Ismagil Habibullin

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 Cn, n = 1, 2 ,... , of this integrable DSG model, focusing explicitly on the contribution of the defect point to each Cn. The bridging condition across the defect, defined through the Backlund transformation is found to induce creation or annihilation of a soliton by the defect point or its preservation with a phase shift.

Boundary conditions for integrable equations

The problem of constructing boundary conditions for nonlinear equations compatible with higher symmetries is considered. In particular, this problem is discussed for the sine - Gordon, Jiber - Shabat, Liouville and KdV equations. New results are obtained for the last two ones. The boundary condition for the KdV contains two arbitrary constants. The substitution maps it onto the boundary condition with linear dependence on t for the potentiated KdV.

Characteristic Algebras of Fully Discrete Hyperbolic Type Equations

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An eective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.

Symmetries of boundary problems

Abstract Boundary problems for integrable equations are considered from the viewpoint of Lies symmetry approach. A symmetry test allowing to verify whether the given boundary condition is compatible with integrability or not is proposed. New examples of integrable boundary problems are found for differential-difference equations.

Boundary Value Problems for Integrable equations compatible with the symmetry algebra

Boundary value problems for integrable nonlinear partial differential equations are considered from the symmetry point of view. Families of boundary conditions compatible with the Harry‐Dym, KdV, and mKdV equations and the Volterra chain are discussed. We also discuss the uniqueness of some of these boundary conditions.

INTEGRABLE BOUNDARY CONDITIONS FOR THE TODA LATTICE

The problem of construction of the boundary conditions for the Toda lattice compatible with its higher symmetries is considered. It is demonstrated that this problem is reduced to finding the differential constraints consistent with the ZS-AKNS hierarchy. A method of their construction is offered based on the Backlund transformations. It is shown that the generalized Toda lattices corresponding to the non-exceptional Lie algebras of finite growth can be obtained by imposing one of the four simplest integrable boundary conditions on both ends of the lattice. This fact allows, in particular, the solution of the reduction problem of the series A Toda lattices into the series D lattices. Deformations of the found boundary conditions are presented which lead to the Painleve-type equations.

Method for searching higher symmetries for quad-graph equations

A generalized symmetry integrability test for discrete equations on the square lattice is studied. Integrability conditions are discussed. A method for searching higher symmetries (including non-autonomous ones) for quad-graph equations is suggested based on characteristic vector fields.

Boundary value problems compatible with symmetries

Abstract Boundary value problems for nonlinear differential equations are considered from the point of view of symmetry. In addition to the known ones new families of boundary conditions are found for integrable equations like the Harry-Dym, KdV and mKdV.

On Darboux-integrable semi-discrete chains

A differential-difference equation with unknown t(n, x) depending on the 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 DxF = 0 and DI = I, where Dx 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 a general solution to Darboux-integrable chains is discussed and such solutions are found for a class of chains.

On the classification of Darboux integrable chains

We study a differential-difference equation of the form tx(n+1)=f(t(n),t(n+1),tx(n)) with unknown t=t(n,x) depending on x and n. The equation is called a 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±1),t(n±2),…,tx(n),txx(n),…, such that DxF=0 and DI=I, where Dx 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 give 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).