共查询到20条相似文献,搜索用时 15 毫秒
1.
Solving Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces via an Improved Variable Separation Approach 总被引:1,自引:0,他引:1
LIDe-Sheng LUOCheng-Xin ZHANGHong-Qing 《理论物理通讯》2004,42(1):1-3
Starting from Baecklund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x, t} and {y, t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach. 相似文献
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3.
A nonlinear transformation and some
multi-solition solutions for the (2+1)-dimensional generalized
Broer-Kaup (GBK) system is first given by using the homogeneous
balance method. Then starting from the nonlinear transformation,
we reduce the (2+1)-dimensional GBK system to a simple linear
evolution equation. Solving this equation, we can obtain some new
explicit exact solutions of the original equations
by means of the extended hyperbola function method. 相似文献
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An extended functional transformation method and its application in some evolution equations 总被引:1,自引:0,他引:1
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In this paper, an extended functional transformation is given to solve some nonlinear evolution equations. This function, in fact,is a solution of the famous KdV equation, so this transformation
gives a transformation between KdV equation and other soliton equations. Then many new exact solutions can be given by virtue of the solutions of KdV equation. 相似文献
6.
An explicit N-fold Darboux transformation for evolution equations determined by general 2×2 AKNS system is constructed. By using the Darboux transformation, the solutions of the evolution equations are reduced to solving alinear algebraic system, from which a unified and explicit formulation of 2N-soliton solutions for the evolution equation are given. Furthermore, a reduction technique for MKdV equation is presented, and an N-fold Darboux transformation of MKdV hierarchy is constructed through the reduction technique. A Maple package which can entirely automatically output the exact N-soliton solutions of the MKdV equation is developed. 相似文献
7.
An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density ∣u∣2 is conserved. We also present an integrable vector Gross–Pitaevskii system with a parabolic potential, where the total particle density ${\sum }_{j=1}^{n}| {u}_{j}{| }^{2}$ is conserved. These equations are related to nonisospectral scalar and vector nonlinear Schrödinger equations. Infinitely many conservation laws are obtained. Gauge transformations between the standard isospectral nonlinear Schrödinger equations and the conserved Gross–Pitaevskii equations, both scalar and vector cases are derived. Solutions and dynamics are analyzed and illustrated. Some solutions exhibit features of localized-like waves. 相似文献
8.
HUANGDing-Jiang ZHANGHong-Qing 《理论物理通讯》2004,42(3):325-328
By using the extended homogeneous balance method, a new auto-Ba^ecklund transformation(BT) to the generalized Kadomtsew-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-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 KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively. 相似文献
9.
Bilinear Bcklund transformation and explicit solutions for a nonlinear evolution equation
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The bilinear form of two nonlinear evolution equations are
derived by using Hirota derivative. The B\"{a}cklund transformation
based on the Hirota bilinear method for these two equations are
presented, respectively. As an application, the explicit solutions
including soliton and stationary rational solutions for these two
equations are obtained. 相似文献
10.
In this paper, we consider distribution solutions to the aggregation equation ${\rho_{t} + \mathrm{div}(\rho \mathbf{u} ) = 0, \; \mathbf{u} = -\nabla V * \rho}$ in ${\mathbb{R}^{d}}$ , where the density ρ concentrates on a co-dimension one manifold. We show that an evolution equation for the manifold itself completely determines the dynamics of such solutions. We refer to such solutions aggregation sheets. When the equation for the sheet is linearly well-posed, we show that the fully non-linear evolution is also well-posed locally in time for the class of bi-Lipschitz surfaces. Moreover, we show that if the initial sheet is C 1 then the solution itself remains C 1 as long as it remains Lipschitz. Lastly, we provide conditions on the kernel ${g(s) = -\frac{\mathrm{d}V}{\mathrm{d}s}}$ that guarantee the solution remains a bi-Lipschitz surface globally in time, and construct explicit solutions that either collapse or blow up in finite time when these conditions fail. 相似文献
11.
W. R. Chappell 《Journal of statistical physics》1970,2(3):267-277
We show that in the case of a dilute gas of neutral particles kinetic equations for autocorrelation functions such as $$\left\langle {\hat f\left( {r,v,t} \right)\hat f\left( {r\prime v\prime ,t\prime } \right)} \right\rangle ,where\hat f\left( {r,v,t} \right) = \sum {_{i = 1}^N } \delta \left( {r - r_i \left( t \right)} \right)\delta \left( {v - v_i \left( {tt} \right)} \right)$$ , can be obtained in a very simple manner by the use of the truncated BBGKY hierarchy. The resulting equations correspond to the low-density limit of the results of van Leeuwen and Yip. Moreover, the derivation does not make use of the Bogoliubov adiabatic approximation, and therefore includes non-Markovian effects which can be important in describing light scattering from gases and the collisional narrowing of atomic dipole radiation. The resulting equations in the long-wavelength limit correspond to the non-Markovian Boltzmann equation for the self-correlation part and the non-Markovian, linearized Boltzmann equation for the total autocorrelation function. 相似文献
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Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation 总被引:1,自引:0,他引:1
Certain generalizations of one of the classical Boussinesq-type equations, $$u_{tt} = u_{xx} - (u^2 + u_{xx} )_{xx} $$ are considered. It is shown that the initial-value problem for this type of equation is always locally well posed. It is also determined that the special, solitary-wave solutions of these equations are nonlinearly stable for a range of their phase speeds. These two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations. This contrasts with the results of blow-up obtained recently by Kalantarov and Ladyzhenskaya for the same type of equation, and casts additional light upon the results for the original version (*) of this class of equations obtained via inverse-scattering theory by Deift, Tomei and Trubowitz. 相似文献
14.
辅助方程法已构造了非线性发展方程的有限多个新精确解. 本文为了构造非线性发展方程的无穷序列类孤子精确解, 分析总结了辅助方程法的构造性和机械化性特点. 在此基础上,给出了一种辅助方程的新解与Riccati方程之间的拟Bäcklund变换. 选择了非线性发展方程的两种形式解,借助符号计算系统 Mathematica,用改进的(2+1) 维色散水波系统为应用实例,构造了该方程的无穷序列类孤子新精确解. 这些解包括无穷序列光滑类孤子解, 紧孤立子解和尖峰类孤立子解. 相似文献
15.
On the Riemann–Hilbert problem of a generalized derivative nonlinear Schrödinger equation 总被引:1,自引:0,他引:1
In this work,we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schr?dinger(DNLS)equation.By establishing a matrix Riemann-Hilbert problem and reconstructing potential function q(x,t)from eigenfunctions{Gj(x,t,η)}3/1 in the inverse problem,the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed.Moreover,we also obtain that the spectral functions f(η),s(η),F(η),S(η)are not independent of each other,but meet an important global relation.As applications,the generalized DNLS equation can be reduced to the Kaup-Newell equation and Chen-Lee-Liu equation on the half-line. 相似文献
16.
G. Z. Tu 《Letters in Mathematical Physics》1980,4(4):347-355
Let $$\begin{gathered} u^* = u + \in \eta (x,{\text{ }}t,{\text{ }}u), \hfill \\ \hfill \\ \hfill \\ x^* = x + \in \xi (x, t, u{\text{),}} \hfill \\ \hfill \\ \hfill \\ {\text{t}}^{\text{*}} = {\text{ }}t + \in \tau {\text{(}}x,{\text{ }}t,{\text{ }}u), \hfill \\ \end{gathered}$$ be an infinitesimal invariant transformation of the evolution equation u t =H(x,t,u,?u/?x,...,? n :u/?x n . In this paper we give an explicit expression for \(\eta ^{X^i }\) in the ‘determining equation’ $$\eta ^T = \sum\limits_{i = 1}^n {{\text{ }}\eta ^{X^i } {\text{ }}\frac{{\partial H}}{{\partial u_i }} + \eta \frac{{\partial H}}{{\partial u_{} }} + \xi \frac{{\partial H}}{{\partial x}} + \tau } \frac{{\partial H}}{{\partial t}},$$ where u i =? i u/?x i . By using this expression we derive a set of equations with η, ξ, τ as unknown functions and discuss in detail the cases of heat and KdV equations. 相似文献
17.
We consider Burgers equation forced by a brownian in space and white noise in time process \(\partial_{t}u+\frac{1}{2}\partial_{x}(u)^{2}=f(x,t)\), with \(E(f(x,t)f(y,s))=\frac{1}{2}(|x|+|y|-|x-y|)\*\delta(t-s)\) and we show that there exist intrinsic statistical solutions that are Lévy processes at any given positive time. We give the evolution equation for the characteristic exponent of such solutions; in particular we give the explicit solution in the case u 0(x)=0. 相似文献
18.
In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations. 相似文献
19.
With the aid of a modified similarity transformation we have obtained exact energy eigenvalues of the generalized Dirac-Coulomb equation. This equation consists of the time component of the Lorentz 4-vector potential Vv(T) = -A1/r, and a Lorentz scalar potential Vs(r) = -A2/r. The transformed radial equations are so simple that their solutions are inferred from the conventional solutions of the Schrödinger-Coulomb equation. 相似文献
20.
A stochastic model for a first-order metabolizing system which was studied in the deterministic sense by Branson and others is formulated and a detailed study of the random integral equation arising in the probabilistic model is presented. The equation is used to describe the evolution in time of the amount of metabolite present in the system. Specifically we present a study of the random integral equation of the Volterra type given by $$M\left( {t; \omega } \right) = M\left( {0; \omega } \right)e^{ - et} + \int_0^t {R\left( {\tau ; \omega } \right) e^{ - e\left( {t - \tau } \right)} d\tau , } t \geqslant 0$$ whereM(t; ω) is an unknown random function giving the amount of metabolite in the system at time t ≥ 0. This equation can be expressed in the general form $$x\left( {t; \omega } \right) = h\left( {t; \omega } \right) + \int_0^t {k\left( {t, \tau ; \omega } \right) f\left( {\tau , x\left( {\tau ; \omega } \right)} \right) d\tau } t \geqslant 0$$ which is of a type whose theoretical aspects have recently been studied by the present authors using as a basis the techniques of probabilistic functional analysis. Conditions are derived under which there exists a unique random solution to the above equation. The usefulness of the model is illustrated using computer simulation by considering a one-organ model, an organ-heart model, and a multicompartment model. 相似文献