Robert L. Pego

Asymptotic stability of solitary waves

AbstractWe show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation

On the Dynamics of Fine Structure

\partial _t u + u\partial _x u + \partial _x^3 u = 0 ,

On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening

is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations,

Solitary waves on Fermi–Pasta–Ulam lattices: IV. Proof of stability at low energy

\partial _t u + \partial _x f(u) + \partial _x^3 u = 0 .

An unconstrained Hamiltonian formulation for incompressible fluid flow

In particular, we study the case wheref(u)=up+1/(p+1),p=1, 2, 3 (and 30, withf∈C4). The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4.) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations.

Solitary waves on Fermi–Pasta–Ulam lattices: III. Howland-type Floquet theory

This paper is the first in a series to address questions of qualitative behaviour, stability and rigorous passage to a continuum limit for solitary waves in one-dimensional non-integrable lattices with the Hamiltonian with a generic nearest-neighbour potential V. Here we establish that for speeds close to sonic, unique single-pulse waves exist and the profiles are governed by a continuum limit valid on all length scales, not just the scales suggested by formal asymptotic analysis. More precisely, if the deviation of the speed c from the speed of sound cs = (V??(0))1/2 is cs2/24 then as 0 the renormalized displacement profile (1/2)rc(/) of the unique single-pulse wave with speed c, qj+1(t)-qj(t) = rc(j-ct), is shown to converge uniformly to the soliton solution of a KdV equation containing derivatives of the potential as coefficients, -rx+rxxx+12(V???(0)/V??(0)) r rx = 0. Proofs involve (a) a new and natural framework for passing to a continuum limit in which the above KdV travelling-wave equation emerges as a fixed point of a renormalization process, (b) careful singular perturbation analysis of lattice Fourier multipliers and (c) a new Harnack inequality for nonlinear differential-difference equations.