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Lax pairs informed neural networks solving integrable systems

Journal of Computational Physics
Lax pairs are one of the most important features of integrable system. In this work, we propose the Lax pairs informed neural networks (LPNNs) tailored for the integrable systems with Lax pairs by designing novel network architectures and loss functions,
Yong Chen
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Lax pairs

Boletim da Sociedade Brasileira de Matemática, 1984
In the pioneering paper [Commun. Pure Appl. Math. 21, 467-490 (1968; Zbl 0162.411)], \textit{P. Lax} stated a condition under which certain one parameter families of operators \(\{\) L(t)\(\}\) are isospectral, i.e., all the L(t) have the same spectrum.
Neto, Hermano Frid, Thayer, F. Javier
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A note on the Lax pairs for Painlevéequations

Journal of Physics A: Mathematical and General, 1999
The Painlevé equations PI-PVI are six genuine nonlinear second-order differential equations such that the only movable singularities of their solutions are poles. By definition, movable singular are points which change when the initial conditions are changed.
Kapaev, A. A., Hubert, E.
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A Lax pair for Kowalevski's top

Physica D: Nonlinear Phenomena, 1987
We show that on each level surface of the invariants, the equations of the Kowalevski top are equivalent to a Neumann system describing the motion of a mass point on the sphere \(S^ 2:| p| =1\) under the influence of a force -Qp. This allows us to write a global Lax pair for the Kowalevski system and to show that Kowalevski's original reduction of the ...
Haine, L., Horozov, E.
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THE LAX PAIR OF A GENERALIZED THIRRING MODEL

International Journal of Modern Physics A, 1999
The system of coupled nonlinear partial differential equations called the Massive Thirring Model is reviewed. In particular it is analyzed in the chiral fermion version, which is extended by introducing a local gauge symmetry in place of the usual global symmetry. This is done by minimally coupling the fermions with a SU L(2) ⊗ SU R(2) gauge potential.
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Generalized Lax pairs for the computation of semi-invariants

49th IEEE Conference on Decision and Control (CDC), 2010
A Lax pair is a classical tool for the computation of first integrals of continuous-time nonlinear systems. Semi-invariants extend the concept of first integral and generalize the concept of the pair (eigenvalue, left eigenvector) of a linear mapping to the nonlinear framework, whence play the role of basic bricks for the computation of Lyapunov ...
MENINI, LAURA, TORNAMBE', ANTONIO
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Banach Algebras Associated to Lax Pairs

Reports on Mathematical Physics, 2015
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Lax-Pair-FIND: Discovering Lax pair from scarce data via deep learning

Chaos: An Interdisciplinary Journal of Nonlinear Science
With the flourishing development of data science and machine learning, significant progress has been made in solving forward and inverse problems of partial differential equations (PDEs) and discovering mathematical equations that describe physical systems.
Shuning Lin, Yong Chen
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Lax Pair Equations and Connes-Kreimer Renormalization

Communications in Mathematical Physics, 2010
Lax pairs are often used to generate solutions to PDEs. Generally speaking, solutions of finite type to a given integrable PDE can be reduced to solving a system of ODEs or alternatively by finding a Birkhoff factorization. (For solutions of infinite type, more likely one must first solve a system of ODEs and then carry out a Birkhoff factorization ...
Baditoiu, G., Rosenberg, S.
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Isospectral flow and lax pairs

Physics Letters A, 1984
Abstract Certain non-linear differential equations may be considered as compatibility conditions for a linear system of equations, that is, as the vanishing-curvature condition for some connection. It is shown how one can obtain this connection from an isospectral-flow condition using a Foldy-Wouthuysen transformation.
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