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Ismagil Habibullin

Publications and source records attributed to Ismagil Habibullin.

At least 19 recordsLinked to original sources
arXiv2019

Integrable boundary conditions for the Hirota-Miwa equation and Lie algebras

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 functions for the local conservation laws are found for the systems corresponding to $D^{(2)}_3$. An example of the higher symmetry is presented.

nlin.SI↗
arXiv2018

Invariant manifolds for the hyperbolic type integrable equations and their applications

We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution $u$. Then we construct a differential (respectively, difference) equation compatible with the linearized equation for any choice of $u$. This equation defines a surface called a generalized invariant manifold. In a sense, the manifold generalizes the symmetry, which is also a solution to the linearized equation. In this paper, we concentrate on continuous and discrete models of hyperbolic type. It is known that such kind equations have two hierarchies of symmetries, corresponding to the characteristic directions. We have shown that properly chosen generalized invariant manifold allows one to construct recursion operators that generate these symmetries. It is surprising that both recursion operators are related to different parametrizations of the same invariant manifold. Therefore, knowing one of the recursion operators for the hyperbolic type integrable equation (having no pseudo-constants) we can immediately find the second one.

nlin.SI↗
arXiv2017

Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings

The main goal of the article is testing a new classification algorithm. To this end we apply it to a relevant problem of describing the integrable cases of a subclass of two-dimensional lattices. By imposing the cut-off conditions $u_{-1}=c_0$ and $u_{N+1}=c_1$ we reduce the lattice $u_{n,xy}=α(u_{n+1},u_n,u_{n-1})u_{n,x}u_{n,y}$ to a finite system of hyperbolic type PDE. Assuming that for each natural $N$ the obtained system is integrable in the sense of Darboux we look for $α$. To detect the Darboux integrability of the hyperbolic type system we use an algebraic criterion of Darboux integrability which claims that the characteristic Lie rings of such a system must be of finite dimension. We prove that up to the point transformations only one lattice in the studied class passes the test. The lattice coincides with the earlier found Ferapontov-Shabat-Yamilov equation. The one-dimensional reduction $x=y$ of this lattice passes also the symmetry integrability test.

nlin.SI↗
arXiv2017

On a method for constructing the Lax pairs for integrable models via quadratic ansatz

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 form. The method is illustrated by the well-known KdV equation and the modified Volterra chain. New Lax pairs are found. The formal asymptotic expansions for their eigenfunctions are constructed around the singular values of the spectral parameter. By applying the method of the formal diagonalization to these Lax pairs the infinite series of the local conservation laws are obtained for the corresponding nonlinear models.

nlin.SI↗
arXiv2016

Discretization of Liouville type nonautonomous equations preserving integrals

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known Goursat list for which the integrals of minimal order are of the order less than or equal to two we presented a list of corresponding semi-discrete versions. The list contains new examples of non-autonomous Darboux integrable chains.

nlin.SI↗
arXiv2013

Formal diagonalization of the discrete Lax operators and construction of conserved densities and symmetries for dynamical systems

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 "matrix" NLS and "vector" derivative NLS equations.

nlin.SI↗
arXiv2012

Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ these systems are proved to be integrable. For the systems corresponding to the algebras $A_2$, $A_1^{(1)}$, $A_2^{(2)}$ generalized symmetries are found. For the systems $A_2$, $B_2$, $C_2$, $G_2$, $D_3$ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to $A_N$, $B_N$, $C_N$, $A^{(1)}_1$, $D^{(2)}_N$ are presented.

nlin.SI↗
arXiv2011

Cartan matrices and integrable lattice Toda field equations

Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets of integrals in both directions are found. For the simple Lie algebras of the classical series $A_N$, $B_N$, $C_N$ and affine algebras of series $D^{(2)}_N$ the corresponding systems are supplied with the Lax representation.

nlin.SI↗
arXiv2011

On Darboux Integrable Semi-Discrete Chains

Differential-difference equation $\frac{d}{dx}t(n+1,x)=f(x,t(n,x),t(n+1,x),\frac{d}{dx}t(n,x))$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$ is studied. We call an equation of such kind Darboux integrable, if there exist two functions (called integrals) $F$ and $I$ of a finite number of dynamical variables such that $D_xF=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator: $Dp(n)=p(n+1)$. It is proved that the integrals can be brought to some canonical form. A method of construction of an explicit formula for general solution to Darboux integrable chains is discussed and for a class of chains such solutions are found.

nlin.SI↗
arXiv2009

Complete list of Darboux Integrable Chains of the form $t_{1x}=t_x+d(t,t_1)$

We study differential-difference equation of the form $$ \frac{d}{dx}t(n+1,x)=f(t(n,x),t(n+1,x),\frac{d}{dx}t(n,x)) $$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$. Equation of such kind is called Darboux integrable, if there exist two functions $F$ and $I$ of a finite number of arguments $x$, $\{t(n\pm k,x)\}_{k=-\infty}^\infty$, ${\frac{d^k}{dx^k}t(n,x)}_{k=1}^\infty$, such that $D_xF=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator: $Dp(n)=p(n+1)$. Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function $f$ is of the special form $f(u,v,w)=w+g(u,v)$.

nlin.SI↗
arXiv2008

On the Classification of Darboux Integrable Chains

We study differential-difference equation of the form $t_{x}(n+1)=f(t(n),t(n+1),t_x(n))$ with unknown $t=t(n,x)$ depending on $x$, $n$. The equation is called Darboux integrable, if there exist functions $F$ (called an $x$-integral) and $I$ (called an $n$-integral), both of a finite number of variables $x$, $t(n)$, $t(n\pm 1)$, $t(n\pm 2)$, $...$, $t_x(n)$, $t_{xx}(n)$, $...$, such that $D_xF=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator: $Dp(n)=p(n+1)$. The Darboux integrability property is reformulated in terms of characteristic Lie algebras that gives an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial $x$-integrals is given in the case when the function $f$ is of the special form $f(x,y,z)=z+d(x,y)$.

nlin.SI↗
arXiv2008

Hydrodynamic type integrable equations on a segment and a half-line

The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions of multi-field systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semi-line are presented.

nlin.SI↗
arXiv2007

Quantum and classical integrable sine-Gordon model with defect

Defects which are predominant in a realistic model, usually spoil its integrability or solvability. We on the other hand show the exact integrability of a known sine-Gordon field model with a defect (DSG), at the classical as well as at the quantum level based on the Yang-Baxter equation. We find the associated classical and quantum R-matrices and the underlying q-algebraic structures, analyzing the exact lattice regularized model. We derive algorithmically all higher conserved quantities $C_n, n=1,2,...$ of this integrable DSG model, focusing explicitly on the contribution of the defect point to each $C_n$. The bridging condition across the defect, defined through the Bäcklund transformation is found to induce creation or annihilation of a soliton by the defect point or its preservation with a phase shift.

hep-th↗
arXiv2007

Integrable nonlinear equations on a circle

The concept of integrable boundary value problems for soliton equations on $\mathbb{R}$ and $\mathbb{R}_+$ is extended to bounded regions enclosed by smooth curves. Classes of integrable boundary conditions on a circle for the Toda lattice and its reductions are found.

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