Nonlinear Dynamics of a System of Particle-Like Wavepackets

Abstract

This work continues our studies of nonlinear evolution of a system of wavepackets. We study a wave propagation governed by a nonlinear system of hyperbolic PDE's with constant coefficients with the initial data being a multi-wavepacket. By definition a general wavepacket has a well defined principal wave vector, and, as we proved in previous works, the nonlinear dynamics preserves systems of wavepackets and their principal wave vectors. Here we study the nonlinear evolution of a special class of wavepackets, namely particle-like wavepackets. A particle-like wavepacket is of a dual nature: on one hand, it is a wave with a well defined principal wave vector, on the other hand, it is a particle in the sense that it can be assigned a well defined position in the space. We prove that under the nonlinear evolution a generic multi-particle wavepacket remains to be a multi-particle wavepacket with a high accuracy, and every constituting single particle-like wavepacket not only preserves its principal wave number but also it has a well-defined space position evolving with a constant velocity which is its group velocity. Remarkably the described properties hold though the involved single particle-like wavepackets undergo nonlinear interactions and multiple collisions in the space. We also prove that if principal wavevectors of multi-particle wavepacket are generic, the result of nonlinear interactions between different wavepackets is small and the approximate linear superposition principle holds uniformly with respect to the initial spatial positions of wavepackets.

Wave-Corpuscle Mechanics for Electric Charges

“…In contrast to Abraham and Lorentz models as well as quantum Abraham models with de Broglie-Bohm laws of quantum motion introduced by M. Kiessling in [26] we do not define a charge as an object with prescribed geometry, the dynamics and shape of a wave corpuscle is governed by a nonlinear Klein-Gordon or a nonlinear Schrödinger equation in relativistic and nonrelativistic cases respectively and a wave-corpuscle is defined as a special type of solutions to these equations. The ideal wave-corpuscle wave function ψ(t, x) defined by (4), (5) together with the corresponding EM field forms an exact solution to the relevant Euler-Lagrange field equations describing an accelerating dressed charge. The point charge momentum p(t) turns out to be exactly equal to the total momentum of the charge as a wave-corpuscle and its electromagnetic field.…”

“…In our approach such a principle is the exact balance of all forces for the resting dressed charge via the static charge equilibrium equation (15). As to a possibility of spatially localized excitations such as wave-packets to maintain their basic properties when they propagate in a dispersive medium with a nonlinearity we refer to our work [3][4][5].…”

Short Pulses Approximations in Dispersive Media

“…This framework is somehow a simplified version of the functional setting of [6] (where the SVEA is not made) and has also proved useful in the study of wave-packets [5] or localized solutions [22].…”

“…Some of these results are briefly recalled below; their common point is that they all abandon the SVEA, because the widely accepted "practical rule" (4) The SVEA (2) is valid if |∇U 0 | ∞ ≪ 1 ε is enforced when the pulses get very small. Alterman and Rauch [1,2,3] modeled short pulses by replacing the fast oscillating term in the initial condition by a fast decaying one; more precisely, they modified the initial condition for (1) as follows: (5) u |t=0 = U 0 (x)e i k·x ε + c.c.…”

“…This general framework allows one to consider initial data of the form (1) -for which the bounded variation measure is obviously U 0 (x)δ (ω(k),k) -and of the form (5). A generalization of (6) is then derived and rigorously justified; this equation however has the drawback of being quite complicated because the transport operator ∂ t + c g · ∇ must be replaced by a nonlocal operator modeling the fact that the group velocity c g depends on the frequency in dispersive media.…”