We consider a nonlinear damped hyperbolic equation in R, 1 ≤ n ≤ 4, depending on a positive parameter ǫ. If we set ǫ = 0, this equation reduces to the well-known Kolmogorov-Petrovski-Piskunov equation. We remark that, after a change of variables, this hyperbolic equation has the same family of one-dimensional travelling waves as the KPP equation. Using various energy functionals, we show that, ...

The connection between motion of space or plane curves and integrable equations has drawn wide interest in the past and many results have been obtained. The pioneering work is due to Hasimoto where he showed in [1] that the nonlinear Schrödinger equation describes the motion of an isolated non-stretching thin vortex filament. Lamb [2] used the Hasimoto transformation to connect other motions of...

Propagation of a Brillouin precursor in a Lorentz dispersive medium is considered. The precursor is excited by a sine modulated initial signal, with its envelope described by a hyperbolic tangent function. The purpose of the paper is to show how the rate of growth of the initial signal affects the form of the Brillouin precursor. Uniform asymptotic approach, pertinent to coalescing saddle point...

In this paper, we will give a global existence theorem in one-dimensional thermoelasticity with second sound in R. For this purpose, we first give the decay rates of the linearized equations with the help of Fourier sine and cosine transformation and the local existence theorem using theorems for quasi-linear symmetric hyperbolic systems. Then we establish some estimates in L, L and L∞ norms to...

We consider the inverse problem for the second order hyperbolic equation in a bounded domain in R n with lower order terms depending analytically on the time variable. We prove that the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism and a gauge transformation.

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashi...