共查询到20条相似文献,搜索用时 62 毫秒
1.
《Journal of Nonlinear Mathematical Physics》2013,20(2):209-229
Abstract We describe Jacobi’s method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order superintegrable if there are 2n ? 1 (classically) functionally independent second-order symmetry operators. (The 2n ? 1 is the maximum possible number of such symmetries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions. 相似文献
2.
The observation that the soliton-like solutions of a given second-order nonlinear differential equation define the separatrix of the equivalent autonomous system is used to obtain the one-soliton solutions for theφ 4 theories (the usual and the one with the wrong sign of the mass term), theφ 6, theφ 8, the sine-Gordon theories and the KdV equation. Transformations are given which transform the sine-Gordon equation into an equation belonging to theφ 2n class of theories. A procedure is evolved for obtaining the two-soliton solutions for the sine-Gordon theory without the use of Backlund transformations; it is suggested that this procedure may be useful for investigating the existence of similar solutions for theories of the polynomial type. 相似文献
3.
A primary branch solution (PBS) is defined as a solution with m independent n ? 1 dimensional arbitrary functions for an m order n dimensional partial differential equation (PDE). PBSs of arbitrary first order scalar PDEs can be determined by using Lie symmetry group approach companying with the introduction of auxiliary fields. Because of the intrusion of the arbitrary function, the PBSs have abundant and complicated structure. Usually, PBSs are implicit solutions. In some special cases, explicit solutions such as the instanton (rogue wave like) solutions may be obtained by suitably fixing the arbitrary function of the PBS. 相似文献
4.
Konstantinos N. Anagnostopoulos Mark J. Bowick Albert Schwarz 《Communications in Mathematical Physics》1992,148(3):469-485
The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on pointsV
1 andV
2 in the big cell Gr(0) of the Sato Grassmannian Gr. This is a consequence of a well-defined continuum limit in which the string equation has the simple form matrices of differential operators. These conditions onV
1 andV
2 yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraintsL
n
(n0), whereL
n
annihilate the two modified-KdV -functions whose product gives the partition function of the Unitary Matrix Model. 相似文献
5.
为了构造非线性发展方程的无穷序列复合型类孤子新解, 进一步研究了G'(ξ)/G(ξ) 展开法. 首先, 给出一种函数变换, 把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题. 然后, 利用Riccati方程解的非线性叠加公式, 获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解. 在此基础上, 借助符号计算系统Mathematica, 构造了改进的(2+1)维色散水波系统和(2+1)维色散长波方程的无穷序列复合型类孤子新精确解.
关键词:
G'(ξ)/G(ξ)展开法')" href="#">G'(ξ)/G(ξ)展开法
非线性叠加公式
非线性发展方程
复合型类孤子新解 相似文献
6.
In this paper, coupled Higgs field equation and Hamiltonian amplitude equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equation and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the coupled Higgs equation and Hamiltonian amplitude equation using (G??/G)-expansion method, where G?=?G(??) satisfies a second-order linear ordinary differential equation (ODE). The travelling wave solutions expressed by hyperbolic, trigonometric and the rational functions are obtained. 相似文献
7.
Patrick J. McCarthy 《Letters in Mathematical Physics》1980,4(1):39-43
Dirac's equation is a first-order auto-Bäcklund transformation for the Klein-Gordon equation in 4 variables, this equation being second-order in each of the 4 variables. Here first-order auto-Bäcklund transformations are given for any linear equation of arbitrary (possibly different) order in each of n variables (n arbitrary). This class of equations includes, for example, any of the linearised versions of the KdV hierarchy or equations, and the n-dimensional diffusion equation. 相似文献
8.
9.
S. D. Katore A. Y. Shaikh D. V. Kapse S. A. Bhaskar 《International Journal of Theoretical Physics》2011,50(9):2644-2654
The exact solutions of the field equations are obtained by using the gamma law equation of state p=(γ−1)ρ in which the parameter γ depends on scale factor R. The fundamental form of γ(R) is used to analyze a wide range of phases in cosmic history: inflationary phase and radiation-dominated phase. The corresponding
physical interpretations of cosmological solutions are also discussed in the framework of (n+2) dimensional space time. 相似文献
10.
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
11.
本文为了构造非线性发展方程的无穷序列尖峰精确解,给出了Riccati方程的Bäcklund 变换和解的非线性叠加公式,并借助符号计算系统Mathematica,用Degasperis-Procesi方程为应用实例,构造了无穷序列尖峰孤立波解和无穷序列尖峰周期解.
关键词:
Riccati方程
解的非线性叠加公式
尖峰孤立波解
Degasperis-Procesi 方程 相似文献
12.
The solutions of the nonlinear matrix equation in the Atiyah-Hitchin-Drifeld-Manin (AHDM) construction that determine the
Yang-Mills self-dual fields with topological charge k = 4 for symplectic gauge groups are discussed. In the case of Sp(n), n > 2, it is possible to use a procedure that was proposed earlier for generating solutions with k = 3. It is shown that for SU(2) = Sp(1) the AHDM matrix can be generated by using cubic equation solutions with coefficients
that depend on 8k — 3 parameters. 相似文献
13.
The slow group-velocity pulse in fiber described by nonlinear Schrödinger equation were demonstrated and investigated extensively. We derive a more generalized nonlinear Schrödinger equation as the superposition of monochromatic waves and numerically study the propagations of 2.5-fs fundamental and 5-fs second-order solitons. It is found that, for a slow-group velocity fiber, the magnitude of time shift is related with the group velocity and the more generalized NLSE is more suitable than the conventional generalized NLSE. When the pulse is slow down to 50% of normal group velocity (c/n0), the effect of the higher nonlinear terms is significant. 相似文献
14.
In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the
K(m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method
is proposed. 相似文献
15.
《Journal of Nonlinear Mathematical Physics》2013,20(2):235-250
In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler–Lagrange equations. They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton–Jacobi theory and theory of integration of variational equations. In this paper, Lepage equivalents of second-order Euler–Lagrange quasi-linear PDE's are characterised explicitly. A closed (n + 1)-form uniquely determined by the Euler–Lagrange form is constructed, and used to find a geometric solution of the inverse problem of the calculus of variations. 相似文献
16.
17.
In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these
fundamental solutions to represent solutions of the Cauchy problem and to prove L
p
− L
q
estimates for the solutions of the equation with and without a source term. 相似文献
18.
《Journal of Geometry and Physics》1996,20(1):54-76
This paper defines nth order Jacobi fields to be solutions to a second-order nonlinear differential equation defined by the Jacobi map. nth order Jacobi fields arise naturally as acceleration vector fields of geodesic variations. As a main theorem we prove necessity and sufficiency conditions for an nth order Jacobi field to be the acceleration vector field of a variation of geodesics normal to a submanifold. An m geodesic, m ≥ 2, is a smooth curve whose mth covariant derivative vanishes. We prove an index theorem giving bounds for the total m focal multiplicity along an m geodesic m normal to a submanifold in a flat manifold. 相似文献
19.
A system of two nonlinear differential equations for sublattice angles is proposed to describe the spin orientation distribution
in a planar antiferromagnet with uniform antisymmetric exchange in a magnetic field. This system involves the initial symmetry
of the problem and is reduced to a single delay differential equation. The solutions of this system are parameterized by the
initial condition imposed on the angle of one sublattice at the hyperbolic singular point of the phase space. The numerical
analysis of the stability boundary of soliton solutions demonstrates that the transition to the commensurate phase takes place
outside the region where the stochastic solutions appear and is accompanied by the magnetization jump Δm ∼ 10−1
m. 相似文献
20.
《Journal of Nonlinear Mathematical Physics》2013,20(3-4):292-300
Abstract The Madelung representation ψ = u exp(iv) is considered for the d’Alembert equation □ n ψ?F (|ψ|)ψ = 0 to develop a technique for finding exact solutions. We classify the nonlinear function F for which the amplitude and phase of the d’Alembert equation are related to the solutions of the compatible d’Alembert–Hamiltonian system. The equations are studied in n-dimensional Minkowski space. 相似文献