Lagrangian formalism in perturbed nonlinear Klein–Gordon equations

Lagrangian Formalism in Perturbed Nonlinear Klein-Gordon Equations

We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term we are able to write the Lagrangian density and then, calculate the Lagrangian as a function of collective variables. We use the Lagrangian formalism togethe...

Lagrangian for nonlinear perturbed heat and wave equations

Soliton interactions in perturbed nonlinear Schrödinger equations.

We use multiscale perturbation theory in conjunction with the inverse scattering transform to study the interaction of a number of solitons of the cubic nonlinear Schrödinger equation under the influence of a small correction to the nonlinear potential. We assume that the solitons are all moving with the same velocity at the initial instant; this maximizes the effect each soliton has on the oth...

Flow Invariance for Perturbed Nonlinear Evolution Equations

Let X be a real Banach space, J = [0, a] ⊂ IR, A : D(A) ⊂ X → 2 \ ∅ an m-accretive operator and f : J × X → X continuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed sets K ⊂ X for the evolution system u′ +Au f(t, u) on J = [0, a]. More generally, we provide conditions under which this evolution system has mild solut...

Oscillation criteria for perturbed nonlinear dynamic equations

Keywords--Oscil lat ion, Second-order nonlinear dynamic equation, Time scale, Riccati transformation technique, Positive solution. 1. I N T R O D U C T I O N The theory of t ime scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis [1] in order to unify continuous and discrete analysis. Not only can this theory of so-called "dynamic equations" unif...