We prove global well-posedness of the short-pulse equation with small initial data in Sobolev spaceHs for an integer s ≥ 2. Our analysis relies on local well-posedness results of Schäfer & Wayne [12], the correspondence of the short-pulse equation to the sine– Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and...

Usin the Painlev test, it is shown that the only interablc nonlinear Klein-Gordon equations ux,=f(u) with f a linear combination of exponentials are the Liouville, sine-Gordon (or sinh-Gordon) and Mikhailov equations. In particular, the double sine-Gordon equation is not interable.

We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modi-ed Korteweg{de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely integrable hierarchies of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectr...

We describe and characterize rigorously the chaotic behavior of the sine– Gordon equation. The existence of invariant manifolds and the persistence of homoclinic orbits for a perturbed sine–Gordon equation are established. We apply a geometric method based on Mel’nikov’s analysis to derive conditions for the transversal intersection of invariant manifolds of a hyperbolic point of the perturbed ...

This paper presents the perturbation theory for the double–sine–Gordon equation. We received the system of differential equations that shows the soliton parameters modification under perturbation’s influence. In particular case λ = 0 the results of the research transform into well-known perturbation theory for the sine–Gordon equation.

We describe and characterize rigorously the homoclinic structure of the perturbed sine{ Gordon equation under periodic boundary conditions. The existence of invariant manifolds for a perturbed sine{Gordon equation is established. Mel'nikov method, together with geometric analysis are used to assess the persistence of the homoclinic orbits under bounded and time-periodic perturbations.

We have studied the solutions of the combined sine–cosine-Gordon Equation found by Wazwaz [A.M. Wazwaz, Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method, Appl. Math. Comput. 177 (2006) 755] using the variable separated ODE method. These solutions can be transformed into a new form. We have derived the relation between t...

The relationship between the soliton dynamics provided by the classical sine-Gordon and massive Thirring models is exhibited. Solitons are characterized as classical relativistic particles through the consideration of their associated canonical realizations of the Poincare group. It is shown that the soliton in the massive Thirring model determines two different kinds of relativistic particles ...