 06207 Spyridon Kamvissis and Gerald Teschl
 Stability of Periodic Soliton Equations under Short Range Perturbations
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Jul 24, 06

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Abstract. We consider the stability of (quasi)periodic solutions of soliton equations under
short range perturbations
and give a complete description of the long time asymptotics in this situation.
We show that, apart from the phenomenon of the solitons travelling on the quasiperiodic
background, the perturbed solution asymptotically approaches a modulated
solution.
We use the Toda lattice as a model but the same methods and ideas are applicable to
all soliton equations in one space dimension.
More precisely, let $g$ be the genus of the hyperelliptic Riemann surface associated with
the unperturbed solution. We show that the $n/t$pane contains $g+2$ areas where
the perturbed solution is close to a quasiperiodic solution in the same isospectral
torus. In between there are $g+1$ regions where the perturbed solution is asymptotically
close to a modulated lattice which undergoes a
continuous phase transition (in the Jacobian variety) and which interpolates
between these isospectral solutions. In the special case of the free solution ($g=0$) the
isospectral torus consists of just one point and we recover the classical result.
Both the solutions in the isospectral torus and the phase transition are explicitly
characterized in terms of Abelian integrals on the underlying hyperelliptic Riemann surface.
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