Article

**J. d'Analyse Math. 116, 163-218 (2012)**[DOI: 10.1007/s11854-012-0005-7]

## Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions

### Alice Mikikits-Leitner and Gerald Teschl

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of
solutions of the Korteweg-de Vries equation which are decaying perturbations of a quasi-periodic
finite-gap background solution. We compute a nonlinear dispersion relation and show that the

*x/t*plane splits into*g+1*soliton regions which are interlaced by*g+1*oscillatory regions, where*g+1*is the number of spectral gaps.In the soliton regions the solution is asymptotically given by a number of solitons travelling on top of finite-gap solutions which are in the same isospectral class as the background solution. In the oscillatory region the solution can be described by a modulated finite-gap solution plus a decaying dispersive tail. The modulation is given by a phase transition on the isospectral torus and is, together with the dispersive tail, explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve.

** MSC2000:** Primary 35Q53, 37K40; Secondary 35Q15, 37K20

**Keywords:** *Riemann-Hilbert problem, Korteweg-de Vries equation, Solitons*

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