Toda Lattice Equation Research Articles

Pseudo grid-based physics-informed convolutional-recurrent network solving the integrable nonlinear lattice equations

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model’s extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model’s extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.

From Algebro Geometric Solutions of the Toda Equation to Sato Formulas

We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the Toda lattice equation from the general formulation in terms of abelian functions when the gaps tend to points. This degeneracy allows us to recover the Sato formulas without using inverse scattering theory or geometric or representation theoretic methods.

Solving the relativistic Toda lattice equation via the generalized exponential rational function method

Solving the relativistic Toda lattice equation via the generalized exponential rational function method

Solving the (2+1)-Dimensional Derivative Toda Equation

The main objectives of this work are to deduce the Lax pair for the Modified Toda lattice hierarchy and to find whether there exists a transformation between (2+1)-dimensional derivative Toda lattice equation and Modified Toda equation. By a new spectral problem, we obtain the Modified Toda lattice hierarchy, furthermore we give the Hirota’s form solution for the Modified Toda lattice equation, thus the Hirota’s solution for the (2+1)-dimensional derivative Toda equation can also be attained correspondingly.

1/N expansion of the D3-D5 defect CFT at strong coupling

We consider four dimensional U(N) mathcal{N} = 4 SYM theory interacting with a 3d mathcal{N} = 4 theory living on a codimension-one interface and holographically dual to the D3-D5 system without flux. Localization captures several observables in this dCFT, including its free energy, related to the defect expectation value, and single trace frac{1}{2} -BPS composite scalars. These quantities may be computed in a hermitian one-matrix model with non-polynomial single-trace potential. We exploit the integrable Volterra hierarchy underlying the matrix model and systematically study its 1/N expansion at any value of the ’t Hooft coupling. In particular, the strong coupling regime is determined — up to non-perturbative exponentially suppressed corrections — by differential relations that constrain higher order terms in the 1/N expansion. The analysis is extended to the model with SU(N) gauge symmetry by resorting to the more general Toda lattice equations.

Exact strong coupling results in mathcal{N} = 2 Sp(2N) superconformal gauge theory from localization

We apply the localization technique to compute the free energy on four-sphere and the circular BPS Wilson loop in the four-dimensional mathcal{N} = ∈ superconformal Sp(2N) gauge theory containing vector multiplet coupled to four hypermultiplets in fundamental representation and one hypermultiplet in rank-2 antisymmetric representation. This theory is unique among similar mathcal{N} = ∈ superconformal models that are planar-equivalent to mathcal{N} = ∆ SYM in that the corresponding localization matrix model has the interaction potential containing single-trace terms only. We exploit this property to show that, to any order in large N expansion and an arbitrary ’t Hooft coupling λ, the free energy and the Wilson loop satisfy Toda lattice equations. Solving these equations at strong coupling, we find remarkably simple expressions for these observables which include all corrections in 1/N and 1/ sqrt{lambda } . We also compute the leading exponentially suppressed mathcal{O}left({e}^{-sqrt{lambda }}right) corrections and consider a generalization to the case when the fundamental hypermultiplets have a non-zero mass. The string theory dual of this mathcal{N} = ∈ gauge theory should be a particular orientifold of AdS5 × S5 type IIB string and we discuss the string theory interpretation of the obtained strong-coupling results.

The κ-Deformed Calogero-Leyvraz Lagrangians and Applications to Integrable Dynamical Systems.

The Calogero-Leyvraz Lagrangian framework, associated with the dynamics of a charged particle moving in a plane under the combined influence of a magnetic field as well as a frictional force, proposed by Calogero and Leyvraz, has some special features. It is endowed with a Shannon "entropic" type kinetic energy term. In this paper, we carry out the constructions of the 2D Lotka-Volterra replicator equations and the N=2 Relativistic Toda lattice systems using this class of Lagrangians. We take advantage of the special structure of the kinetic term and deform the kinetic energy term of the Calogero-Leyvraz Lagrangians using the κ-deformed logarithm as proposed by Kaniadakis and Tsallis. This method yields the new construction of the κ-deformed Lotka-Volterra replicator and relativistic Toda lattice equations.

A systematic construction of integrable delay-difference and delay-differential analogues of soliton equations

We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotka–Volterra, Toda lattice (TL), and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional TL equations. Each of the delay-difference and delay-differential equations has the N-soliton solution, which depends on the delay parameter and converges to an N-soliton solution of a known soliton equation as the delay parameter approaches 0.

Integrability and geometry of the Wynn recurrence

We show that the Wynn recurrence (the missing identity of Frobenius of the Padé approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev–Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Padé theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions.

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to 3×3 matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known 3×3 Lax pair of this equation, the discrete generalized (m,3N−m)-fold Darboux transformation was constructed for the first time and extended from the 2×2 Lax pair to the 3×3 Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants.

Two new integrable differential-difference hierarchies related to the relativistic Toda lattice equation and their soliton fusion or fission

Starting from a discrete spectral problem, two new integrable differential-difference hierarchies are presented and their Lax pairs are given, respectively. The first hierarchy is related to the relativistic Toda lattice equation. Darboux transformations for the first systems in the two hierarchies are derived, respectively, accordingly their explicit solutions are obtained. By selecting appropriate parameters, we can see the phenomena of soliton fission or soliton fusion.

N-Fold Darboux Transformation and Soliton Solutions for The relativistic Toda Lattice Equation

Under investigation in this paper is the relativistic Toda lattice (RTL) equation which is a deformation of the Toda lattice (TL) equation and may simulate many physical phenomena. The N -fold Darboux transformation (DT) of the RTL equation is constructed in terms of determinants. Compared with the usual 1-fold DT, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. Based on the N-fold DT, we obtain the N-fold explicit solutions from initial solutions. The figures of one-, two-, three-and four-soliton solutions with proper parameters are presented to illustrate the propagation of solitary waves, and elastic interactions between or among the two-, three- and four-soliton solutions are discussed: solitonic shapes and amplitudes have not changed after the interaction. What is more, the relationship between the structures of solutions and the parameters is generated with N = 1, from which we find that the 1-fold solutions may be one-soliton solutions or periodic solutions. Results in this paper might be helpful for understanding and interpreting some physical phenomena.

Integrability, discrete kink multi-soliton solutions on an inclined plane background and dynamics in the modified exponential Toda lattice equation

Integrability, discrete kink multi-soliton solutions on an inclined plane background and dynamics in the modified exponential Toda lattice equation

Crossing bridges with strong Szeg\u0151 limit theorem

We develop a new technique for computing a class of four-point correlation functions of heavy half-BPS operators in planar mathcal{N} = 4 SYM theory which admit factorization into a product of two octagon form factors with an arbitrary bridge length. We show that the octagon can be expressed as the Fredholm determinant of the integrable Bessel operator and demonstrate that this representation is very efficient in finding the octagons both at weak and strong coupling. At weak coupling, in the limit when the four half-BPS operators become null separated in a sequential manner, the octagon obeys the Toda lattice equations and can be found in a closed form. At strong coupling, we exploit the strong Szegő limit theorem to derive the leading asymptotic behavior of the octagon and, then, apply the method of differential equations to determine the remaining subleading terms of the strong coupling expansion to any order in the inverse coupling. To achieve this goal, we generalize results available in the literature for the asymptotic behavior of the determinant of the Bessel operator. As a byproduct of our analysis, we formulate a Szegő-Akhiezer-Kac formula for the determinant of the Bessel operator with a Fisher-Hartwig singularity and develop a systematic approach to account for subleading power suppressed contributions.

A generalized integrable lattice hierarchy associated with the Toda and modified Toda lattice equations: Hamiltonian representation, soliton solutions

In this paper, we give an integrable four-field lattice hierarchy associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. The Hamiltonian representation of the four-field lattice hierarchy are constructed by using the trace identity. We consider reductions of our hierarchy, and find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, and some new integrable three-field lattice hierarchies. In addition, the modified Toda lattice equation is studied, the figures of one-, two-soliton and periodic solutions with properly parameters are shown graphically to illustrate the propagation of solitary waves, and we find that the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The elastic and inelastic interactions among the two-soliton for the modified Toda lattice equation are also discussed. Furthermore, we generate the relationships between the structures of exact solutions and parameters, and present soliton solutions with free parameters. The results in this paper might be helpful for interpreting certain physical phenomena.

A class of solutions of the two-dimensional Toda lattice equation

A method is proposed to systematically generate solutions of the two-dimensional Toda lattice equation in terms of previously known solutions ϕ(x,y) of the two-dimensional Laplace's equation. The solitonic solution of Nakamura's [J. Phys. Soc. Jpn. 52 (1983) 380] is shown to correspond to one particular choice of ϕ(x,y).

Tropical limit of matrix solitons and entwining Yang\u2013Baxter maps

We consider a matrix refactorization problem, i.e., a “Lax representation,” for the Yang–Baxter map that originated as the map of polarizations from the “pure” 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang–Baxter equation, but satisfies a mixed version of the Yang–Baxter equation together with the Yang–Baxter map. Such maps have been called “entwining Yang–Baxter maps” in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is not in general a Yang–Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang–Baxter equation holds, by exploring the pure 3-soliton solution in the “tropical limit,” where the 3-soliton interaction decomposes into 2-soliton interactions. Here, this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang–Baxter maps as in the KP case, indicating a kind of universality.

N-fold Darboux transformations and exact solutions of the combined Toda lattice and relativistic Toda lattice equation

In this paper, the N-fold Darboux transformation (DT) of the combined Toda lattice and relativistic Toda lattice equation is constructed in terms of determinants. Comparing with the usual 1-fold DT of equations, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications of the N-fold DT, we derive two kinds of N-fold explicit exact solutions from two different seed solutions and plot the figures with properly parameters to illustrate the propagation of solitary waves. What’s more, we present the relationships between the structures of exact solutions parameters with $$N=1$$ , from which we find the 1-fold solutions may be one soliton solutions or periodic solutions and the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The results in this paper might be helpful for interpreting certain physical phenomena.

Explicit Solutions of Integrable Variable-coeffcient Cylindrical Toda Equations

Integrable cylindrical Toda lattice equations are proposed by utilizing a generalized version of the dressing method. A compatibility condition is given which insures that these equations are integrable. Further, soliton solutions for new type equations are shown in explicit forms, including one soliton solution and two soliton solutions, respectively.

Trigonal Toda Lattice Equation

In this article, we give the trigonal Toda lattice equation,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ - {1 \\over 2}{{{d^3}} \\over {d{t^3}}}q\\ell (t)\\, = \\,{e^{q\\ell + 1(t)}}\\, + \\,{{\\rm{e}}^{q\\ell + {\\zeta _3}^{(t)}}}\\, + \\,{{\\rm{e}}^{{q_\\ell } + \\,\\zeta {{_3^2}^{(t)}}}}\\, - \\,3{{\\rm{e}}^{q\\ell {\\rm{(}}t{\\rm{)}}}}$$\\end{document}for a lattice point £ ∊ ℤ[ζ3] as a directed 6-regular graph where \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\zeta _3}\\, = \\,{{\\rm{e}}^{{\\rm{2\\pi }}\\sqrt { - 1/3} }}$$\\end{document}, and its elliptic solution for the curve y(y − s) = x3, (s ≠ 0).