With the exception of the harmonic oscillator, quantum wave-packets usually spread as time evolves. This is due to the non-linear character of the classical equations of motion which makes the various components of the wave-packet evolve at various frequencies. We show here that, using the nonlinear resonance between an internal frequency of a system and an external periodic driving, it is poss...

the homogeneous balance method can be used to construct exact traveling wave solutions of nonlinear partial differential equations. in this paper, this method is used to construct newsoliton solutions of the (3+1) jimbo--miwa equation.

We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation.

The purpose of the present paper is to establish the local energy decay estimates and dispersive estimates for 3-dimensional wave equation with a potential to the initial-boundary value problem on exterior domains. The geometrical assumptions on domains are rather general, for example non-trapping condition is not imposed in the local energy decay result. As a by-product, Strichartz estimates i...

1 Introduction Let Ω be a smooth manifold of dimension d ≥ 2 with C ∞ boundary ∂Ω, equipped with a Riemannian metric g. Let ∆ g be the Laplace-Beltrami operator associated to g on Ω, acting on L 2 (Ω) with Dirichlet boundary condition. Let 0 < T < ∞ and consider the wave equation with Dirichlet boundary conditions:    (∂ 2 t − ∆ g)u = 0 on Ω × [0, T ], u| t=0 = u 0 , ∂ t u| t=0 = u 1 , u| ∂Ω...

Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose–Einstein condensates. As in the classical theory of hyperbolic equations where a nonconvex flux leads to nonclassical solution structures, a nonconvex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth orde...

Both one-dimensional in the horizontal direction (1DH, dispersive and non-dispersive) two-dimensional (2DH) axisymmetric (approximate, analytical solutions are derived for water waves generated by moving atmospheric pressures. For 1DH, three wave components can be identified: locked propagating with speed of pressure, $C_p$ , two free opposite directions respective celerity, according to linear...

From the outset, mathematicians and physicists have been concerned with the ways in which solutions to this equation can be associated to classical particle motion, and the ways in which they cannot: the classical dynamical behavior of particles is intermixed with dispersive spreading and interference phenomena in quantum theory. More recently, nonlinear Schrödinger equations have come to play ...

A direct rational exponential scheme is offered to construct exact multi-soliton solutions of nonlinear partial differential equation. We have considered the Calogero–Bogoyavlenskii–Schiff equation and KdV equation as two concrete examples to show efficiency of the method. As a result, one wave, two wave and three wave soliton solutions are obtained. Corresponding potential energy of the solito...