Search

We consider a class of complex manifolds constructed as multiplicative quiver varieties associated with a cyclic quiver extended by an arbitrary number of arrows starting at a new vertex. Such varieties admit a Poisson structure, which is obtained by quasi-Hamiltonian reduction. We construct several families of Poisson subalgebras inside the coordinate ring of these spaces, which we use to obtain degenerately integrable systems. We also extend the Poisson center of these algebras to maximal abelian Poisson algebras, hence defining Liouville integrable systems. By considering a suitable set of local coordinates on the multiplicative quiver varieties, we can derive the local Poisson structure explicitly. This allows us to interpret the integrable systems that we have constructed as new generalizations of the spin Ruijsenaars–Schneider system with several types of spin variables. [ABSTRACT FROM AUTHOR]

We show that any SU (2) BPS monopole of charge k with rotational spatial dihedral symmetry is gauge equivalent to the Nahm data obtained from affine Toda equations of C l (1) type when k = 2 l or A 2 (l − 1) (2) type when k = 2 l − 1 . [ABSTRACT FROM AUTHOR]

In this paper, we search for some approaches for generating (1+1)-dimensional, (2+1)-dimensional and (3+1)-dimensional integrable equations by making use of various Lie algebras and the corresponding loop algebras under the frame of the Tu scheme. The well-known modified KdV equation, the heat conduction equation, the nonlinear Schrödinger equation, the (2+1)-dimensional cylindrical dissipative Zaboloskaya–Khokhlov equation and the (3+1)-dimensional heavenly equation are obtained, respectively. In addition, some new isospectral integrable hierarchies and their nonisospectral integrable hierarchies are singled out. All the Lie algebras and their loop algebras presented in the paper can be extensively applied to investigate other integrable hierarchies of evolution equations. [ABSTRACT FROM AUTHOR]

We formulate an inverse scattering transformation for the focusing Hirota equation with asymmetric boundary conditions, which means that the limit values of the solution at spatial infinities have different amplitudes. For the direct problem, we do not use Riemann surfaces, but instead analyze the branching properties of the scattering problem eigenvalues. The Jost eigenfunctions and scattering coefficients are defined as single-valued functions on the complex plane, and their analyticity properties, symmetries, and asymptotics are obtained, which are helpful in constructing the corresponding Riemann–Hilbert problem. On an open contour, the inverse problem is described by a Riemann–Hilbert problem with double poles. Finally, for comparison purposes, we consider the initial value problem with one-sided nonzero boundary conditions and obtain the formulation of the inverse scattering transform by using Riemann surfaces. [ABSTRACT FROM AUTHOR]

A theory of partial separability for classical Hamiltonian systems is proposed in the context of Haantjes geometry. As a general result, we show that the knowledge of a non-semisimple symplectic-Haantjes manifold for a given Hamiltonian system is sufficient to construct sets of coordinates (called Darboux-Haantjes coordinates) that allow both the partial separability of the associated Hamilton-Jacobi equations and the block-diagonalization of the operators of the corresponding Haantjes algebra. We also introduce a novel class of Hamiltonian systems, characterized by the existence of a generalized Stäckel matrix, which by construction are partially separable. They widely generalize the known families of partially separable Hamiltonian systems. The new systems can be described in terms of semisimple but non-maximal-rank symplectic-Haantjes manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we explore some recent advancements in enumerative algebraic geometry, focusing particularly on the role of quantum K-theory of quiver varieties as viewed through the lens of integrable systems. We highlight a number of conjectures and open questions. [ABSTRACT FROM AUTHOR]

In light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits f (r , θ) = c in polar coordinates (where c is the parameter varying along the family of orbits), which are generated by planar potentials V = V (r , θ) . The corresponding family of orbits can be uniquely represented by the "slope function" γ = f θ f r . By using the basic partial differential equation of the planar inverse problem, which combines families of orbits and potentials, we apply a new methodology in order to find specific potentials, e.g., V = A (r) + B (θ) or V = H (γ) and one-dimensional potentials, e.g., V = A (r) or V = G (θ) . In order to determine such potentials, differential conditions on the family of orbits f (r , θ) = c are imposed. If these conditions are fulfilled, then we can find a potential of the above form analytically. For the given families of curves, such as ellipses, parabolas, Bernoulli's lemniscates, etc., we find potentials that produce them. We present suitable examples for all cases and refer to the case of straight lines. [ABSTRACT FROM AUTHOR]

The Maxwell-Bloch system of equations with inhomogeneous broadening is studied, and the local and global well-posedness of the corresponding initial-boundary value problem is established by taking advantage of the integrability of the system and making use of the corresponding inverse scattering transform (IST). A key ingredient in the analysis is the L2{L}^{2}-Sobolev bijectivity of the direct and IST established by Xin Zhou for the focusing Zakharov-Shabat problem.

We use the Hamiltonian reduction method to construct the Ruijsenaars dual systems to generalized Toda chains associated with the classical Lie algebras of types B , C , D . The dual systems turn out to be the B, C and D analogues of the rational goldfish model, which is, as in the type A case, the strong coupling limit of rational Ruijsenaars systems. We explain how both types of systems emerge in the reduction of the cotangent bundle of a Lie group and provide the formulae for dual Hamiltonians. We compute explicitly the higher Hamiltonians of goldfish models using the Cauchy–Binet theorem. [ABSTRACT FROM AUTHOR]

We prove that both local and non-local formulations of the Degasperis–Procesi equation possess a pseudospherical nature. As a result, solutions determined by Cauchy problems with non-trivial initial data and a minimal specific regularity define a coframe for a pseudospherical metric within a designated strip. While the region is entirely described by the initial condition, the one-forms are determined by the corresponding solutions. Moreover, we establish that any non-trivial initial condition yields a second fundamental form, which is locally determined within the strip. The surface may collapse within a finite-height strip when the initial condition leads to wave breaking. We determine conditions for the coframe to be defined on the upper plane. We investigate possible integrability structures related to the triad of fundamental one-forms. [ABSTRACT FROM AUTHOR]

According to Zuo and an unpublished work of Bertola, there is a two-index series of Dubrovin–Frobenius manifold structures associated to a B-type Coxeter group. We study the relations between these structures for the different values of these indices. We show that part of the data of such Dubrovin–Frobenius manifold indexed by (k, l) can be recovered by the (k + r , l + r) Dubrovin–Frobenius manifold.Continuing the program of Basalaev et al. (J Phys A: Math Theor 54:115201, 2021) we associate an infinite system of commuting PDEs to these Dubrovin–Frobenius manifolds and show that these PDEs extend the dispersionless BKP hierarchy. [ABSTRACT FROM AUTHOR]

We study the bilinearization of N = 1 Supersymmetric Coupled Dispersionless (SUSY-CD) integrable system by introducing transformations defined in terms of bosonic and fermionic superfields. From the bosonic and fermionic superfield bilinear equations, we obtain the superfield soliton solutions of SUSY-CD integrable system. [ABSTRACT FROM AUTHOR]

A semidiscrete short-pulse equation (sdSPE) is presented via a proposed Lax pair. A multicomponent sdSPE is derived using Lax matrices. The standard binary Darboux transformation (SBDT) is employed by constructing the Darboux matrices from particular eigenvector solutions of the generalized Lax pair not only in the direct space but also in its adjoint space. Explicit expressions of the first- and second-order nontrivial quasi-Grammian loop solutions of the multicomponent sdSPE are computed, by iterating its SBDT. It is also shown that quasi-Grammian loop solutions reduce to the elementary loop solutions by applying reduction of spectral parameters. [ABSTRACT FROM AUTHOR]

We rigorously prove that the S = 1 / 2 anisotropic Heisenberg chain (XYZ chain) with next-nearest-neighbor interaction, which is anticipated to be non-integrable, is indeed non-integrable in the sense that this system has no nontrivial local conserved quantity. Our result covers some important models including the Majumdar–Ghosh model, the Shastry–Sutherland model, and many other zigzag spin chains as special cases. These models are shown to be non-integrable while they have some solvable energy eigenstates. In addition to this result, we provide a pedagogical review of the proof of non-integrability of the S = 1 / 2 XYZ chain with Z magnetic field, whose proof technique is employed in our result. [ABSTRACT FROM AUTHOR]

In this paper, we examine the role of the Jacobi last multiplier in the context of two-dimensional oscillators. We first consider two-dimensional unit-mass oscillators admitting a separable Hamiltonian description, i.e., H = H 1 + H 2 , where H 1 and H 2 are the Hamiltonians of two one-dimensional unit-mass oscillators. It is shown that there exists a third functionally-independent first integral Θ , ensuring superintegrability. Various examples are explicitly worked out. We then consider position-dependent-mass oscillators and the Bateman pair, where the latter consists of a pair of dissipative linear oscillators. Quite remarkably, the Bateman pair is found to be superintegrable, despite admitting a Hamiltonian which cannot be separated into two isolated (non-interacting) one-dimensional oscillators. [ABSTRACT FROM AUTHOR]

This paper introduces a (3 + 1)-dimensional dispersionless integrable system, utilizing a Lax pair involving contact vector fields, in alignment with methodologies presented by Sergyeyev in 2014. Significantly, it is shown that the proposed system serves as an integrable (3 + 1)-dimensional generalization of the well-studied (2 + 1)-dimensional dispersionless Davey–Stewartson system. This way, an interesting new example on integrability in higher dimensions is presented, with potential applications in analyzing three-dimensional nonlinear waves across various fields, including oceanography, fluid dynamics, plasma physics, and nonlinear optics. Importantly, the integrable nature of the system suggests that established techniques like the study of symmetries, conservation laws, and Hamiltonian structures could be applicable. [ABSTRACT FROM AUTHOR]

Some characteristics of stationary flows of the Sawada–Kotera system lend themselves to generalization, producing a large class of separable Lagrangian systems with two degrees of freedom. All of these systems come in couples that have the same equations of motion, although they are not related by a gauge transform. Some nonpolynomial examples are provided. [ABSTRACT FROM AUTHOR]

We give explicit formulas for the number of meromorphic differentials on C P 1 with two zeros and any number of residueless poles and for the number of meromorphic differentials on C P 1 with one zero, two poles with unconstrained residue and any number of residueless poles, in terms of the orders of their zeros and poles. These are the only two finite families of differentials on C P 1 with vanishing residue conditions at a subset of poles, up to the action of PGL (2 , C) . The first family of numbers is related to triple Hurwitz numbers by simple integration and we show its connection with the representation theory of SL 2 (C) and the equations of the dispersionless KP hierarchy. The second family has a very simple generating series, and we recover it through surprisingly involved computations using intersection theory of moduli spaces of curves and differentials. [ABSTRACT FROM AUTHOR]

Some characteristics of stationary flows of the Sawada–Kotera system lend themselves to generalization, producing a large class of separable Lagrangian systems with two degrees of freedom. All of these systems come in couples that have the same equations of motion, although they are not related by a gauge transform. Some nonpolynomial examples are provided.

In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control ([1], [2]). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $. In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in [2]). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by ([3], [4]) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.

We study the dynamics of a spatial four-body problem where the bodies maintain a rhombus-shape configuration at all times: two of the bodies of equal mass move in the horizontal plane symmetrically with respect to the origin while another pair of bodies of equal mass move symmetrically opposed along the vertical axis. The bodies interact via the Jacobi potential, an attractive binary potential of the form - 1 / x 2 , where x is the distance between the particles. We use appropriate transformations to blow up total collision into a manifold pasted onto the phase space for all levels of energy. We find that the topology of the total collision manifold changes as the angular momentum varies, and also that the dynamics on the collision manifold changes as a function of angular momentum and the mass ratio. We also give a qualitative description of the global flow of the problem for negative energy. This description utilizes knowledge concerning the flow on and near the collision manifold, the presence of an additional integral of motion, and takes advantage of the time-reversing symmetry inherent in the system of equations. [ABSTRACT FROM AUTHOR]

We study solutions of the reflection equation related to the quantum affine algebra U q ( s l n ^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of U q ( s l n ^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained. [ABSTRACT FROM AUTHOR]

We consider the relaxation of an initial non-equilibrium state in a one-dimensional fluid of hard rods. Since it is an interacting integrable system, we expect it to reach the Generalized Gibbs Ensemble (GGE) at long times for generic initial conditions. Here we show that there exist initial conditions for which the system does not reach GGE even at very long times and in the thermodynamic limit. In particular, we consider an initial condition of uniformly distributed hard-rods in a box with the left half having particles with a singular velocity distribution (all moving with unit velocity) and the right half particles in thermal equilibrium. We find that the density profile for the singular component does not spread to the full extent of the box and keeps moving with a fixed effective speed at long times. We show that such density profiles can be well described by the solution of the Euler equations almost everywhere except at the location of the shocks, where we observe slight discrepancies due to dissipation arising from the initial fluctuations of the thermal background. To demonstrate this effect of dissipation analytically, we consider a second initial condition with a single particle at the origin with unit velocity in a thermal background. We find that the probability distribution of the position of the unit velocity quasi-particle has diffusive spreading which can be understood from the solution of the Navier–Stokes (NS) equation of the hard rods. Finally, we consider an initial condition with a spread in velocity distribution for which we show convergence to GGE. Our conclusions are based on molecular dynamics simulations supported by analytical arguments. [ABSTRACT FROM AUTHOR]

We consider the classical integrable trigonometric Landau–Lifshitz models constructed by means of quantum -matrices that also satisfy the associative Yang–Baxter equation. It is shown that a field analogue of the trigonometric Calogero–Moser–Sutherland model is gauge equivalent to the Landau–Lifshitz model that arises from the Antonov–Hasegawa–Zabrodin trigonometric nonstandard -matrix. The latter generalizes Cherednik's -vertex -matrix in the case to the case of . An explicit change of variables between the models is obtained. [ABSTRACT FROM AUTHOR]

We consider chiral-type systems admitting a Lax representation with values in a real or complex semisimple Lie algebra such that an additional regularity condition is satisfied (one of the matrices is a regular element of the Lie algebra). We prove that for a chiral-type system with vanishing torsion and a nonvanishing curvature, the existence of at least one pointwise cosymmetry is a necessary condition for the regular Lax representation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel class of magnetic curves and call it pseudo‐solitonic magnetic curves by considering the connection between two significant types of integrable equations, that is, the nonlinear heat system/the nonlinear Schrödinger equation and moving space curves in the Minkowski 3‐space. Later, we construct a very special class of soliton surfaces and call it pseudo‐solitonic magnetic surfaces by considering the effects of the pseudo‐solitonic magnetic flows in the Minkowski 3‐space. These curves and surfaces are very useful since they not only contain geometric and physical features in their inner forms but also their constructions rely on a simple geometric procedure compared to other models. [ABSTRACT FROM AUTHOR]

We describe all metrics geodesically compatible with a gl -regular Nijenhuis operator L. The set of such metrics is large enough so that a generic local curve γ is a geodesic for a suitable metric g from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from L preserves the property of γ to be a g-geodesic. This implies that every metric g geodesically compatible with L gives us a finite-dimensional reduction of this PDE system. We show that its restriction onto the set of g-geodesics is naturally equivalent to the Poisson action of R n on the cotangent bundle generated by the integrals coming from geodesic compatibility. [ABSTRACT FROM AUTHOR]

In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group G in which the control functions belong to the Lie algebra of a compact Lie subgroup K of G and we investigate conditions under which the time optimal solutions of this compactified system are “approximately” time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control ( [1], [2]). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair (G, K) under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of K. In the second part of the paper we applied our results to the quantum systems known as Icing n-chains (introduced in [2]). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by ( [3], [4]) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains. [ABSTRACT FROM AUTHOR]

In this article, some exact, atypical, topological soliton solutions of the integrable Kundu–Mukherjee–Naskar (KMN) equation have been derived. Those soliton solutions can have atypical shapes like U shaped soliton, W shaped soliton, three hump soliton, periodic soliton etc. on the x - y plane. This property is quite unusual for integrable systems, because solitons usually travel along a line with constant amplitude and velocity. The property of curvature of the solitons on x - y plane occur due to the presence of an arbitrary nonlinear function in the argument of the solution. Such arbitrary function in the soliton solution can exist because of the current like nonlinearity and Galilean covariance present in the KMN equation. These solutions are quite uncommon in the field of integrable systems; that may be used for better mathematical modelling of physical systems enriching the field of nonlinear optics. [ABSTRACT FROM AUTHOR]

We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form, and in the tropical case, the existence of a conserved quantity. We show in certain cases that the orbits are unbounded. The tropical dynamics are related to matrix mutation, from the theory of cluster algebras. We are able to show that in certain special cases, the tropical map is periodic. We also explain how our dynamics imply the asymptotic sign-coherence observed by Gekhtman and Nakanishi in the two-dimensional situation. [ABSTRACT FROM AUTHOR]

We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A 2 (1) . The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair L , M . [ABSTRACT FROM AUTHOR]

Nowadays, breather solutions are generally accepted models of rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multisoliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact N‐soliton solution converging asymptotically to the plane wave at large number of solitons N. On the example of the Peregrine, Akhmediev, Kuznetsov–Ma, and Tajiri–Watanabe breathers, we show that constructed with our method multisoliton solutions, being localized in space with characteristic width proportional to N, are practically indistinguishable from the breathers in a wide region of space and time at large N. Our method makes it possible to build solitonic models with the same dynamical properties for the higher order rational and super‐regular breathers, and can be applied to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems. The constructed multisoliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronization conditions represents a challenging problem for future studies. [ABSTRACT FROM AUTHOR]

We study the motion of a test particle in a conservative force field. In the framework of the 2D inverse problem of Newtonian dynamics, we find 2D potentials that produce a preassigned monoparametric family of regular orbits f (x , y) = c on the x y -plane (where c is the parameter of the family of orbits). This family of orbits can be represented by the "slope function" γ = f y f x uniquely. A new methodology is applied to the basic equation of the planar inverse problem in order to find potentials of a special form, i.e., V = F (x + y) + G (x − y) , V = F (x + i y) + G (x − i y) and V = P (x) + Q (y) , and polynomial ones. According to this methodology, we impose differential conditions on the family of orbits f (x , y) = c. If they are satisfied, such a potential exists and it is found analytically. For known families of curves, e.g., circles, parabolas, hyperbolas, etc., we find potentials that are compatible with them. We offer pertinent examples that cover all the cases. The case of families of straight lines is referred to. [ABSTRACT FROM AUTHOR]

We show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura group. [ABSTRACT FROM AUTHOR]

Lagrangian multiforms provide a variational framework to describe integrable hierarchies. The case of Lagrangian 1-forms covers finite-dimensional integrable systems. We use the theory of Lie dialgebras introduced by Semenov-Tian-Shansky to construct a Lagrangian 1-form. Given a Lie dialgebra associated with a Lie algebra g and a collection H k , k = 1 , ⋯ , N , of invariant functions on g ∗ , we give a formula for a Lagrangian multiform describing the commuting flows for H k on a coadjoint orbit in g ∗ . We show that the Euler–Lagrange equations for our multiform produce the set of compatible equations in Lax form associated with the underlying r-matrix of the Lie dialgebra. We establish a structural result which relates the closure relation for our multiform to the Poisson involutivity of the Hamiltonians H k and the so-called double zero on the Euler–Lagrange equations. The construction is extended to a general coadjoint orbit by using reduction from the free motion of the cotangent bundle of a Lie group. We illustrate the dialgebra construction of a Lagrangian multiform with the open Toda chain and the rational Gaudin model. The open Toda chain is built using two different Lie dialgebra structures on sl (N + 1) . The first one possesses a non-skew-symmetric r-matrix and falls within the Adler–Kostant–Symes scheme. The second one possesses a skew-symmetric r-matrix. In both cases, the connection with the well-known descriptions of the chain in Flaschka and canonical coordinates is provided. [ABSTRACT FROM AUTHOR]

The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so-called post-Newtonian expansions of the general relativistic description of black hole binaries. In this paper we study the general question of whether there exist relativistic binaries that display Kepler-like dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem indeed exists for relativistic systems with a Hamiltonian of a Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians conserving a Laplace – Runge – Lenz-like vector are related in this way to Kepler. [ABSTRACT FROM AUTHOR]

We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A2(1). The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair L,M.

Abstract This article reviews the recent developments in the theory of generalised hydrodynamics (GHD) with emphasis on the repulsive one-dimensional Bose gas. We discuss the implications of GHD on the mechanisms of thermalisation in integrable quantum many-body systems as well as its ability to describe far-from-equilibrium behaviour of integrable and near-integrable systems in a variety of quantum quench scenarios. We outline the experimental tests of GHD in cold-atom gases and its benchmarks with other microscopic theoretical approaches. Finally, we offer some perspectives on the future direction of the development of GHD.

This paper reviews the generalized Hamiltonian system and its connection to contact transformations. The generalized Hamiltonian system is related to Herglotz variational principle in the same way in which the Hamiltonian system is related to the classical variational principle. We prove a criterion for the integrability of the generalized Hamiltonian system in terms of a complete set of first integrals, and a method of generating such first integrals. These results are due to Gustav Herglotz. [ABSTRACT FROM AUTHOR]

A complete embedding is a symplectic embedding $\iota :Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $\iota (K)$ inside $M$. Under the assumption that the torsion exponents of $K$ are bounded, we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions. [ABSTRACT FROM AUTHOR]

We investigate the integrability of a family of pentagram maps proposed by M. Glick and P. Pylyavskyy, which share the property that each generates an infinite configuration of points and lines with four points on each line. Their Lax representations with spectral parameters are constructed. It’s important to emphasize that the proofs we provide are presented at a physical level of rigor, acknowledging that the periodicity aspect requires further meticulous treatment to establish complete mathematical rigor. Moreover, our investigation reveals a connection between these pentagram-type maps and the discrete KP equation. Finally, their continuous limits are considered. [ABSTRACT FROM AUTHOR]

The Kundu-Mukherjee-Naskar equation holds significant relevance as a nonlinear model for investigating intricate wave phenomena in fluid and optical systems. This study uncovers new optical soliton solutions for the KMN equation by employing analytical techniques that utilize combined elliptic Jacobian functions. The solutions exhibit mixtures of distinct Jacobian elliptic functions, offering novel insights not explored in prior KMN equation research. Visual representations in the form of 2D ContourPlots elucidate the physical behaviors and properties of these newly discovered solution forms. The utilization of symbolic computations facilitated the analytical derivation of these solutions, offering a deeper understanding of the nonlinear wave dynamics governed by the KMN equation. These employed techniques showcase the potential for future analytical advancements in unraveling the complex soliton landscape of the multifaceted KMN model. The findings provide valuable insights into the intricacies of soliton behavior within this nonlinear system, offering new perspectives for analysis and exploration in areas such as fiber optic communications, ocean waves, and fluid mechanics. Maple symbolic packages have enabled us to derive analytical results. [ABSTRACT FROM AUTHOR]

This paper primarily focuses on the development of a novel category of magnetic curves. While we explore conventional techniques, viewing these curves through a geometric lens and considering the electromagnetic principles of physics, our emphasis shifts towards investigating integrability conditions and solitonic behaviors. Consequently, we introduce diverse varieties of hybrid magnetic curves linked to the modified non-linear Schrödinger (mNLS) equation within the context of three-dimensional Minkowski space. We also scrutinize alterations in energy distribution and the pseudo-parallel transport of magnetic and electric vector lines associated with flows, enabling the construction of a new class of hybrid magnetic soliton surfaces. Finally, we employ the conformable fractional method to derive specific solution families for these integrable systems, presenting them visually. [ABSTRACT FROM AUTHOR]

We consider the Becchi, Rouet, Stora and Tyutin (BRST) invariant effective action of the non-abelian BF topological theory in two dimensions with gauge group S l (2 , R). By considering different gauge fixing conditions, the zero-curvature field equation gives rise to several well known integrable equations. We prove that each integrable equation together with the associated ghost field evolution equation, obtained from the BF theory, is a BRST invariant system with an infinite sequence of BRST invariant conserved quantities. We construct explicitly the systems and the BRST transformation laws for the Korteweg-de Vries (KdV) sequence (including the KdV, mKdV and CKdV equations) and Harry Dym integrable equation. [ABSTRACT FROM AUTHOR]