Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2101.03964

On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems

Arno Kuijlaars·Alexander Tovbis
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states $u(z)$ in the spectral theory of soliton gases for the one dimensional integrable focusing Nonlinear Schrödinger Equation (fNLS) and for the Korteweg de Vries (KdV) equation. Our proofs are based on ideas and methods of potential theory. In particular, we show that the minimizing (positive) measure for certain energy functional is absolutely continuous and its density $u(z)\geq 0$ solves the required integral equation. In a similar fashion we show that $v(z)$, the temporal analog of $u(z)$, is the difference of densities of two absolutely continuous measures. Together, integral equations for $u,v$ represent nonlinear dispersion relation for the fNLS soliton gas. We also discuss smoothness and other properties of the obtained solutions. Finally, we obtain exact solutions of the above integral equations in the case of a KdV condensate and a bound state fNLS condensate. Our results is a first step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis, Phys. Rev. E, 2020. It is expected that the presented ideas and methods will be useful for studying similar classes of integral equation describing, for example, breather gases for the fNLS, as well as soliton gases of various integrable systems.

Explore related subjects

math-phmath.MPnlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Arno Kuijlaars, Alexander Tovbis. 2021-01-11. On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems. https://doi.org/10.1088/1361-6544%2Fac20a5

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

KEEP EXPLORING

Related papers

arXiv2026

A Functorial Theory of Defects in Abelian Chern-Simons Theory

Recent work has constructed Abelian Chern-Simons theories as categorical TQFTs, allowing us to naturally incorporate categorical defects and construct defect extensions of Abelian Chern-Simons TQFTs. We first identify the Turaev-Viro realizations of Abelian Chern-Simons theory in the center and doubled pointed modular cases, clarifying the distinction between single bulk realizations and canonical doubled ones. Alternatively, the Alterfold construction supplies the associated topological boundaries, domain walls, and condensation sectors, establishing an explicit Alterfold/Chern-Simons dictionary. We show that the finite quadratic module is the invariant controlling the bulk theory, its topological symmetries, orientation-reversal invariance, and defects. We further show that multicomponent Abelian BF theory arises as the extended TQFT of an off-diagonal Abelian Chern-Simons theory, placing it naturally within the same extended framework. Finally, we demonstrate that recently proposed Abelian Chern-Simons dualities do not define a genuine TQFT duality. These results provide a concrete model for defects in Abelian topological orders and suggest a route toward the non-Abelian case.

math-ph↗
arXiv2026

Gradient nature of Laplacian growth

For a class of growth processes of Laplacian type in the plane, we suggest an interpretation as a ``gradient descent'' in the space of smooth closed curves. More precisely, we show that boundary of a growing domain moves along a gradient of a certain functional in the space of curves. In the simplest cases this functional is $\log (1/r)$, where $r$ is the external conformal radius of the growing domain.

math-ph↗
arXiv2026

Entanglement-Inducing Quantum Markov Processes

We introduce a new model for a system of interacting bosons placed in an array of sites. At its core is a nonlinear, nonlocal evolution equation, which we have dubbed the Schrödinger-Dirichlet equation. The construction is closely related to the Bose-Hubbard model and to a specific type of generalized bosons. In contrast to conventional mean-field closures, the resulting nonlinear dynamics need not preserve product structure and can generate entanglement from initially separable states. The relevant methods of analysis are based on harmonic analysis for the multiplicative group of positive rationals.

math-ph↗