We extend the least action principle to continuum systems. The data for the new principle consists of the intensity of the wave (or rather the wave action) at two instances of time. We define an appropriate Lagrangian, and formulate a variational problem in terms of it. The critical points of the functional are used to determine the wave’s phase. The theory is applicable to the semiclassical li...

Abstract: The Korteweg-deVries equation, used to describe long and solitary wave propagation on shallow water, later called soliton, besides describing phenomena ranging from the dynamics of plasmas to tsunami can be implemented into Cellular Neural Networks (CNNs), analog circuit which processes signals in real time with the aim to create a completely innovative generation of trasmission lines...

In this paper, the modified simple equation method is used to construct exact periodic and soliton solutions of some nonlinear partial differential equations. Exact solutions of the nonlinear Schrödinger equation, the Hamiltonian amplitude equation, Klein-Gordon equation in 1+2 dimension, the coupled Klein-Gordon equation, the (2 + 1)-dimensional long-wave-short-wave resonance interaction equat...

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L∞(R), n ≥ 4, satisfying V (x) = O(〈x〉−(n+1)/2−ǫ), ǫ > 0.

Abstract. A fully discrete a priori analysis of the finite element heterogenenous multiscale method (FE-HMM) introduced in [A. Abdulle, M. Grote, C. Stohrer, Multiscale Model. Simul. 2014] for the wave equation with highly oscillatory coefficients over long time is presented. A sharp a priori convergence rate for the numerical method is derived for long time intervals. The effective model over ...

We consider a U(1)-invariant nonlinear Klein-Gordon equation in dimension n ≥ 1, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges as t →±∞ to the two-dimensional set of all “nonlinear eigenfunctions” of the form φ(x)e−iωt . This global attraction is caused by ...

Since space charge waves on a particle beam exhibit both dispersive and nonlinear character, solitary waves or solitons are possible. Dispersive, nonlinear wave propagation in high current beams is found to be similar to ion-acoustic waves in plasmas with an analogy between Debye screening and beam pipe shielding. Exact longitudinal solitary wave propagation is found for potentials associated w...

Abstract With the help of bifurcation theory dynamical differential system and maple software, we shall devote ourselves to research travelling wave solutions bifurcations (2 + 1)-dimensional dissipative long equation. The study for equation derives a planar Hamiltonian system. Based on phase portraits, obtain exact explicit expressions some bounded traveling important singular solutions, under...