Distributional solutions to a generalized wave equation for gravity waves on deep water

The equations governing the linearized small amplitude approximation for gravity waves on deep water can be reformulated by the introduction of a cross-surface differential operator, H, which acts like a square-root of the two-dimensional Laplacian. This yields a single scalar equation for the amplitude of the wave-like motion off a horizontal static surface resulting in a mixed initial and boundary value problem for the wave operator, ∂_tt + c²H. The pressure impulse response for an unperturbed static fluid will be calculated via a formal eigenfunction expansion and it will be shown that this yields a distributional solution. Then, the mixed problem will be generalized to allow for distributional data where the initial data is injected into the non-homogeneous term. By employing eigenfunction representations for distributions with compact support it will be shown that a formal eigenfunction expansion also yields a distributional solution to this generalized mixed problem.     

Exact solutions of the Kudryashov-Sinelshchikov equation

The Kudryashov-Sinelshchikov equation for describing the pressure waves in liquid with gas bubbles is studied. New exact solutions of this equation are found. Modification of truncated expansion method is used for obtaining exact solution of this equation.     

一个曲面波方程的精确解

Abstract. Some exact travelling wave solutions and rational travelling wave solutions of a sur-face wave equation in a convecting fluid are given in this paper.     

A mixed problem for the equation of internal gravity waves in an infinite cylindrical domain

The existence of a unique classical solution to the mixed problem for the equation describing internal gravity waves in a cylindrical domain is proved. The behavior of the solution is studied at t → +∞.     

On asymptotic solutions of the renewal equation

Consider the renewal equation in the form (¹) $\text{u(t) = g(t) + ∝}{}_{\mathrm{o}}{}^{\mathrm{t}}\text{u(t ? τ) ?(τ) dτ}$, where $\text{?(t)}$ is a probability density on [0, ∞) and lim_{t → ∞}g(t) = g₀. Asymptotic solutions of (¹) are given in the case when f(t) has no expectation, i.e., $\text{∝}{}_0{}^{\mathrm{\infty }}\text{t?(t)dt = ∞}$. These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation.     

一类Camassa-Holm型方程的不变子空间及其精确解

研究一类三阶CH型方程的解的问题.利用不变子空间理论,讨论了方程在不同参变量取值情况下所允许的不变子空间,从而得出了此类方程的两个特殊的精确解.所得结果不仅描述了该方程的一些特性,而且丰富了文献中关于此类CH型方程的内容.     

Exact three-wave solutions for the KP equation

A new type of three-wave solution, periodic two-solitary-wave solutions, for (1 + 2)D Kadomtsev-Petviashvili (KP) equation is obtained using the extended three-soliton method and with the help of Maple.     

Exact solutions to the double Sine-Gordon equation

The double Sine-Gordon equation (DSG) with arbitrary constant coefficients is studied by F-expansion method, which can be thought of as an over-all generalization of the Jacobi elliptic function expansion since F here stands for every one of the Jacobi elliptic functions (even other functions). We first derive three kinds of the generic solutions of the DSG as well as the generic solutions of the Sine-Gordon equation (SG), then in terms of Appendix A, many exact periodic wave solutions, solitary wave solutions and trigonometric function solutions of the DSG are separated from its generic solutions. The corresponding results of the SG, which is a special case of the DSG, can also be obtained.     

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

Exact solutions of a novel integrable partial differential equation

We consider the derivation of exact solutions of a novel integrable partial differential equation (PDE). This equation was introduced with the aim that it mirror properties of the second Painlevé equation (P_II), and it has the remarkable property that, in addition to the usual kind of auto-Bäcklund transformation that one would expect of an integrable PDE, it also admits an auto-Bäcklund transformation of ordinary differential equation (ODE) type, i.e., a mapping between solutions involving shifts in coefficient functions, and which is an exact analogue of that of P_II with its shift in parameter.We apply three methods of obtaining exact solutions. First of all we consider the Lie symmetries of our PDE, this leading to a variety of solutions including in terms of the second Painlevé transcendent, elliptic functions and hyperbolic functions. Our second approach involves the use of our ODE-type auto-Bäcklund transformation applied to solutions arising as solutions of an equation analogous to the special integral of P_II. It turns out that our PDE has a second remarkable property, namely, that special functions defined by any linear second order ODE can be used to obtain a solution of our PDE. In particular, in the case of solutions defined by Bessel’s equation, iteration using our ODE-type auto-Bäcklund transformation is possible and yields a chain of solutions defined in terms of Bessel functions. We also consider the use of this transformation in order to derive solutions rational in x. Finally, we consider the standard auto-Bäcklund transformation, obtaining a nonlinear superposition formula along with one- and two-soliton solutions. Velocities are found to depend on coefficients appearing in the equation and this leads to a wide range of interesting behaviours.     

Exact envelope-soliton solutions of a two-dimensional nonlinear wave equation

ExactN-envelope-soliton solutions are obtained, by extending Hirota's procedure, for the twodimensional nonlinear wave Eqn. (1) withq>0, which describes the evolution of the envelope of a train of surface gravity waves on deep water. They are shown to propagate in directions making an angle greater than tan^–1/2 with the propagation direction of the underlying carrier waves. We also point out and discuss the limitations of Hirota's procedure for generating solitonsolutions to problems of more than one spatial dimensions. Envelope-soliton solutions to Eqn. (1) withq<0 are also discussed.

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Sommaire On généralise la méthode de Hirota pour obtenir des solutions exactes àN-solitons et on l'applique à l'équation des ondes nonlinéaires à deux dimensions (1) avecq>0, qui décrit l'évolution de l'enveloppe d'un train d'ondes de gravitation dans un fluide de grande profondeur. Ces solutions se propagent en directions formant un angle plus grand que tan^–1/2 avec la direction de propagation des ondes porteuses fondamentales. On montre aussi que la méthode de Hirota n'est pas capable de produire, en deux dimensions, des solutions exactes aussi générales que dans le cas d'une seule dimension. Enfin on étudie les solutions de l'équation (1) avecq<0.

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Research supported by the Natural Sciences and Engineering Research Council of Canada. The author thanks Dr. G. Tenti for valuable discussions.     

Wick型随机Kadomtsev-Petviashvili方程的精确解

利用Hermite变换和Tanh函数法,研究了Wick型随机Kadomtsev-Petviashvili(KP)方程,得到其三种类型不同的随机精确解.     

Exact solutions of the Liouville equation in multidimensional spaces

Exact solutions of the multidimensional Liouville equation, which are sought in the class of functional forms of the degree n equal to the coordinate space dimension, are constructed based on a special representation of the Laplace and D'Alembert equations. The corresponding Liouville equation solutions are completely described in the coordinate space dimensions d=3,4. For d>4, the general solution form and the method for obtaining algebraic equations for the coefficients of functional n-forms is presented. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 1, pp. 3–19, July, 1999.     

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{?\sigma }$. The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent $\sigma >3$, but the model can be expected to be valid, on heuristic grounds, for $\sigma >\frac{5}{3}$. The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent σ. Here we show that for $\frac{5}{3}<\sigma <2$ the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.     

Exact solutions to difference equation models of Burgers' equation

This article gives exact solutions to several difference equation models of Burgers' equation. The particular cases considered correspond to the diffusion-free, nonlinear steady-state and linear steady-state situations.     

Exact solitary wave solutions of the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the nonlinear dispersive Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.