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The Zero-Dispersion Limit for the Odd Flows in the Focusing Zakharov-Shabat Hierarchy

Nicholas M. Ercolani·Shan Jin·C. David Levermore·Warren D. MacEvoy Jr
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Abstract

We present a numerical and theoretical study of the zero-dispersion limit of the focusing Zakharov-Shabat hierarchy, which includes NLS and mKdV flows as its second and third members. All the odd flows in the hierachy are shown to preserve real-valued data. We establish the zero-dispersion limit of all the nontrivial conserved densities and associated fluxes for these odd flows for a large class of real-valued initial data which includes all ``single hump'' initial data. In particular, it is done for the ``focusing'' mKdV flow. The method is based on the Lax-Levermore KdV strategy, but here it is carried out in the context of a nonselfadjoint spectral problem. We find that after an algebraic transformation the limiting dynamics of the mKdV equation is identical to that of the KdV equation.

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Nicholas M. Ercolani, Shan Jin, C. David Levermore, Warren D. MacEvoy Jr. 2003-02-03. The Zero-Dispersion Limit for the Odd Flows in the Focusing Zakharov-Shabat Hierarchy. https://arxiv.org/abs/nlin/0302003

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