A method of the formal diagonalization of the discrete linear operator with a parameter is studied. In the case when the operator provides a Lax operator for a nonlinear quad system the formal diagonalization method allows one to describe effectively conservation laws and generalized symmetries for this system. Asymptotic representation of the Lax operators eigenfunctions are constructed and infinite series of conservation laws are described for the quad system connected with $A^{(1)}_3$ affine Lie algebra, for the modified discrete Boussinesq system and for the discrete Tzitzeica equation. For a newly found multiquadratic discrete model conservation laws and several generalized symmetries are presented.
An alternative method of constructing the formal diagonalization for the discrete Lax operators is proposed which can be used to calculate conservation laws and in some cases generalized symmetries for discrete dynamical systems. Discrete potential KdV equation, lattice derivative nonlinear Schrödinger equation, dressing chain, Toda lattice are considered as illustrative examples. For the Toda lattice on a quad graph corresponding to the Lie algebra $A_1^{(1)}$ infinite series of conservation laws are described. Systems of quad graph equations are represented including lattice versions of the …
In the paper we derive infinitely many conservation laws for the ABS lattice equations from their Lax pairs. These conservation laws can algebraically be expressed by means of some known polynomials. We also show that H1, H2, H3, Q1, Q2, Q3 and A1 equation in ABS list share a generic discrete Riccati equation.
It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.
We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schrodinger (NLS) equation. We explore the Darboux links of the NLS hierarchy with the hierarchy of Heisenberg model, principal chiral field model as well as with differential-difference integrable systems (including the Toda lattice and differential-difference Heisenberg chain) and integrable partial difference systems. We show that there exists a transformation which formally diagonalises all elements of the Lax-Darboux scheme simultaneously. It provides us with generating func…
We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax h…
In this paper, the semi-discrete Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy is shown in spirit composed by the Ablowitz-Ladik flows under certain combinations. Furthermore, we derive its explicit Lax pairs and infinitely many conservation laws, which are non-trivial in light of continuum limit. Reductions of the semi-discrete AKNS hierarchy are investigated to include the semi-discrete Korteweg-de Vries (KdV), the semi-discrete modified KdV, and the semi-discrete nonlinear Schrödinger hierarchies as its special cases. Finally, under the uniform continuum limit we introduce in the paper, the a…
A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold to the linearization of the given equation. Examples show that such invariant manifold does exist and can effectively be found. Actually it is defined by a quadratic form. As a result we get a nonlinear Lax pair consisting of the linearized equation and the invariant manifold. Our second step consists of finding an appropriate change of the variables to linearize the found nonlinear Lax pair. The desired change of the variables is again defined by a quadratic f…
We derive infinitely many conservation laws for some multi-dimensionally consistent lattice equations from their Lax pairs. These lattice equations are the Nijhoff-Quispel-Capel equation, lattice Boussinesq equation, lattice nonlinear Schrödinger equation, modified lattice Boussinesq equation, Hietarinta's Boussinesq-type equations, Schwarzian lattice Boussinesq equation and Toda-modified lattice Boussinesq equation.
Sequences of canonical conservation laws and generalized symmetries for the lattice Boussinesq and the lattice modified Boussinesq systems are successively derived. The interpretation of these symmetries as differential-difference equations leads to corresponding hierarchies of such equations for which conservation laws and Lax pairs are constructed. Finally, using the continuous symmetry reduction approach, an integrable, multidimensionally consistent system of partial differential equations is derived in relation with the lattice modified Boussinesq system.
We study nonlocal reductions of coupled equations in $1+1$ dimensions of the Heisenberg ferromagnet type. The equations under consideration are completely integrable and have a Lax pair related to a linear bundle in pole gauge. We describe the integrable hierarchy of nonlinear equations related to our system in terms of generating operators. We present some special solutions associated with four distinct discrete eigenvalues of the scattering operator. Using Lax pair diagonalization method, we derive recurrence formulas for the conserved densities and find the first two simplest conserved dens…
We study the Veronese web equation $u_y u_{tx}+ λu_xu_{ty} - (λ+1)u_tu_{xy} =0$ and using its isospectral Lax pair construct two infinite series of nonlocal conservation laws. In the infinite differential coverings associated to these series, we describe the Lie algebras of the corresponding nonlocal symmetries. Finally, we construct a recursion operator and explore its action on nonlocal shadows. The operator provides a new shadow which serves as a master-symmetry.
We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples cons…
Various new two-component systems related to the lattice Schwarzian Boussinesq equation are constructed in a systematic way from conservation laws. Their multidimensional consistency is demonstrated, Lax pairs, symmetries and conservation laws are derived and an auto-Backlund transformation is constructed. Finally, Yang-Baxter maps from these systems are constructed.
In this article, by means of considering an isospectral operator equation which corresponds to the Volterra lattice, and constructing opportune time evolution problems with negative powers of spectral parameter, and using discrete zero curvature representation, negative Volterra flows are proposed. We also propose the mixed Volterra flows, which come from positive and negative volterra flows. From the Lax representation, we demonstrate the existence of infinitely many conservation laws for the two flows and give the corresponding conserved densities and the associated fluxes formulaically. Thu…
Recently, Lembert, Gilson et al proposed a lucid and systematic approach to obtain bilinear Bäcklund transformations and Lax pairs for constant-coefficient soliton equations based on the use of binary Bell polynomials. In this paper, we would like to further develop this method with new applications. We extend this method to systematically investigate complete integrability of nonisospectral and variable-coefficient equations. In addiction, a method is described for deriving infinite conservation laws of nonlinear evolution equations based on the use of binary Bell polynomials. All conserved d…
The hierarchy of the integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of the linearization of these equations and their conservation law in the terms of the solutions of the corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of the linearized equation due to the Noether's theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and the conservation laws that are explicitly expressed through the variables of the…
Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws is discussed. In the generic case, nonlocal conservation laws are also generated. A new integrable equation satisfying the second order integrability conditions is presented and its integrability is established by the construction of symmetries, conservation laws and a 3x3 Lax representation. Finally, the relation of the symmetries of this equation to a generalized Bogoyavlensky lattice and a new integrable lattice a…
New extra series of conserved densities for the polytropic gas model and nonlinear elasticity equation are obtained without any references to the recursion operator or to the Lax operator formalism. Our method based on the utilization of the symmetry operators and allows us to obtain the densities of arbitrary homogenuity dimensions. The nonpolynomial densities with logarithmics behaviour are presented as an example. The special attention is paid for the singular case $(γ=1)$ for which we found new non homogenious solutions expressed in terms of the elementary functions.
Using the lax pair, nonlocal symmetry of the coupled Hirota equation is obtained. By introducing an appropriate auxiliary dependent variable the nonlocal symmetry is successfully localized to a Lie point symmetry. For the closed prolongation, one-dimensional optimal systems and nonlocal conservation laws of the coupled Hirota equation are studied.
Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the u…
The Calogero type matrix discretization scheme is applied to constructing the Lax type integrable discretizations of one wide enough class of nonlinear integrable dynamical systems on functional manifolds. Their Lie-algebraic structure and complete integrability related with co-adjoint orbits on the Markov co-algebras is discussed. It is shown that a set of conservation laws and the associated Poisson structure ensue as a byproduct of the approach devised. Based on the Lie algebras quasi-representation property the limiting procedure of finding the nonlinear dynamical systems on the correspond…
We derive a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the Ablowitz-Ladik (AL) hierarchy. Our methods rely on a recursive approach to the AL hierarchy using Laurent polynomials and on asymptotic expansions of the Green's function of the AL Lax operator, a five-diagonal finite difference operator.
Systems of discrete equations on a quadrilateral graph related to the series $D^{(2)}_N$ of the affine Lie algebras are studied. The systems are derived from the Hirota-Miwa equation by imposing boundary conditions compatible with the integrability property. The Lax pairs for the systems are presented. It is shown that in the continuum limit the quad systems tend to the corresponding systems of the differential equations belonging to the well-know Drinfeld-Sokolov hierarchies. The problem of finding the formal asymptotic expansion of the solutions to the Lax equations is studied. Generating fu…
We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain an infinite-dimensional differential covering over the Przanowski equation.
The article deals with the problem of the integrable discretization of the well-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been suggested earlier in \cite{GHY} which coincide with the corresponding Drinfeld-Sokolov systems in the continuum limit. It was conjectured that the systems in this class are all integrable and the conjecture has been approved by numerous examples. In the present article we study those systems from this class which are related to the algebras $A^{(1)}_{N-1}$. We fou…
Centre for Theoretical Physics and Natural Philosophy, Nakhonsawan Studiorum for Advanced Studies, Mahidol University（马希隆大学）；Department of Mathematics, Khalifa University of Science and Technology（哈利法科学与技术大学）；School of Management Sciences, Maulana Abul Kalam Azad University of Technology（阿卜杜勒·卡拉姆·阿扎德理工大学）
We construct an infinite hierarchy of nonlocal conservation laws for the $ABC$ equation $A u_t\,u_{xy}+B u_x\,u_{ty}+C u_y\,u_{tx} = 0$, where $A,B,C$ are constants and $A+B+C\neq 0$, using a nonisospectral Lax pair. As a byproduct, we present new coverings for the ABC equation. The method of proof of nontriviality of the conservation laws under study is quite general and can be applied to many other integrable multidimensional systems.
Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The well defined regularization of the model is constructed and its Lax type integrability is discussed. A generalized hydrodynamical Riemann type system is considered, infinite hierarchies o…
We consider an integrable generalization of the nonlinear Schrödinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.
We consider an integrable generalization of the sine-Gordon (sG) equation that was earlier derived by one of the authors using bi-Hamiltonian methods. This equation is related to the sG equation in the same way that the Camassa-Holm equation is related to the KdV equation. In this paper we: (a) Derive a Lax pair. (b) Use the Lax pair to solve the initial value problem on the line. (c) Analyze solitons. (d) Show that the generalized sG and sG equations are related by a Liouville transformation. (e) Derive conservation laws. (f) Analyze traveling-wave solutions.
Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation $%D_{t}^{N}u=0$ for arbitrary $N\in \mathbb{Z}_{+}.$ The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless are constructed. Special attention is paid to the cases $%N=2,3$ and N=4 for which the conservation laws, Lax type representations and bi-Hamiltonian structures are analyzed in detail. We also show that the case N=2 is equivalent to a generalized Hunter-Saxton dynamical system, whose integrability …
Learning Lax Pairs: Revisiting the Classical Paradigm
Arizona State University（亚利桑那州立大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）；Georgia Institute of Technology（佐治亚理工学院）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Seoul National University（首尔国立大学）
This paper focuses on the modulation instability, conservation laws and localized wave solutions of the generalized coupled Fokas-Lenells equation. Based on the theory of linear stability analysis, distribution pattern of modulation instability gain G in the (K,k) frequency plane is depicted, and the constraints for the existence of rogue waves are derived. Subsequently, we construct the infinitely many conservation laws for the generalized coupled Fokas-Lenells equation from the Riccati-type formulas of the Lax pair. In addition, the compact determinant expressions of the N-order localized wa…
We construct the hierarchy of a multi-component generalisation of modified KdV equation and find exact solutions to its associated members. The construction of the hierarchy and its conservation laws is based on the Drinfel'd-Sokolov scheme, however, in our case the Lax operator contains a constant non-regular element of the underlying Lie algebra. We also derive the associated recursion operator of the hierarchy using the symmetry structure of the Lax operators. Finally, using the rational dressing method, we obtain the one soliton solution, and we find the one breather solution of general ra…
We discuss the dispersionless Boussinesq type equation, which is equivalent to the Benney-Lax equation, being a system of equations of hydrodynamical type. This equation was discussed in <http://dx.doi.org/doi:10.1088/0305-4470/27/1/013>. The results include: a description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws (cosymmetries). Highly interesting are the appearances of operators that send conservation laws and symmetries to each othe…
We review some of the fundamental notions associated to the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux-Backlund transformation is introduced and employed in order to obtain soliton solutions for specific examp…
We consider the half-wave maps (HWM) equation which is a continuum limit of the classical version of the Haldane-Shastry spin chain. In particular, we explore a many-body dynamical system arising from the HWM equation under the pole ansatz. The system is shown to be completely integrable by demonstrating that it exhibits a Lax pair and relevant conservation lows. Subsequently, the analytical multisoliton solutions of the HWM equation are constructed by means of the pole expansion method. The properties of the one- and two-soliton solutions are then investigated in detail as well as their pole …
In this thesis we study the Darboux transformations related to particular Lax operators of NLS type which are invariant under the action of the so-called reduction group. Specifically, we study the cases of: 1) the nonlinear Schrödinger equation (with no reduction), 2) the derivative nonlinear Schrödinger equation, where the corresponding Lax operator is invariant under the action of the $\mathbb{Z}_2$-reduction group and 3) a deformation of the derivative nonlinear Schrödinger equation, associated to a Lax operator invariant under the action of the dihedral reduction group. These reduction gr…
A construction of conservation laws for chiral models (generalized sigma-models on a two-dimensional space-time continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions. This is achieved via a deformation of the ordinary differential calculus. In particular, the nonlinear Toda lattice results in this way from the linear (continuum) wave equation. The method is applied to several further examples. We also construct Lax pairs and Bäcklund transformations for the class of models considered in this work.
