Stability of Time-Dependent Particlelike Solutions in Nonlinear Field Theories. I

Abstract

The stability of time-dependent particlelike solutions of the form ψ=φ(r)e−iωt is examined for the nonlinear field ∇2ψ−c−2∂2ψ/∂t2=κ2ψ−μ2ψψ*ψ. It is found that such solutions are unstable for all ω.

Stability and instability of standing waves for nonlinear Schrödinger equations

“…We note that ω 0 < ∞. Anderson [1] and Shatah [51] showed that there are stable standing waves for ω close to ω 0 with p = 3, q = 5 and n = 3. However, it…”

“…As we mentioned in Chapter 1, this case was numerically studied by Anderson [1] and Shatah [51] showed that there are stable standing waves for ω close to ω 0 with p = 3, q = 5 and n ≥ 3. Namely, there exists a sequence {ω k } approaching ω 0 , for which e iω k t φ ω k (x) is stable.…”

“…For simplifying the argument, we use the operator H defined by (3.2) and functionals Q and N on X G defined by 1) , and…”

“…When −∆ + V (x) has the first eigenvalue λ 1 , Rose and Weinstein [48] claimed that the standing wave solution e iωt φ ω (x) of (1.1) is stable for ω such that ω > −λ 1 and sufficiently close to −λ 1 . To verify the stability condition of Grillakis, Shatah and Strauss, they investigated the behavior of the function φ ω 2 2 of ω near −λ 1 using the standard bifurcation theory.…”

Some properties ofU(1)gaugedQ-balls

“…According to this method, one should adjust the initial data for the system of equations at one of the boundaries (at the origin of gauged Q-ball in our case) in such a way that the solution satisfies required conditions at the second boundary (for large r in our case). The shooting method is very simple and it is easy to implement it, but it fails to find gauged Q-ball solutions for large r, where the scalar field falls off to zero exponentially, see relations (16), (19). To find a solution for such large values of the coordinate r using the shooting method, one has to fine tune the initial data with very high accuracy, which may even exceed the truncation error of double precision floating-point numbers.…”