Soliton Solution of New (2+1) Dimensional Nonlinear Partial Differential Equations

Using singularity structure analysis, we establish the integrability property of new (2+1) dimensional nonlinear partial differential equations (NPDEs) derived by Maccari from integrable equations through the reduction method. We also derive the bilinear form and one soliton solution is explicitly generated. Finally, we discuss the connection between the system equations and other integrable models.     

Fast Soliton Scattering by Attractive Delta Impurities

We study the Gross–Pitaevskii equation with an attractive delta function potential and show that in the high velocity limit an incident soliton is split into reflected and transmitted soliton components plus a small amount of dispersion. We give explicit analytic formulas for the reflected and transmitted portions, while the remainder takes the form of an error. Although the existence of a bound state for this potential introduces difficulties not present in the case of a repulsive potential, we show that the proportion of the soliton which is trapped at the origin vanishes in the limit.     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

Four-Parameter 1/<Emphasis Type="Italic">r</Emphasis><Superscript>2</Superscript> Singular Short-Range Potential with Rich Bound States and A Resonance Spectrum

We use the tridiagonal representation approach to enlarge the class of exactly solvable quantum systems. For this, we use a square-integrable basis in which the matrix representation of the wave operator is tridiagonal. In this case, the wave equation becomes a three-term recurrence relation for the expansion coefficients of the wave function with a solution in terms of orthogonal polynomials that is equivalent to a solution of the original problem. We obtain S-wave bound states for a new four-parameter potential with a 1/r² singularity but short-range, which has an elaborate configuration structure and rich spectral properties. A particle scattered by this potential must overcome a barrier and can then be trapped in the potential valley in a resonance or bound state. Using complex rotation, we demonstrate the rich spectral properties of the potential in the case of a nonzero angular momentum and show how this structure varies with the parameters of the potential.     

Lump and mixed rogue-soliton solutions to the 2+1 dimensional Ablowitz-Kaup-Newell-Segur equation

In this paper, the 2+1 dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation which obtained from the potential Boiti-Leon-Manna-Pempi nelli (pBLMP) equation, is introduced. Through the bilinear method and ansatz technique, the rational solutions consisting of rogue wave and lump soliton solutions are constructed, where we discuss the condition of guaranteeing the positiveness and analyticity of the lump solutions. The collection of a quadratic function with an exponential function describing rational-exponential solutions is proved, the interaction consisting of one lump and one soliton with fission and fusion phenomena. The second kind of interaction comprises the line rogue wave and soliton solution, which is inelastic. With the usage of the extended homoclinic test approach, the homoclinic breather-wave solution is derived. The characteristics of these various solutions are exhibited and illustrated graphically.     

The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method

A procedure for finding the solutions of the Vakhnenko–Parkes equation by means of the inverse scattering method is described. Both the bound state spectrum and the continuous spectrum are considered in the associated eigenvalue problem. The suggested special form of the singularity function gives rise to periodic solutions. The interaction of a soliton with a one-mode periodic wave is studied.     

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.     

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.     

一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解北大核心CSCD

在本文中,一类新的矩阵型修正Korteweg-de Vries(简记为mmKdV)方程被首次通过RiemannHilbert方法研究,而且,这一方程可通过选取特殊的势矩阵来降阶为我们熟知的耦合型修正Kortewegde Vries方程.从方程对应的Lax对的谱分析入手,作者成功地建立了方程对应的Riemann-Hilbert问题.在无反射势的特殊条件下,mmKdV方程的精确解可由Riemann-Hilbert问题的解给出.而且,基于特殊势矩阵所对应的特殊对称性,作者可以对原有的孤子解进行分类,从而得到一些有趣的解的现象,比如呼吸孤子、钟形孤子等.     

Bilinearization of discrete soliton equations through the singularity confinement test

An elementary and systematic method to construct exact solutions for discrete soliton equations is presented. In this method, the singularity patterns, obtained through the singularity confinement test, give us critical information for bilinearization.     

Optical soliton with damping and frequency chirping in fibre media

Nonlinear Schrödinger (NLS) equation which governs the propagation of single field in the fibre medium with pulse chirping and gain (loss) is considered. The integrability condition of NLS equation is arrived from the linear eigenvalue problem and is studied by using the Painlevé singularity structure analysis. From the Lax pair, soliton solution is constructed by using the Darboux–Bäcklund transformation technique. From the results, we show that the soliton is alive, i.e., pulse area is conserved with the inclusion of damping and pulse chirping effects.     

Chaos, solitons and fractals in (2 + 1)-dimensional KdV system derived from a periodic wave solution

With the help of an extended mapping method and a linear variable separation method, new types of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with two arbitrary functions for (2 + 1)-dimensional Korteweg–de Vries system (KdV) are derived. Usually, in terms of solitary wave solutions and rational function solutions, one can find some important localized excitations. However, based on the derived periodic wave solution in this paper, we find that some novel and significant localized coherent excitations such as dromions, peakons, stochastic fractal patterns, regular fractal patterns, chaotic line soliton patterns as well as chaotic patterns exist in the KdV system as considering appropriate boundary conditions and/or initial qualifications.     

Solitary wave solution for inhomogeneous nonlinear Schrödinger system with loss/gain

This paper deals with the propagation of solitons in real fibres, governed by the system of inhomogeneous nonlinear Schrödinger (INLS) equations. The Painlevé singularity structure analysis is utilized to check for the integrability of the system and from the analysis, the system is found to admit soliton-type lossless wave propagation. The system is transformed to its homogeneous counterpart using a suitable variable transformation and the soliton solutions are obtained through Bäcklund transformation after constructing the explicit Lax pair for the system. The one-soliton solutions are plotted for different choices of inhomogeneity parameters and the evolutionary characteristics of the solutions are analyzed.     

Integrability and singularity structure of coupled nonlinear Schrödinger equations

Multimode propagation of electromagnetic waves in optical fibre is often described by coupled nonlinear Schrödinger (NLS) equations. To understand the integrability properties of such coupled NLS systems, we extend the Painlevé singularity structure analysis of two coupled systems to three coupled systems and identify four integrable sets of parameters. We bilinearize these cases to obtain soliton solutions. The results are extended to N-coupled systems, completing the earlier analysis of Sahadevan, Tamizhmani and Lakshmanan.     

（3＋1）维破碎孤子方程的变量分离解和局域激发模式

多线性分离变量法已成功地应用于诸多（2＋1）维非线性可积系统.将该方法拓展运用于（3＋1）维破碎孤子方程中,获得了含任意函数的变量分离解.通过适当地设定任意函数的形式,得到了（3＋1）维破碎孤子方程丰富的局域激发模式.     

Bright soliton solution of a Gross–Pitaevskii equation

We theoretically and numerically study the bright soliton solutions of a Gross–Pitaevskii equation governing one-dimensional (1D)(cigar-shaped) Bose–Einstein condensates (BEC) trapped in an optical lattice of 1D structure. The analytical expression of bright soliton is derived by using the variational approximation, which completely matches the numerical results with a range of potential’s parameters. Moreover, we determined the parameter domains for the persistence and non-persistence of bright soliton solutions.     

推广的（2＋1）维浅水波方程的精确解及其混沌行为

利用Riccati方程映射法和变量分离法,得到了推广的（2＋1）维浅水波系统的变量分离解（包括孤波解、周期波解和有理函数解）.根据得到的孤波解,构造出了方程的单孤子和双孤子结构,研究了孤子的混沌行为.     

Singularities of the Radon Transform of a Class of Piecewise Smooth Functions in R2

The Radon transform is the mathematical foundation of Computerized Tomography[1](CT).Its important applications includes medical CT,noninvasive test and etc.If one is specially interested in the places at which the image function changed largely such as the interfaces of two different tissues,tissue and ill tissue and the interfaces of two difierent matters,and want to reconstruct the outlines of the interfaces,one should reconstruct the singularities of the image function.The exact inversion of the Radon transform is valid only for smooth function[2].The singularity places of the reconstructed function should be studied specially.The research includes the propagation and inversion of singularity of the Radon transform.If one use convolutionbackprojection method to reconstruct the image function,the reconstructed function become blurring at the singularity places of the original function.M.Jiang and etc[3]developed a blind deconvolution method deblurring reconstructed image.By[4]and following research,we see that one can use a neighborhood data of the singularities of the Radon transform to inverse the singularity of the Radon transform,and therefore the reconstruction is available for some incomplete data reconstructions.     

CTE method to the interaction solutions of Boussinesq–Burgers equations

Under investigation in this paper is the Boussinesq–Burgers equations, which describe the propagation of shallow water waves. Via the truncated Painlevé analysis and the consistent tanh expansion (CTE) method, some exact interaction solutions among different nonlinear excitations such as multiple resonant soliton solutions, soliton–error function waves, soliton–periodic waves, soliton–rational waves, and soliton–potential Burgers waves are explicitly given.     

Exact soliton solutions for the general fifth Korteweg-de Vries equation

With the aid of computer symbolic computation system such as Maple, the extended hyperbolic function method and the Hirota’s bilinear formalism combined with the simplified Hereman form are applied to determine the soliton solutions for the general fifth-order KdV equation. Several new soliton solutions can be obtained if we taking parameters properly in these solutions. The employed methods are straightforward and concise, and they can also be applied to other nonlinear evolution equations in mathematical physics. The article is published in the original.