Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · nlin/0205012

Singular sector of the Burgers-Hopf hierarchy and deformations of hyperelliptic curves

Y. Kodama·B. G. Konopelchenko
View original source ↗Read source document ↗Export metadata ↓

Abstract

We discuss the structure of shock singularities of the Burgers-Hopf hierarchy. It is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy. The stratification is associated with the Birkhoff decomposition of the Grassmannian given by the set of linear spaces spanned by the hierarchy. We then construct integrable hierarchy on each stratum and demonstrate that it describes a deformation of a hyperelliptic curve parametrizing the stratum. The hierarchy is called the hidden Burgers-Hopf hierarchy, and we found the Riemann invarint form and the hodograph solution.

Explore related subjects

nlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Y. Kodama, B. G. Konopelchenko. 2002-10-01. Singular sector of the Burgers-Hopf hierarchy and deformations of hyperelliptic curves. https://doi.org/10.1088/0305-4470%2F35%2F31%2F104

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

KEEP EXPLORING

Related papers

arXiv2026

Elliptic solutions to matrix CKP equation

A class of elliptic solutions to the matrix CKP equation is studied. Equations of motion for their poles and matrix coefficients at the poles are obtained. As in the scalar case, they are of the first order, in contrast to what takes place in the KP and BKP hierarchies, where the equations of motion are of the second order.

nlin.SI↗
arXiv2026

On integrals of non-autonomous dynamical systems in finite characteristic

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlevé equation and the difference second Painlevé equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.

nlin.SI↗
arXiv2026

On equivalence between solution structures of the anti-self-dual Yang--Mills equations

We establish an exact correspondence between the quasi-Wronskian and quasi-Grammian constructions for the anti-self-dual Yang--Mills (ASDYM) equations with complex \(2\times2\) matrix-valued fields. Starting from vacuum wave functions associated with \(2M\) pairwise distinct nonzero constant spectral parameters, we restrict attention to a nonempty local domain on which the corresponding interpolation matrix is invertible. We show that both constructions are governed by the same uniquely solvable matrix-polynomial interpolation problem. In particular, the quasi-Wronskian Darboux polynomial \(T(λ)\) and the quasi-Grammian rational dressing matrix \(W(λ)\) satisfy \[ T(λ)=q_ξ(λ)W(λ), \] where \(q_ξ(λ)\) is the monic scalar polynomial whose zeros are the prescribed poles of \(W(λ)\). This identity yields explicit relations between the corresponding Yang and potential matrices and shows that the two constructions generate identical ASDYM gauge potentials in the same gauge.

nlin.SI↗