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

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.