Symmetry approach in boundary value problems

The problem of construction of the boundary conditions for nonlinear equations is considered compatible with their higher symmetries. Boundary conditions for the sine-Gordon, Jiber-Shabat and KdV equations are discussed. New examples are found for the Jiber-Shabat equation.Comment: 7 pages, LaTe

Complete list of Darboux Integrable Chains of the form t1x=tx+d(t,t1)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 $f(u,v,w)=w+g(u,v)$

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 of the differential constraints consistent with the ZS-AKNS hierarchy. A method of their construction is offered based on the B\"acklund 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 the both ends of the lattice. This fact allows, in particular, to solve the problem of reduction of the series $A$ Toda lattices into the series $D$ ones. Deformations of the found boundary conditions are presented which leads to the Painlev\'e type equations. Key words: Toda lattice, boundary conditions, integrability, B\"acklund transformation, Lie algebras, Painlev\'e equation