On Darboux integrability of discrete 2D Toda lattices

We study the dispersionless limit of the Manakov system, the integrable two-component generalization of the nonlinear Schrödinger equation. We derive the resulting four-component genus-zero Manakov-Whitham system, characterize its hydrodynamic structure, and show that it passes the Haantjes tensor test for integrability. We show that the branch points of the spectral curve associated with plane wave solutions of the Manakov system are the local Riemann invariants of the dispersionless system. We also use the characteristic speeds to classify the baseband modulational stability/instability of the plane waves, and we study a direct linearization of the Manakov system to characterize their finite-wavenumber stability and verify agreement with the Whitham prediction in the long-wave limit. Finally, we validate the predictions by comparing them with the results of direct numerical simulations.

nlin.SI↗

arXiv2026

Geometric, algebraic and analytic properties of $\mathrm{al}_{ab}$ function for hyperelliptic curves of genus $g$

In this paper, we investigate the geometric, algebraic and analytic properties of the hyperelliptic $\mathrm{al}_{ab}$ functions of a hyperelliptic curve $X$ with genus $g$ as the $\mathrm{al}_{ab}$ functions together with the $\mathrm{al}_a$ functions are a generalization of the Jacobi elliptic $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$ functions. We then demonstrate the differential identities of the $\mathrm{al}_{ab}$ function. These identities are novel integrable partial nonlinear differential equations as an extension of the differential identities in terms of the $\mathrm{al}_a$ function known as the hyperelliptic solutions of the modified Korteweg-de Vries equation. Thus, we also show that by the identities, the $\mathrm{al}_{ab}$ function is useful for expressing hyperelliptic solutions to the nonlinear Schrödinger and complex modified Korteweg-de Vries equations in an explicit form as an extension of the elliptic $\mathrm{sn}$ function solutions.

nlin.SI↗

arXiv2026

Equations of state of hydrodynamic type and particle statistics of a Dyson gas in an analytic confining potential

We investigate the equilibrium thermodynamics of a Dyson gas in connection with a set of integrable statistical mechanical observables satisfying the Toda Lattice hierarchy. We prove that in the thermodynamic limit, the integrable observables are state functions satisfying a set algebraic equations of state in closed form, obtained from direct integration of the Toda Lattice hierarchy in the continuum limit. We then explore the connection between regularity and critical behaviour of the state functions and the Dyson gas particle statistics via Monte Carlo simulations. We show that the properties of the integrable observables, such as regularity, multivaluedness, cusp singularities, carry information on the macroscopic particle statistics and its qualitative changes but with some limitations.

nlin.SI↗