Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · nlin/0302003

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
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

nlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The $\mathrm{al}_{ab}$ functions of hyperelliptic curves: a generalization of Jacobi's elliptic functions and explicit solutions of a degenerate Krichever-Novikov equation

The Jacobi elliptic functions $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ and their ratios are expressed by Weierstrass's functions $\mathrm{al}_a=\sqrt{x-b_a}$ of an elliptic curve. For a hyperelliptic curve of genus $g$, Weierstrass's $\mathrm{al}_a$ functions generalize $\mathrm{sn}$, $\mathrm{cn}$ and $\mathrm{dn}$, and the modified Korteweg-de Vries and sine-Gordon equations stand behind them. In this paper we study the $\mathrm{al}_{ab}$ functions, which are defined by the sigma function shifted by the sum of two half-periods and which, together with the ratios of the $\mathrm{al}_a$ functions, generalize the remaining Jacobi elliptic functions. We determine the exact lattice of periods of $\mathrm{al}_{12}$ and show that it is single-valued on the unramified double covering of the Jacobian corresponding to the double covering $\hat{w}^2=(x-b_1)(x-b_2)$ of the curve. We show that $\mathrm{al}_{12}$ is the Wronskian of the band-edge eigenfunctions $\mathrm{al}_1$ and $\mathrm{al}_2$, so that the half-period translations realize Darboux transformations. As the main result, $\mathrm{al}_{12}$, $\mathrm{al}_2/\mathrm{al}_1$ and $1/\mathrm{al}_{12}$ satisfy one and the same degenerate Krichever-Novikov equation, whose polynomial has the double roots $0$ and $\infty$, and the half-periods act on these solutions as the Möbius symmetries of the equation. Thus the Krichever-Novikov equation stands behind the $\mathrm{al}_{ab}$ functions. In genus one these functions reduce to $\mathrm{sn}$, $\mathrm{ns}$ and $\mathrm{dc}$.

nlin.SI↗
arXiv2026

The integer spectrum of a superintegrable quantum field theory

In our previous article CMP 407 (2026) 131, we explored the rich algebraic structures arising from a specific quantum many-body Hamiltonian quartic in the bosonic creation-annihilation operators. This Hamiltonian originated from the works of Gérard and Grellier on classical turbulent energy transfer in a Lax-integrable setting, and its quantum version respects hierarchies of polynomial conservation laws all of which possess purely integer spectra, as does the Hamiltonian itself. The latter feature, known from numerical studies, has previously remained unproved. Here, we present an elementary solution of the quantum Gérard-Grellier Hamiltonian. The eigenstates are labelled by partitions and the corresponding eigenvalues admit simple explicit expressions in terms of the principal hook lengths. The key ingredient in this construction is a specific basis we introduce, inspired by the classical multisoliton coherent states, in which the Hamiltonian is upper-triangular. The expressions for the eigenvalues then immediately follow, while the full eigenbasis can be recovered via a Gram-Schmidt procedure.

nlin.SI↗
arXiv2026

The \texorpdfstring{$\mathbf N$-soliton}{N-soliton} solution formula for the Kaup--Kupershmidt equation

An explicit formula is presented for the $\mathbf N$-soliton solution to the Kaup--Kupershmidt equation, for any positive integer $\mathbf N,$ by using the input data set consisting of $\mathbf N$ distinct positive $p_j$-values and $\mathbf N$ arbitrarily chosen positive $c_j$-values. The formula is derived from the solution to the inverse scattering problem for a third-order linear operator in the reflectionless case with the $p_j$-values related to the bound-state poles of a transmission coefficient and the $c_j$-values related to the bound-state dependency constants. It is indicated how the formula is constructed from the solution to a Riemann--Hilbert problem and from the solution to a Marchenko integral equation used in the formulation of the aforementioned inverse scattering problem.

nlin.SI↗