated energy space Es to consider the well-posedness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in Es for 0<s?1. To obtain the local well-posedness, we present a bilinear estimate in the Bourgain space in the framework of the interpolated energy space. It crucially depends on the dyadic decomposed Strichartz estimate, the fifth order dispersive smoothing effect and maximal estimate.     

模糊关系几何规划

提出一类目标函数为正项式,约束是模糊关系方程的优化问题。阐述模糊关系方程解集的结构以及求解的方法,基于目标函数中每个单项式的指数取值情况讨论最优解,并且给出解决此类优化问题的一个程序,为了说明该方法的有效性给出具体例子。     

On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented that can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial differential equations of Korteweg–de Vries (KdV) type. The new equation, which is referred to as generalized discrete Painlevé equation (GDP), contains various "discrete Painlevé equations" as subcases for special values/limits of the parameters, some of which have already been given in the literature. The general solution of the GDP can be expressed in terms of Painlevé VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Bäcklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.     

On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators

Theoretical and Mathematical Physics - We study the existence and uniqueness of the Kadomtsev–Petviashvili (KP ) hierarchy solutions in the algebra $$\mathcal FCl(S^1,\mathbb K^n)$$ of formal...