(2+1)维色散长波方程的新的类孤子解

应用一种新的修改的代数方法去求解(2+1)维色散长波方程,获得方程的大量新的精确解.这些解包括类孤子解、类周期解、类有理解、类双曲函数解、类Jacobi椭圆函数解等等. 关键词： (2+1)维色散长波方程 类孤子解 类有理解 类双曲函数解 类Jacobi椭圆 函数解     

Some new solutions derived from the nonlinear (2+1)-dimensional Toda equation---an efficient method of creating solutions

This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential--difference equations (NLDDEs) and lattice equation. By using this method via symbolic computation system MAPLE, we obtained abundant soliton-like and/or period-form solutions to the (2+1)-dimensional Toda equation. It seems that solitary wave solutions are merely special cases in one family. Furthermore, the method can also be applied to other nonlinear differential--difference equations.     

Time Dependent Ginzburg-Landau方程的Weierstrass椭圆函数解

将文[22]中提出的求解非线性演化方程的Weierstrass椭圆函数解的一个新方法应用于Time Dependent Ginzburg-Landau方程,获得了该方程的一些新的双周期解,并在退化情形下得到了一些新的精确孤波解.     

Multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas-Lenells equation

In this letter, we investigate multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas-Lenells equation over a nonzero background. First, we obtain 2n-soliton solutions with a nonzero background via n-fold Darboux transformation, and find that these soliton solutions will appear in pairs. Particularly, 2n-soliton solutions consist of n ‘bright' solitons and n ‘dark' solitons. This phenomenon implies a new form of integrability: even integrability. Then interactions between solitons with even numbers and breathers are studied in detail. To our best knowledge, a novel nonlinear superposition between a kink and 2n-soliton is also generated for the first time. Finally, interactions between some different smooth positons with a nonzero background are derived.     

Multiple-pole solutions and degeneration of breather solutions to the focusing nonlinear Schrödinger equation

Based on the Hirota’s method, the multiple-pole solutions of the focusing Schrödinger equation are derived directly by introducing some new ingenious limit methods. We have carefully investigated these multi-pole solutions from three perspectives: rigorous mathematical expressions, vivid images, and asymptotic behavior. Moreover, there are two kinds of interactions between multiple-pole solutions: when two multiple-pole solutions have different velocities, they will collide for a short time; when two multiple-pole solutions have very close velocities, a long time coupling will occur. The last important point is that this method of obtaining multiple-pole solutions can also be used to derive the degeneration of N-breather solutions. The method mentioned in this paper can be extended to the derivative Schrödinger equation, Sine-Gorden equation, mKdV equation and so on.     

非线性耦合标量场方程的新双周期解(Ⅰ)

对于具有丰富物理意义和众多应用价值的非线性耦合标量场方程，利用sinh-Gordon方程展开法，导出了五种新的双周期解，同时在退化的情形得到了新的孤波解和三角函数解.结果说明该方程具有丰富的解的结构，方程描述的物理模型所具有的物理现象仍将是物理学家和数学家进一步探讨的对象. 关键词： 非线性耦合标量场方程 sinh-Gordon方程展开法 双周期解 精确解     

(3+1)维 KdV 方程新的精确解和孤子解

用普通Korteweg-de Vries(KdV)方程作变换,构造(3 1)维KdV方程的解,获得了新的孤子解、Jaoobi椭圆函数解、三角函数解和Weierstrass椭圆函数解.     

Resonance Y-type soliton solutions and some new types of hybrid solutions in the(2+1)-dimensional Sawada–Kotera equation

In this paper, based on N-soliton solutions, we introduce a new constraint among parameters to find the resonance Y-type soliton solutions in (2+1)-dimensional integrable systems. Then, we take the (2+1)-dimensional Sawada–Kotera equation as an example to illustrate how to generate these resonance Y-type soliton solutions with this new constraint. Next, by the long wave limit method, velocity resonance and module resonance, we can obtain some new types of hybrid solutions of resonance Y-type solitons with line waves, breather waves, high-order lump waves respectively. Finally, we also study the dynamics of these interaction solutions and indicate mathematically that these interactions are elastic.     

New exact travelling wave solutions for the Ostrovsky equation

In this Letter, auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equation with the aid of symbolic computation. In order to illustrate the validity and the advantages of the method we choose the Ostrovsky equation. As a result, many new and more general exact solutions have been obtained for the equation.     

Zakharov方程的显式行波解

借助Mathematica软件，采用双函数法和吴文俊消元法，获得了等离子体物理中的重要方程组Zakharov方程的十组行波解，其中包括包络孤波解，孤子解. 关键词： Zakharov方程 孤子解     

构造孤子方程的Weierstrass椭圆函数解的一个新方法

利用具有Weierstrass椭圆函数解的方程，首先获得了投影Riccati方程的两组新解.由于投影Riccati方程可用于多种具孤子解的非线性演化方程的求解，因而得到了一个可以构造这些方程的Weierstrass椭圆函数解的新方法. 关键词： Weierstrass椭圆函数解 投影Riccati方程 非线性演化方程     

非线性耦合标量场方程的新双周期解(Ⅱ)

基于具有双周期解的常微分方程，提出了一种构造非线性微分方程双周期解的新方法，在计算机符号软件帮助下方法可实现机械化.应用此方法于非线性耦合标量场方程，得到了该方程的大量的新精确解. 关键词： 非线性耦合标量场方程 双周期解 精确解     

变系数Whitham-Broer-Kaup方程组的对称、约化及精确解

利用修正的Clarkson-Kruskal直接法对变系数Whitham-Broer-Kaup(VCWBK)方程组进行等价转化,建立了VCWBK方程组与常系数WBK方程组解之间的关系,并得到了常系数WBK方程组的一些对称和相似约化.借助辅助函数法得到了VCWBK方程组的一些新精确解,包括有理函数解、双曲函数的解、三角函数解和Jacobi椭圆函数解.     

扩展的双曲函数法和Zakharov方程组的新精确孤立波解

借助于符号计算软件Maple，利用扩展的双曲函数法求出了Zakharov方程组的精确孤立波解，包括钟状孤立波解、扭结状孤立波解、包络孤立波解、奇性孤立波解和一种新的形式的孤立波解.这种方法也适用于其他非线性波方程.     

From 2-dimensional surfaces to cosmological solutions

We construct perfect fluid metrics with two symmetries by means of a recently developed geometrical method [1]. The Einstein equations are reduced to a single equation for a conformal factor. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI₀ and VII₀. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.     

The soliton-like solutions to the （2＋1）-dimensional modified dispersive water-wave system

By a simple transformation, we reduce the (2 1)-dimensional modified dispersive water-wave system to a simple nonlinear partial differential equation. In order to solve this equation by generalized tanh-function method, we only need to solve a simple system of first-order ordinary differential equations, and by doing so we can obtain many new soliton-like solutions which include the solutions obtained by using the conventional tanh-function method.     

Unsteady analytical solutions of the spherical shallow water equations

A new class of unsteady analytical solutions of the spherical shallow water equations (SSWE) is presented. Analytical solutions of the SSWE are fundamental for the validation of barotropic atmospheric models. To date, only steady-state analytical solutions are known from the literature. The unsteady analytical solutions of the SSWE are derived by applying the transformation method to the transition from a fixed cartesian to a rotating coordinate system. Fundamental examples of the new unsteady analytical solutions are presented for specific wind profiles. With the presented unsteady analytical solutions one can provide a measure of the numerical convergence in the case of a temporally evolving system. An application to the atmospheric model PLASMA shows the benefit of unsteady analytical solutions for the quantification of convergence properties.     

Non‐Volkov solutions for a charge in a plane wave

As it is known, a set of solutions of the Klein‐Gordon and Dirac equations with a plane‐wave field was found for the first time by Volkov. We construct new solutions of these equations different from the Volkov ones. In particular, the new solutions are characterized by quantum numbers different from Volkov solutions. In fact, our result is based on the demonstration that the transversal charge motion in a plane wave can be mapped by a special quantum transformation to transversal free particle motion. Similarly, we find new sets of solutions of the Klein‐Gordon and Dirac equations with the combined electromagnetic field.     

Multi-bump, self-similar, blow-up solutions of the Ginzburg-Landau equation

For the Ginzburg-Landau equation (GL), we establish the existence and local uniqueness of two classes of multi-bump, self-similar, blow-up solutions for all dimensions 2<d<4 (under certain conditions on the coefficients in the equation). In numerical simulation and via asymptotic analysis, one class of solutions was already found; the second class of multi-bump solutions is new.In the analysis, we treat the GL as a small perturbation of the cubic nonlinear Schrödinger equation (NLS). The existence result given here is a major extension of results established previously for the NLS, since for the NLS the construction only holds for d close to the critical dimension d=2.The behaviour of the self-similar solutions is described by a nonlinear, non-autonomous ordinary differential equation (ODE). After linearisation, this ODE exhibits hyperbolic behaviour near the origin and elliptic behaviour asymptotically. We call the region where the type of behaviour changes the mid-range. All of the bumps of the solutions that we construct lie in the mid-range.For the construction, we track a manifold of solutions of the ODE that satisfy the condition at the origin forward, and a manifold of solutions that satisfy the asymptotic conditions backward, to a common point in the mid-range. Then, we show that these manifolds intersect transversely. We study the dynamics in the mid-range by using geometric singular perturbation theory, adiabatic Melnikov theory, and the Exchange Lemma.     

Factor structure of the Tomimatsu‐Sato solutions and their recurrence relations

We consider stationary axisymmetric vacuum solutions of Einstein's equations for which the Ernst potential is rational in prolate spheroidal coordinates. Extending an earlier study we show that there are several new expressions which are factorizable. In particular, we concentrate on the Tomimatsu‐Sato solutions and their recurrence relations. Various continuum limits of the recurrence relations will be discussed.