Periodic solutions of a singularly perturbed delay differential equation

A singularly perturbed differential delay equation of the form

(1)     

Travelling solitary wave solutions to a compound KdV-Burgers equation

Explicit travelling solitary wave solutions to a compound KdV-Burgers equation are obtained by using an automated method. A two-dimensional generalization is discussed.     

Existence of perturbed solitary wave solutions to a model equation for water waves

We prove the existence of travelling wave solutions to a fifth order partial differential equation, which is a formal asymptotic approximation for water waves with surface tension. These travelling waves are arbitrarily small perturbations of solitary waves, but are not solitary waves themselves, because they approach small amplitude oscillations for large values of the independent variable. This result suggests that for Bond numbers less than one third, there are branches of travelling wave solutions to the water wave equations, which are perturbations of supercritical elevation solitary waves, and which bifurcate from Froude number one and Bond number one third.     

Solitary wave solution to a singularly perturbed generalized Gardner equation with nonlinear terms of any order

This paper is concerned with the existence of travelling wave solutions to a singularly perturbed generalized Gardner equation with nonlinear terms of any order. By using geometric singular perturbation theory and based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of solitary wave solutions of this equation is proved when the perturbation parameter is sufficiently small. The numerical simulations verify our theoretical analysis.     

Asymptopic solution for a class of semilinear singularly perturbed fractional differential equation

This paper considers a class of boundary value problems for the semilinear singularly perturbed fractional differential equation.Under the suitable conditions,first,the outer solution of the original problem is obtained;secondly,using the stretched variable and the composing expansion method the boundary layer is constructed;finally,using the theory of differential inequalities the asymptotic behaviour of solution for the problem is studied and the uniformly valid asymptotic estimation is discussed.     

一类奇摄动非线性方程Robin问题激波的位置

利用匹配法构造了一类奇摄动非线性方程Robin问题激波解的渐近表示式.得出了激波在边界附近出现的条件.     

Travelling solitary wave solutions for the generalized Burgers--Huxley equation with nonlinear terms of any order

In this paper, the travelling wave solutions for the generalized Burgers--Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved.     

Instability of oscillatory shock profile solutions of the generalized Burgers-KdV equation

Oscillatory shock profile solutions of the Burgers-KdV equation are studied in the limit as the viscosity → 0. A rigorous proof of their instability is obtained by showing that the linearized operator about such a solution has many unstable eigenvalues for sufficiently small . The result is obtained by applying a topological method introduced by Alexander, Gardner and Jones to extract spectral information about the perturbed wave with slightly positive viscosity from particular “pieces” of the underlying wave which are approximated by a solitary wave solution of the reduced equation with no viscous dissipation.     

一类广义扰动KdV-Burgers方程的同伦近似解

通过构造一个同伦映射, 研究了一类广义扰动KdV-Burgers方程. 在引入典型无扰动任意次广义KdV-Burgers方程扭状孤立波解的基础上, 研究了扰动方程的具有任意精度的近似解,指出了近似解级数的收敛性, 最后利用不动点定理,进一步说明近似解的有效性,并对精度进行了讨论. 关键词： 广义扰动KdV-Burgers方程 同伦映射 渐近方法 近似解     

扰动Vakhnenko方程物理模型的行波解

研究了一类扰动Vakhnemko方程.给出了改进的渐近方法.首先, 对原模型系统对应的典型方程得到对应的行波解.其次, 引入一个泛函, 建立迭代关系式,将求解非线性问题转化为求解一系列的迭代序列.然后, 逐次地求出对应的解的近似式, 最后,得到了原扰动Vakhnemko模型行波解的任意次精度的近似展开式,并讨论了它的精度. 关键词： 泛函 行波解 Vakhnemko方程     

Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity

In this Letter, we investigate the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity. All explicit expressions of the bounded traveling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded traveling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Exact and numerical solutions to the perturbed sine-Gordon equation

We present numerical and analytic solutions to the perturbed sine-Gordon equation, which models long Josephson tunnel junctions. We make comparisons between numerical results and results obtained from perturbational methods. We present unstable, analytic kink solutions to the equation and further a solution, which is an array of kinks, corresponding to a solution, where the current through the junction is larger than the critical current.     

Solitary wave solutions for the regularized long-wave equation

The regularized long-wave equation has been solved numerically using the collocation method based on the Adams-Moulton method for the time integration and quintic B-spline functions for the space integration. The method is tested on the problems of propagation of a solitary wave and interaction of two solitary waves. The three conserved quantities of motion are calculated to determine the conservation properties of the proposed algorithm. The L _?? error norm is used to measure the difference between exact and numerical solutions. A comparison with the previously published numerical methods is performed.     

Travelling wave-like solutions of the Zakharov-Kuznetsov equation with variable coefficients

Travelling wave-like solutions of the Zakharov-Kuznetsov equation with variable coefficients are studied using the solutions of Raccati equation. The solitary wave-like solution, the trigonometric periodic wave solution and the rational wave solution are obtained with a constraint between coefficients. The property of the solutions is numerically investigated. It is shown that the coefficients of the equation do not change the wave amplitude, but may change the wave velocity.      

Multiple-mode wave solutions in Vakhnenko equation

A new type of wave solutions, called as multiple-mode waves, which can be expressed in the superposition forms of more than two types of single-mode waves of Vakhnenko equation have been investigated in this Letter. A new general method for obtaining the multiple-mode waves is proposed, based on which four cases of the possible forms of wave solutions with two-mode have been derived. The explicit expressions of the two-mode waves as well as the existence conditions have been presented, which may be the nonlinear combinations between periodic waves, solitons, compactons, etc., with different wave speeds, respectively. It is pointed out that more complicated multiple-mode waves with more than three single-mode waves can be derived accordingly, which can be used to reveal the evolution of interactions between different types of waves, especially between various solitons.     

Multipole solutions of the wave equation

The wave equation is solved by the operator separation method proposed in V. V. Zashkvara and N. N. Tyndyk, Zh. Tekh. Fiz. 61(4), 148 (1991) [Sov. Phys. Tech. Phys. 36, 456 (1991)]. Solutions describing the evolution of circular-multipole fields are obtained in a cylindrical coordinate system. Zh. Tekh. Fiz. 68, 9–14 (June 1998) Deceased.     

Existence of traveling wave solutions of a high-order nonlinear acoustic wave equation

In this Letter, we present an analytical study of a high-order acoustic wave equation in one dimension, and reformulate a previously given equation in terms of an expansion of the acoustic Mach number. We search for non-trivial traveling wave solutions to this equation, and also discuss the accuracy of acoustic wave equations in terms of the range of Mach numbers for which they are valid.     

Travelling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms

Applying the general projective Riccati equations method, we consider the exact travelling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms using symbolic computation.From our results, we not only can successfully recover some previously known travelling wave solutions found by using various tanh methods, but also can obtain some new formal solutions. The solutions obtained include kink-shaped solitons, bell-shaped solitons, singular solitons and periodic solutions.