We explore new symmetries in two-component third-order Burgers' type systems in (1+1)-dimension using Wang's O-scheme. We also find a master symmetry for a (2+1)-dimensional Davey-Stewartson type system. These results shed light on the behavior of these equations and help us understand their integrability properties. Our approach offers a practical method for identifying symmetries, contributing to the study of integrable systems in mathematics and physics.
We first obtain the master symmetry of the differential-difference KP equation.Then we show how this master symmetry, through sl(2;C)-representation of the equation, can construct generators of time dependent symmetries.
In this paper, we first give a short account on the indecomposable sl(2,C) modules in the Bernstein-Gelfand-Gelfand (BGG) category O. We show these modules naturally arise for homogeneous integrable nonlinear evolutionary systems. We then develop an approach to construct master symmetries for such integrable systems. This method naturally enables us to compute the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For the new integrable equatio…
An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows $(λ_t=λ^l, l\ge0)$ from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.
An approach to master symmetries of lattice equations is proposed by the use of discrete zero curvature equation. Its key is to generate non-isospectral flows from the discrete spectral problem associated with a given lattice equation. A Volterra-type lattice hierarchy and the Toda lattice hierarchy are analyzed as two illustrative examples.
We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero, Moral and Velázquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] and their action on the tau-functions of the hierarchy is described. The use of the CMV matrices turns out to be crucial for obtaining a Lax pair representation of the master symmetries. The AL hierarchy seems to be the fir…
We give a general account of nonlocal symmetries in symmetric space models and their relation to the AdS/CFT correspondence. In particular, we study a master symmetry which generates the spectral parameter and acts as a level-raising operator on the classical Yangian generators. The master symmetry extends to an infinite tower of symmetries with nonlocal Casimir elements as associated conserved charges. We discuss the algebraic properties of these symmetries and establish their role in explaining the recently observed one-parameter deformation of holographic Wilson loops. Finally, we provide a…
We study the temporal equation of radiating stars by using three powerful methods for the analysis of nonlinear differential equations. Specifically, we investigate the global dynamics for the given master ordinary differential equation to understand the evolution of solutions for various initial conditions as also to investigate the existence of asymptotic solutions. Moreover, with the application of Lie's theory, we can reduce the order of the master differential equation, while an exact similarity solution is determined. Finally, the master equation possesses the Painlevé property, which me…
The properties of a system of n = 3 coupled oscillators with linear terms in the velocities (magnetic terms) depending in two parameters are studied. We proved the existence of a bi-Hamiltonian structure arising from a non-symplectic symmetry, as well the existence of master symmetries and additional integrals of motion (weak superintegrability) for certain particular values of the parameters.
We present easily verifiable sufficient conditions of time-independence and commutativity for local and nonlocal symmetries for a large class of homogeneous (1+1)-dimensional evolution systems. In contrast with the majority of known results, the verification of our conditions does not require the existence of master symmetry or hereditary recursion operator for the system in question. We also give simple sufficient conditions for the existence of infinite sets of time-independent symmetries for homogeneous (1+1)-dimensional evolution systems within slightly modified master symmetry approach.
All quasilinear NLS-type systems with fourth order symmetry are listed. Vector generalizations of some of them are constructed. Local master symmetries for several systems from the list are found.
A sequence of canonical conservation laws for all the Adler-Bobenko-Suris equations is derived and is employed in the construction of a hierarchy of master symmetries for equations H1-H3, Q1-Q3. For the discrete potential and Schwarzian KdV equations it is shown that their local generalized symmetries and non-local master symmetries in each lattice direction form centerless Virasoro type algebras. In particular, for the discrete potential KdV, the structure of its symmetry algebra is explicitly given. Interpreting the hierarchies of symmetries of equations H1-H3, Q1-Q3 as differential-differen…
In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by using discrete non-point transformations. We use both non-invertible linearizable transformations and non-point transformations invertible on solutions of the discrete equation. As a result, we get several series of new examples of discrete equations along with their generalized symmetries and master symmetries. The generalized symmetries constructed give new i…
We study solutions of the KdV equation governed by a stationary equation for symmetries from the non-commutative subalgebra, namely, for a linear combination of the master-symmetry and the scaling symmetry. The constraint under study is equivalent to a sixth order nonautonomous ODE possessing two first integrals. Its generic solutions have a singularity on the line $t=0$. The regularity condition selects a 3-parameter family of solutions which describe oscillations near $u=1$ and satisfy, for $t=0$, an equation equivalent to degenerate $P_5$ equation. Numerical experiments show that in this fa…
In the paper we investigate integrability characteristics for the dispersionless Kadomtsev-Petviashvili hierarchy. These characteristics include symmetries, Hamiltonian structures and conserved quantities. We give a Lax triad to construct a master symmetry and a hierarchy of non-isospectral dispersionless Kadomtsev-Petviashvili flows. These non-isospectral flows, together with the known isospectral dispersionless Kadomtsev-Petviashvili flows, form a Lie algebra, which is used to derive two sets of symmetries for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy. By means of the m…
The Volterra lattice admits two non-Abelian analogs that preserve the integrability property. For each of them, the stationary equation for non-autonomous symmetries defines a constraint that is consistent with the lattice and leads to Painlevé-type equations. In the case of symmetries of low order, including the scaling and master-symmetry, this constraint can be reduced to second order equations. This gives rise to two non-Abelian generalizations for the discrete Painlevé equations dP$_1$ and dP$_{34}$ and for the continuous Painlevé equations P$_3$, P$_4$ and P$_5$.
We construct the additional symmetries and derive the Adler-Shiota-van Moerbeke formula for the two-component BKP hierarchy. We also show that the Drinfeld-Sokolov hierarchies of type D, which are reduced from the two-component BKP hierarchy, possess symmetries written as the action of a series of linear Virasoro operators on the tau function. It results in that the Drinfeld-Sokolov hierarchies of type D coincide with Dubrovin and Zhang's hierarchies associated to the Frobenius manifolds for Coxeter groups of type D, and that every solution of such a hierarchy together with the string equation…
We introduce a single tau function that represents the CKP hierarchy into a generalized Hirota "bilinear" equation. The actions on the tau function by additional symmetries for the hierarchy are calculated, which involve strictly more than a central extension of the $w^C_\infty$-algebra. As an application, for Drinfeld-Sokolov hierarchies of type C that are equivalent to certain reductions of the CKP hierarchy, their Virasoro symmetries are proved to be non-linearizable when acting on the tau function.
Two-component second and third-order Burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries. New symmetry integrable systems with their master symmetries are obtained. Some third order systems are observed to possess conservation laws. Bi-Poisson structures of systems possessing conservation laws are given.
This paper is a continuation of our previous work "Six-vertex model and non-linear differential equations I. Spectral problem" in which we have put forward a method for studying the spectrum of the six-vertex model based on non-linear differential equations. Here we intend to elaborate on that approach and also discuss properties of the spectrum unveiled by the aforementioned differential formulation of the transfer matrix's eigenvalue problem. In particular, we intend to demonstrate how this differential approach allows one to study continuous symmetries of the transfer matrix's spectrum thro…
Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra conf_3. We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of conf_3 are classified and the co…
Conditional Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra conf_3. The corresponding representations of the parabolic and almost-parabolic subalgebras of conf_3 are classified and the complete list of conditionally invariant semi-linear Schrödinger equations is obtained. Applications to the phase-orderi…
We present a generalized study and characterization of the integrability properties of the derivative non-linear Schrödinger equation in 1+1 dimensions. A Lax pair is derived for this equation by means of a Miura transformation and the singular manifold method. This procedure, together with the Darboux transformations, allow us to construct a wide class of rational soliton-like solutions. Lie classical symmetries have also been computed and similarity reductions have been analyzed and discussed.
