Multidimensional Toda Lattices: Continuous and Discrete Time

Abstract

Abstract. In this paper we present multidimensional analogues of both the continuousand discrete-time Toda lattices. The integrable systems that we consider here have two or more space coordinates. To construct the systems, we generalize the orthogonal polynomial approach for the continuous and discrete Toda lattices to the case of multiple orthogonal polynomials.

Nonautonomous ultradiscrete hungry Toda lattice and a generalized box-ball system

“…Similar calculations for each pair of the spectral transformations (11) also yield ( − ( 1 ) 1 ) ( 1 +1, 2 , 1 +1, 2 ) ( ) = ( 1 , 2 , 1 +1,2 )̃ ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 )( ) =̃ ( 1 +1, 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 ) ( ), ( − ( 1 ) 1 ) ( 1 , 2 , 1 +1, 2 ) ( ) = ( 1 , 2 , 1 +1, 2 )̃ ( 1 , 2 +1, 1 , 2 ) ( 1 , 2 +1, 1 , 2 ) ( ) =̃ ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 ) ( 1 , 2 +1, 1 , 2 ) ( ), ( 1 +1, 2 , 1 , 2 ) ( ) =̃ ( 1 +1, 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 +1) ( 1 , 2 , 1 , 2 +1) ( ) = ( 1 , 2 , 1 , 2 )̃ ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 +1) ( ) ( 1 , 2 , 1 , 2 ) ( ) =̃ ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 +1) ( 1 , 2 +1, 1 , 2 +1) ( ) = ( 1 , 2 , 1 , 2 )̃ ( 1 , 2 +1, 1 , 2 ) ( 1 , 2 +1, 1 , 2 +1) ( ), ( − ( 1 ) 1 ) ( 1 , 2 , 1 +1, 2 ) ( ) =̃ ( 1 , 2 , 1 +1, 2 )̃ ( 1 , 2 , 1 , 2 +1) ( 1 , 2 , 1 , 2 +1) ( ) =̃ ( 1 , 2 , 1 , 2 )̃ ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 +1) ( ),whose elements give the relations( 1 , 2 , 1 +1, 2 ) +̃ ( 1 , 2 , 1 , 2 ) +1 = ( 1 , 2 , 1 , 2 ) +1 +̃ ( 1 +1, 2 , 1 , 2 ) , (13a) ( 1 , 2 , 1 +1, 2 ) ̃ ( 1 , 2 , 1 , 2 ) = ( 1 , 2 , 1 , 2 ) ̃ ( 1 +1, 2 , 1 , 2 ) , (13b) ( 1 , 2 +1, 1 , 2 ) + ( 1 , 2 , 1 +1, 2 ) −1 =̃ ( 1 , 2 , 1 , 2 ) + ( 1 , 2 , 1 , 2 ) , (14a) ( 1 , 2 +1, 1 , 2 ) ( 1 , 2 , 1 +1, 2 ) =̃ ( 1 , 2 , 1 , 2 ) +1 ( 1 , 2 , 1 , 2 ) , (14b) ( 1 , 2 , 1 , 2 +1) +̃ ( 1 +1, 2 , 1 , 2 ) −1 = ( 1 , 2 , 1 , 2 ) +̃ ( 1 , 2 , 1 , 2 ) , (15a) ( 1 , 2 , 1 , 2 +1) ̃ ( 1 +1, 2 , 1 , 2 ) = ( 1 , 2 , 1 , 2 ) +1̃ ( 1 , 2 , 1 , 2 ) , (15b) ( 1 , 2 , 1 , 2 +1) +̃ ( 1 , 2 , 1 , 2 ) = ( 1 , 2 , 1 , 2 ) +̃ ( 1 , 2 +1, 1 , 2 ) , (16a) ( 1 , 2 , 1 , 2 +1) ̃ ( 1 , 2 , 1 , 2 ) +1 = ( 1 , 2 , 1 , 2 ) +1̃ ( 1 , 2 +1, 1 , 2 ) , (16b) ( 1 , 2 , 1 , 2 +1) +̃ ( 1 , 2 , 1 +1, 2 ) −1 =̃ ( 1 , 2 , 1 , 2 ) +̃ ( 1 , 2 , 1 , 2 ) , (17a) ( 1 , 2 , 1 , 2 +1) ̃ ( 1 , 2 , 1 +1, 2 ) =̃ ( 1 , 2 , 1 , 2 ) +1̃ ( 1 , 2 , 1 , 2 )(17b)for = 0, 1, 2, … with the boundary condition (12c) and( 1 , 2 , 1 , 2 ) −1 = 0 for all 1 , 2 , 1 , 2 ∈ ℤ.In these discrete equations, the parameters( 1 ) 1 and ( 2 ) 2 do not appear explicitly. The parameters are, in fact, embedded into boundary conditions as follows.For equations (13) Subtraction of (10a) from (10c) yields the relation ( − ( 1 )1 )( 1 , 2 , 1 +1, 2 ) ( ) = ( 1 +1, 2 , 1 , 2 ) ( ) + ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 ) ( ),(18)where ( 1 , 2 , 1 , 2 ) ≔̃ ( 1 , 2 , 1 , 2 ) − ( 1 , 2 , 1 , 2 ) .…”

“…It is known that equations (2) are derived from the discrete Toda lattice through ultradiscretization [15]. The right side of figure 1 shows an example of the time evolution of the ultradiscrete Toda lattice (2), in which the initial values are chosen to correspond to the initial state of the original BBS on the left side. We can introduce some extended rules to the original BBS.…”

“…( 1 +1,2 , 1 , 2 ) ( ) = ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 ) ( ), (11a) ( 1 , 2 −1, 1 , 2 ) ( ) = ( 1 , 2 −1, 1 , 2 ) ( 1 , 2 , 1 , 2 ) ( ), (11b) ( − ( 1 ) 1 ) ( 1 , 2 , 1 +1, 2 ) ( ) =̃ ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 ) ( ), (11c) ( 1 , 2 , 1 , 2 −1) ( ) =̃ ( 1 , 2 , 1 , 2 −1) ( 1 , 2 , 1 , 2 ) ( ). (11d)From (11a) and (11b), we have( 1 +1, 2 , 1 , 2 ) ( ) = ( 1 +1, 2 , 1 , 2 ) ( 1 , 2 +1, 1 , 2 ) ( 1 , 2 +1, 1 , 2 ) ( ) = ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 ) ( 1 , 2 +1, 1 , 2 ) ( ),whose each element gives( 1 +1, 2 , 1 , 2 ) ( ) − ( 1 , 2 +1, 1 , 2 ) +1 ( ) = ( ( 1 , 2 +1, 1 , 2 ) + ( 1 +1, 2 , 1 , 2 ) −1 ) ( 1 , 2 +1, 1 , 2 ) ( ) + ( 1 , 2 +1, 1 , 2 ) −1 ( 1 +1, 2 , 1 , 2 ) −1 ( 1 , 2 +1, 1 , 2 ) −1 ( ) = ( ( 1 , 2 , 1 , 2 ) + ( 1 , 2 , 1 , 2 ) ) ( 1 , 2 +1, 1 , 2 ) ( ) + ( 1 , 2 , 1 , 2 ) ( 1 , 2 , 1 , 2 ) −1 ( 1 , 2 +1, 1 , 2 ) −1( ) for = 0, 1, 2, … .…”

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

“…There were several results in this direction. In [8,9], equations (1.10) and (1.11) were combined to obtain the electro-magnetic Schrödinger operator defined on 2 (Z d + ). These operators were symmetrized but only in very special cases.…”

Selfadjoint Jacobi matrices on graphs and multiple orthogonal polynomials

“…, ), принадлежащих всей ре-шётке Z + , уместно рассматривать с точки зрения спектральной теории мно-гомерных дискретных операторов. Некоторые шаги в этом направлении были сделаны в [20,21,22], где, в частности, многомерный дискретный электромаг-нитный оператор Шредингера был получен путём усреднения рекурренций (1.7). Однако общих классов разностных потенциалов { ⃗ , , ⃗ , } =1 , коэффи-циентов, допускающих симметризацию, приводящую к самосопряжённому в 2 (Z + ) электромагнитному оператору Шредингера, пока получить не удалось.…”