Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.14996

Multivariable Painleve'-II equation: connection formulas for arbitrary system size

Nikolai A. Sinitsyn

nlin.SIcond-mat.dis-nnmath-phquant-phSubmitted Sep 14, 2026• Announced Sep 15, 2026• Revised Oct 6, 2026 (v3)
v1 · Sep 14, 2026v2 · Sep 28, 2026v3 · Oct 6, 2026
arXivHTMLPDF

Abstract

Connection formulas for the asymptotic solutions of a system of n> 1 coupled Painleve'-II equations with symmetry-breaking parameters are written explicitly. An asymptotically exact WKB approach to these formulas relies on the quantum-mechanical independent crossing approximation for an explicitly time-dependent Schroedinger equation.

Comments
45 pages with figures. v3 removed discussion of corrections

Related papers

New exact quasi-classical asymptotic beyond WKB approximation and beyond Maslov formal expansion

Jaykov Foukzon, Менькова Елена Романовна, Alexander A. Potapov +1 more

New exact quasi-classical asymptotic of solutions to the n -dimensional Schrödinger equation beyond WKB-theory and beyond Maslov's canonical operator theory is presented. Quantum jump nature is considered successfully. We pointed out that an explanation of quantum jumps can be found to result from Colombeau solutions of the Schrödinger equation alone without additional postulates. Such jumps, does not depend on any single measurement of particle position q ( t ) at instant t and completely obtained only from Schrodinger equation, without reference to any phenomenological master-equation of Lindblad form.

Journal of Physics Conference Series•Sep 21, 2015•1 citation

Multivariable Painleve'-II equation: connection formulas for asymptotic solutions

N. A. Sinitsyn

For an integrable generalization of the Painleve'-II equation (P-II) to a system of coupled equations with symmetry breaking terms, an asymptotically exact WKB analysis is applied to obtain connection formulas for solutions at different infinities. The analysis relies on an exact solution of the quantum mechanical Demkov--Osherov model (DOM), revealing a possible deeper relation between classical integrable systems and solvable multistate Landau--Zener models. An application of the connection formulas to the problem of unstable vacuum decay during a second-order phase transition provides precise scaling of the number of excitations, including subdominant contributions.

math-phCross-listedarXiv:2603.22470•Mar 25, 2026

Quasiadiabatic solution to the two-level problem

R. T. Robiscoe

We consider a two-level system where a pair of discrete quantum states of arbitrary widths and separation are coupled by a general time-dependent external field with a pulselike matrix element, $V(t)$. Using a transformation suggested by Robinson [Phys. Rev. A 24, 2239 (1981)] and applying the WKB approximation, a quasiadiabatic solution is constructed for the system's transition amplitude. This solution is useful in estimating transition amplitudes in the asymptotic region for a wide variety of smoothly varying pulses $V(t)$.

Physical Review A•Feb 1, 1982•6 citations

Completely Integrable Equation for the Quantum Correlation Function of Nonlinear Schrödinger Equation

Takeo Kojima, V. E. Korepin, N. A. Slavnov

arXiv:hep-th/9612252•Communications in Mathematical Physics•Nov 1, 1997•25 citations

Nonlinear Luttinger Liquid from Painlev\'e IV

Tom Price, Dmitry L. Kovrizhin, Austen Lamacraft

We show that exact time dependent single particle Green function in the Imambekov-Glazman theory of nonlinear Luttinger liquids can be written, for any value of the Luttinger parameter, in terms of a particular solution of the Painleve IV equation. Our expression for the Green function has a form analogous to the celebrated Tracy-Widom result connecting the Airy kernel with Painleve II. The asymptotic power law of the exact solution as a function of a single scaling variable $x/\sqrt{t}$ agrees with the mobile impurity results. The full shape of the Green function in the thermodynamic limit is recovered with arbitrary precision via a simple numerical integration of a nonlinear ODE.

•Jun 28, 2016•0 citations

Exact integral formulas and asymptotics for the correlations in the quantum many body system

Peter J. Forrester

Physics Letters A•Aug 1, 1993•42 citations
Connection formulas for the asymptotic solutions of a system of n> 1 coupled Painleve'-II equations with symmetry-breaking parameters are written explicitly. An asymptotically exact WKB approach to these formulas relies on the quantum-mechanical independent crossing approximation for an explicitly time-dependent Schroedinger equation.
New exact quasi-classical asymptotic of solutions to the n -dimensional Schrödinger equation beyond WKB-theory and beyond Maslov's canonical operator theory is presented. Quantum jump nature is considered successfully. We pointed out that an explanation of quantum jumps can be found to result from Colombeau solutions of the Schrödinger equation alone without additional postulates. Such jumps, does not depend on any single measurement of particle position q ( t ) at instant t and completely obtained only from Schrodinger equation, without reference to any phenomenological master-equation of Lindblad form.
For an integrable generalization of the Painleve'-II equation (P-II) to a system of coupled equations with symmetry breaking terms, an asymptotically exact WKB analysis is applied to obtain connection formulas for solutions at different infinities. The analysis relies on an exact solution of the quantum mechanical Demkov--Osherov model (DOM), revealing a possible deeper relation between classical integrable systems and solvable multistate Landau--Zener models. An application of the connection formulas to the problem of unstable vacuum decay during a second-order phase transition provides precise scaling of the number of excitations, including subdominant contributions.
We consider a two-level system where a pair of discrete quantum states of arbitrary widths and separation are coupled by a general time-dependent external field with a pulselike matrix element, V(t)V(t)V(t). Using a transformation suggested by Robinson [Phys. Rev. A 24, 2239 (1981)] and applying the WKB approximation, a quasiadiabatic solution is constructed for the system's transition amplitude. This solution is useful in estimating transition amplitudes in the asymptotic region for a wide variety of smoothly varying pulses V(t)V(t)V(t).
We show that exact time dependent single particle Green function in the Imambekov-Glazman theory of nonlinear Luttinger liquids can be written, for any value of the Luttinger parameter, in terms of a particular solution of the Painleve IV equation. Our expression for the Green function has a form analogous to the celebrated Tracy-Widom result connecting the Airy kernel with Painleve II. The asymptotic power law of the exact solution as a function of a single scaling variable x/tx/\sqrt{t}x/t​ agrees with the mobile impurity results. The full shape of the Green function in the thermodynamic limit is recovered with arbitrary precision via a simple numerical integration of a nonlinear ODE.