Lax Pair Formulation Research Articles

ANALYTICAL SOLUTIONS OF THE NONLOCAL NONLINEAR SCHRÖDINGER-TYPE EQUATIONS

In physics, nonlinear equations are applіed to characterize the varied phenomena. Usually, the nonlinear equations are presented by nonlinear partial differential equations, that can be received as conditions for the compatibility of two linear differentіal equations, named the Lax pairs. The presence of the Lax pair determines integrability for the nonlinear partial differentіal equation. Linked to this development was the realization that certаіn coherent structures, known as solіtons, which play a fundamental role in nonlinear phenomena as lattice dynamics, nonlinear optіcs, and fluіd mechanics. One of the famous equations is the nonlinear Schrödinger equation which is associated with various physical phenomena in nonlinear optics and Bose-Einstein condensates. This equation allows the Lax pair thus it is integrable. This work investigates nonlocal nonlinear Schrödinger-type equations with PT symmetry. Nonlocal nonlinear equations arise in various physical contexts as fluid dynamics, condensed matter physics, optics, and so on. We introduce the Lax pair formulation for the nonlocal nonlinear Schrödinger-type equations. The method of the Darboux transformation is applied to receive analytical solutions.

Non-Abelian Toda-type equations and matrix valued orthogonal polynomials

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail.

AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials

The aim of this paper is to derive a kind of integrable hierarchies with four potentials, which possess bi-Hamiltonian structures, from reduced AKNS matrix spectral problems. The associated recursion operators are worked out explicitly. The Lax pair formulation and the trace identity are basic tools in the analysis. Two nonlinear examples in the resulting integrable hierarchies are integrable nonlinear Schrödinger type equations and integrable modified Korteweg–de Vries type equations.

Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps

We present two non-equivalent families of hierarchies of non-Abelian compatible maps and we provide their Lax pair formulation. These maps are associated with families of hierarchies of non-Abelian Yang-Baxter maps, which we provide explicitly. In addition, these hierarchies correspond to integrable difference systems with variables defined on edges of an elementary cell of the graph, that in turn lead to hierarchies of difference systems with variables defined on vertices of the same cell. In that respect we obtain the non-Abelian lattice-modified Gel’fand-Dikii hierarchy, together with the explicit form of a non-Abelian hierarchy that we refer to as the lattice-NQC (or lattice-) Gel’fand-Dikii hierarchy.

Lax pair formulation for the open boundary Osp(1∣2) spin chain

Based on the Lax pair formulation, we study the integrable conditions of the Osp(1∣2) spin chain with open boundaries. We consider both the non-graded and graded versions of the Osp(1∣2) chain. The Lax pair operators M ± for the boundaries can be induced by the Lax operator M j for the bulk of the system. They correspond to the reflection equations (RE) and the Yang–Baxter equation, respectively. We further calculate the boundary K-matrices for both the non-graded and graded versions of the model with open boundaries. The double row monodromy matrix and transfer matrix of the spin chain have also been constructed. The K-matrices obtained from the Lax pair formulation are consistent with those from Sklyanin’s RE. This construction is another way to prove the quantum integrability of the Osp(1∣2) chain. We find that the Lax pair formulation has advantages in dealing with the boundary terms of the supersymmetric model.

The elliptical vortices, integrable Ermakov structure, Schrödinger connection, and Lax pair in the compressible Navier–Stokes equation

AbstractIn this paper, we investigate the (2+1)‐dimensional compressible Navier–Stokes equation with density‐dependent viscosity coefficients. We introduce a novel power‐type elliptic vortex ansatz and thereby obtain a finite‐dimensional nonlinear dynamical system. The latter is shown to not only have an underlying integrable Ermakov structure of Hamiltonian type, but also admit a Lax pair formulation and associated stationary nonlinear Schrödinger connection. In addition, we construct a class of elliptical vortex solutions termed pulsrodons corresponding to pulsating elliptic warm core eddies and discuss their dynamical behaviors. These solutions have recently found applications in geography, and oceanic and atmospheric dynamics.

Analytic (non)integrability of Arutyunov-Bassi-Lacroix model

We use the notion of the gauge/string duality and discuss the Liouvillian (non) integrability criteria for string sigma models in the context of recently proposed Arutyunov-Bassi-Lacroix (ABL) model in Arutyunov et al. (2021) [31]. Our analysis complements those previous results due to numerical analysis as well as Lax pair formulation. We consider a winding string ansatz for the deformed torus Tk(λ1,λ2,λ) which can be interpreted as a system of coupled pendulums. Our analysis reveals the Liouvillian nonintegrability of the associated sigma model. We also obtain the generalized decoupling limit and confirm the analytic integrability for the decoupled sector.

On nomalized differentials on spectral curves associated with the sinh-Gordon equation

<p style='text-indent:20px;'>The spectral curve associated with the sinh-Gordon equation on the torus is defined in terms of the spectrum of the Lax operator appearing in the Lax pair formulation of the equation. If the spectrum is simple, it is an open Riemann surface of infinite genus. In this paper we construct normalized differentials on this curve and derive estimates for the location of their zeroes, needed for the construction of angle variables.

On the Integrability of theBenjamin‐OnoEquation on the Torus

In this paper we prove that the Benjamin‐Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: this equation admits global Birkhoff coordinates on the spaceof real‐valued, 2π‐periodic,L2‐integrable functions of mean 0. These are coordinates that allow us to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all thesolutions of the Benjamin‐Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin‐Ono equation and on the use of a generating functional, which encodes the entire Benjamin‐Ono hierarchy. © 2020 Wiley Periodicals, Inc.

On the spectral problem associated with the time-periodic nonlinear Schr\xf6dinger equation

According to its Lax pair formulation, the nonlinear Schrödinger (NLS) equation can be expressed as the compatibility condition of two linear ordinary differential equations with an analytic dependence on a complex parameter. The first of these equations—often referred to as the x-part of the Lax pair—can be rewritten as an eigenvalue problem for a Zakharov–Shabat operator. The spectral analysis of this operator is crucial for the solution of the initial value problem for the NLS equation via inverse scattering techniques. For space-periodic solutions, this leads to the existence of a Birkhoff normal form, which beautifully exhibits the structure of NLS as an infinite-dimensional completely integrable system. In this paper, we take the crucial steps towards developing an analogous picture for time-periodic solutions by performing a spectral analysis of the t-part of the Lax pair with a periodic potential.

Generalised Born-Infeld models, Lax operators and the mathrm{T}overline{mathrm{T}} perturbation

Surprising links between the deformation of 2D quantum field theories induced by the composite mathrm{T}overline{mathrm{T}} operator, effective string models and the AdS/CFT correspondence, have recently emerged. The purpose of this article is to discuss various classical aspects related to the deformation of 2D interacting field theories. Special attention is given to the sin(h)-Gordon model, for which we were able to construct the mathrm{T}overline{mathrm{T}} -deformed Lax pair. We consider the Lax pair formulation to be the first essential step toward a more satisfactory geometrical interpretation of this deformation within the integrable model framework.Furthermore, it is shown that the 4D Maxwell-Born-Infeld theory, possibly with the addition of a mass term or a derivative-independent potential, corresponds to a natural extension of the 2D examples. Finally, we briefly comment on 2D Yang-Mills theory and propose a modification of the heat kernel, for a generic surface with genus p and n boundaries, which fully accounts for the mathrm{T}overline{mathrm{T}} contribution.

Noncommutative Painlevé Equations and Systems of Calogero Type

All Painlev\'e equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of $\beta$-models. An almost two decade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlev\'e correspondence. In this paper we answer in the affirmative by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlev\'e equation.

The Nahm–Schmid equations and hypersymplectic geometry

We explore the geometry of the Nahm-Schmid equations, a version of Nahm's equations in split signature. Our discussion ties up different aspects of their integrable nature: dimensional reduction from the Yang--Mills anti-self-duality equations, explicit solutions, Lax-pair formulation, conservation laws and spectral curves, as well as their relation to hypersymplectic geometry.

The elliptic sinh-Gordon equation in a semi-strip

Abstract We study the elliptic sinh-Gordon equation posed in a semi-strip by applying the so-called Fokas method, a generalization of the inverse scattering transform for boundary value problems. Based on the spectral analysis for the Lax pair formulation, we show that the spectral functions can be characterized from the boundary values. We express the solution of the equation in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are defined by the spectral functions. Moreover, we derive the global algebraic relation that involves the boundary values. In addition, it can be verified that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists if the spectral functions defined by the boundary values satisfy this global relation.

Chiral asymmetry in propagation of soliton defects in crystalline backgrounds

By applying Darboux-Crum transformations to a Lax pair formulation of the Korteweg-de Vries (KdV) equation, we construct new sets of multi-soliton solutions to it as well as to the modified Korteweg-de Vries (mKdV) equation. The obtained solutions exhibit a chiral asymmetry in propagation of different types of defects in crystalline backgrounds. We show that the KdV solitons of pulse and compression modulation types, which support bound states in semi-infinite and finite forbidden bands in the spectrum of the perturbed quantum one-gap Lame system, propagate in opposite directions with respect to the asymptotically periodic background. A similar but more complicated picture also appears for the multi-kink-antikink mKdV solitons that propagate with a privileged direction over topologically trivial or topologically nontrivial crystalline background in dependence on position of energy levels of the trapped bound states in the spectral gaps of the associated Dirac system. An exotic N=4 nonlinear supersymmetric structure incorporating Lax-Novikov integrals of a pair of perturbed Lame systems is shown to underlie the Miura-Darboux-Crum construction. It unifies the KdV and mKdV solutions, detects the defects and distinguishes their types, and identifies the types of crystalline backgrounds.

From Toda to KdV

For periodic Toda chains with a large number N of particles we consider states which are N−2-close to the equilibrium and constructed by discretizing arbitrary given C2−functions with mesh size N−1. Our aim is to describe the spectrum of the Jacobi matrices LN appearing in the Lax pair formulation of the dynamics of these states as N → ∞. To this end we construct two Hill operators H±—such operators come up in the Lax pair formulation of the Korteweg–de Vries equation—and prove by methods of semiclassical analysis that the asymptotics as N → ∞ of the eigenvalues at the edges of the spectrum of LN are of the form where are the eigenvalues of H±. In the bulk of the spectrum, the eigenvalues are o(N−2)-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to LN.

N = 2 $$ \mathcal{N}=2 $$ super-EYM coloured black holes from defective Lax matrices

We construct analytical supersymmetric coloured black hole solutions, i.e. non-Abelian black hole solutions that have no asymptotic non-Abelian charge but do have non-Abelian charges on the horizon that contribute to the Bekenstein-Hawking entropy, to two SU(3)-gauged N=2 d= supergravities. The analytical construction is made possible due to the fact that the main ingredient is the Bogomol'nyi equation, which under the assumption of spherical symmetry admits a Lax pair formulation. The Lax matrix needed for the coloured black holes must be defective which, even though it is the non-generic and less studied case, is a minor hindrance.

Schrödinger equation with time-dependent mass function and associated generalized KdV equation

We show that a time-dependent disturbance term could be embedded into a Korteweg–de Vries (KdV) equation. This is accomplished through a Lax pair formulation of a linearized scattering problem corresponding to the Schrödinger equation. A new generalized KdV equation is obtained by considering the time-dependent mass function. Choosing initial data for the stationary Schrödinger potential, we solve the direct scattering problem. Then, in the inverse scattering problem, the Gel’fand–Levitan integral equation is solved to derive the solution for our new generalized KdV equation. Finally, an elliptic functional form is chosen for the mass function to obtain an elliptic soliton solution.

The anisotropic λ-deformed SU(2) model is integrable

The all-loop anisotropic Thirring model interpolates between the WZW model and the non-Abelian T-dual of the anisotropic principal chiral model. We focus on the SU(2) case and we prove that it is classically integrable by providing its Lax pair formulation. We derive its underlying symmetry current algebra and use it to show that the Poisson brackets of the spatial part of the Lax pair, assume the Maillet form. In this way we procure the corresponding r and s matrices which provide non-trivial solutions to the modified Yang–Baxter equation.

Lax pair formulation in the simultaneous presence of boundaries and defects

Inspired by recent results on the effect of integrable boundary conditions on the bulk behavior of an integrable system, and in particular on the behavior of an existing defect, we systematically formulate the Lax pairs in the simultaneous presence of integrable boundaries and defects. The respective sewing conditions as well as the relevant equations of motion on the defect point are accordingly extracted. We consider a specific prototype i.e. the vector nonlinear Schrödinger model to exemplify our construction. This model displays a highly non-trivial behavior and allows the existence of two distinct types of boundary conditions based on the reflection algebra or the twisted Yangian.