The modified tanh-coth function method is used to obtain new exact travelling wave solutions for Zhiber-Shabat equation and the related equations: Liouville equation, sinh-Gordon equation, Dodd-Bullough-Mikhailov equation, and Tzitzeica-Dodd-Bullough equation. Exact travelling wave solutions for each equation are derived and expressed in terms of hyperbolic functions, trigonometric functions an...

In this paper we study the existence and uniqueness of solutions of a certain Fredholm type Riemann-Liouville integral equation of two variables by using Banach contraction principle.

The phase flow method was originally introduced in [28] which can efficiently compute the autonomous ordinary differential equations. In [13], it was generalized to solve the Hamiltonian system where the Hamiltonian contains discontinuous functions, for example discontinuous potential or wave speed. However, both these works require the flow map constructed on an invariant manifold. This could ...

This paper concerns the derivation of a Fokker-Planck equation for the correlation of two high frequency wave fields propagating in two different random media. The mismatch between the random media need be small, on the order of the wavelength, and their correlation length need be large relative to the wavelength. The loss of correlation caused by the mismatch in the random media is quantified ...

We review the relation between the matrix model and Liouville approaches to two-dimensional gravity as elaborated by Moore, Seiberg and Staudacher. Then, based on the supersymmetric Liouville formulation and the discrete eigenvalue model proposed by Alvarez-Gaumé, Itoyama, Mañes and Zadra, we extend the previous relation to the supersymmetric case. The minisuperspace approximation for the super...

An inverse spectral problem is studied for the non-selfadjoint matrix Sturm–Liouville differential equation on the half-line. We give a formulation of the inverse problem, prove the corresponding uniqueness theorem and provide a constructive procedure for the solution of the inverse problem by the method of spectral mappings. The obtained results are natural generalizations of the classical res...

This paper is concerned with the construction and study of specific finite and infinite integral transforms arising from a singular Sturm-Liouville problem on a half line, which is connected to the hydrogen atom equation. The paper goes further to describe fully the image of some spaces of square integrable functions with respect to some measure under the finite and infinite integral transforms...

An analogue to the theory of Riesz potentials and Liouville operators in R for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As ...

  Abstract.   The Sturm-Liouville boundary value problem of the multi-order fractional differential equation  is studied. Results on the existence of solutions are established. The analysis relies on a weighted function space and a fixed point theorem. An example is given to illustrate the efficiency of the main theorems.

We build a family of entire solutions to the Allen-Cahn equation in RN+1 for N ≥ 3, whose level set approaches the higher dimensional catenoid in a compact region and has two logarithmic ends governed by the solutions to the Liouville equation.