共查询到19条相似文献,搜索用时 265 毫秒
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通过把十二个Jacobi椭圆函数分类成四组,提出了新的广泛的Jacobi椭圆函数展开法,利用这一方法求得了非线性发展方程的丰富的Jacobi椭圆函数双周期解.当模数m→0或1时,这些解退化为相应的三角函数解或孤立波解和冲击波解.
关键词:
非线性发展方程
Jacobi椭圆函数
双周期解
行波解 相似文献
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李德生 《原子与分子物理学报》2006,23(5):933-937
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. 相似文献
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借助于符号计算软件Maple,通过一种构造非线性偏微分方程(组)更一般形式精确解的直接方法即改进的代数方法,求解(2+1) 维 Broer-Kau-Kupershmidt方程,得到该方程的一系列新的精确解,包括多项式解、指数解、有理解、三角函数解、双曲函数解、Jacobi 和 Weierstrass 椭圆函数双周期解.
关键词:
代数方法
(2+1) 维 Broer-Kau-Kupershmidt 方程
精确解
行波解 相似文献
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为了获得sine-Gordon型方程的无穷序列精确解,给出三角函数型辅助方程和双曲函数型辅助方程及其Bäcklund变换和解的非线性叠加公式,借助符号计算系统Mathematica,构造了sine-Gordon方程、mKdV-sine-Gordon方程、(n+1)维双sine-Gordon方程和sinh-Gordon方程的无穷序列新精确解.其中包括无穷序列三角函数解、无穷序列双曲函数解、无穷序列Jacobi椭圆函数解和无穷序列复合型解.
关键词:
sine-Gordon型方程
解的非线性叠加公式
辅助方程
无穷序列精确解 相似文献
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本文为了获得非线性发展方程新的无穷序列精确解,给出了几种辅助方程的Böcklund变换和解的非线性叠加公式,并构造了一些非线性发展方程新的无穷序列精确解,其中包括无穷序列Jacobi椭圆函数解、无穷序列双曲函数解和无穷序列三角函数解.该方法在构造非线性发展方程无穷序列精确解方面具有普遍意义.
关键词:
辅助方程法
解的非线性叠加公式
无穷序列解
非线性发展方程 相似文献
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扩展了最近提出的F展开方法以构造非线性演化方程更多的精确解,即将F展开法中的一阶非线性常微分方程和单变量的有限幂级数代之以类似的一阶常微分方程组和两个变量的有限幂级数,这两个变量是一阶常微分方程组的解分量.作为例子,用扩展的F展开法解非线性Schroedinger方程,得到了很丰富的包络形式的精确解,特别是以两个不同的Jacobi椭圆函数表示的解.显然,扩展的F展开方法也可以解其他类型的非线性演化方程. 相似文献
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Many physical systems can be successfully modelled using equations that admit the soliton solutions. In addition, equations with soliton solutions have a significant mathematical structure. In this paper, we study and analyze a three-dimensional soliton equation, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. Indeed, solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour. We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time. Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function, elliptic functions, elementary trigonometric and hyperbolic functions solutions of the equation. Besides, various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique. These solutions comprise dark soliton, doubly-periodic soliton, trigonometric soliton, explosive/blowup and singular solitons. We further exhibit the dynamics of the solutions with pictorial representations and discuss them. In conclusion, we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula. We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new. 相似文献
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Soliton,breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints 下载免费PDF全文
《中国物理 B》2021,30(6):60202-060202
The nonlinear Schro¨dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schro¨dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schro¨dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method. 相似文献
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In this paper, based on new auxiliary nonlinear ordinary differential equation with a sixth-degree nonlinear term, we study the (2 1)-dimensional Davey-Stewartson equation and new types of travelling wave solutions are obtained, which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations. 相似文献
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HUANG Ding-Jiang ZHANG Hong-Qing 《理论物理通讯》2004,42(9)
By using the extended homogeneous balance method, a new auto-Backlund transformation(BT) to thegeneralized Kadomtsev-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of theauto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations,which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KPequation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results ofthese equations are also given respectively. 相似文献
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In this paper, a discrete KdV equation that is related to the famous continuous KdV equation is studied. First, an integrable discrete KdV hierarchy is constructed, from which several new discrete KdV equations are obtained. Second, we correspond the first several discrete equations of this hierarchy to the continuous KdV equation through the continuous limit. Third, the generalized (m, 2N − m)-fold Darboux transformation of the discrete KdV equation is established based on its known Lax pair. Finally, the diverse exact solutions including soliton solutions, rational solutions and mixed solutions on non-zero seed background are obtained by applying the resulting Darboux transformation, and their asymptotic states and physical properties such as amplitude, velocity, phase and energy are analyzed. At the same time, some soliton solutions are numerically simulated to show their dynamic behaviors. The properties and results obtained in this paper may be helpful to understand some physical phenomena described by KdV equations. 相似文献
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With the aid of the symbolic computation, we improve Xie's algorithm [F. Xie, Z.Y. Yan, H. Zhang, Phys. Lett. A 285 (2001) 76], and present a new extended method. Based on the new general ansatz (3), we successfully solve a compound KdV-MKdV equation, and obtain some special solutions which contain soliton solutions, and periodic solutions. The method can also be applied to other nonlinear partial differential equations. 相似文献
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Figen Kangalgil 《Physics letters. A》2008,372(11):1831-1835
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. 相似文献
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In this paper we present the full classification of symmetry-invariant solutions for the Gibbons–Tsarev equation. Then we use these solutions to construct explicit expressions for two-component reductions of Benney’s moments equations, to get solutions of Pavlov’s equation, and to find integrable reductions of the Ferapontov–Huard–Zhang system, which describes implicit two-phase solutions of the dKP equation. 相似文献