| Abstract: |
| The nonlinear Schr\odinger and Korteweg--De Vries equations are both universal amplitude equations for one-dimensional nonlinear dispersive waves. As a consequence of their complete integrability, they both have an associated infinite hierarchy of nonlinear PDEs: the NLS and KdV hierarchies.
We present an approximation of certain renormalized conserved quantities for the NLS hierarchy with nonzero boundary condition, in the long-wave regime, by the energies of the KdV hierarchy.
We use this to prove that long-wave solutions to the NLS hierarchy with nonzero boundary condition are approximated by two solutions to KdV hierarchies on a certain timescale. |
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