Algebraic properties of quasilinear two-dimensional lattices connected with integrability

Abstract

In the paper we discuss a classification method for nonlinear integrable equations with three independent variables based on the notion of the integrable reductions. We call an equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied in the framework of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we study the quasilinear lattices of the form

On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras

“…In §3 for the lattice of general form (1.1) a necessary integrability condition is derived which might be useful for further investigations. In §4 the efficiency of the algorithm is illustrated by presenting the results obtained earlier in [23], [24]. We note that, in contrast to the symmetry classification, in the framework of this approach we do not use non-local variables.…”

“…Theorem 4 Up to point transformations there are three essentially different versions of the chain (4.9) passing the necessary integrability test: 1) the chain (4.9) reduces to the known Ferapontov-Shabat-Yamilov chain (see [9,27]) Equations 2) and 3) are found in our articles [23,24]. It is proved in [25] that in the periodically closed case u n = u n+2 the lattice 2) admits higher symmetries.…”

“…It is proved in [25] that in the periodically closed case u n = u n+2 the lattice 2) admits higher symmetries. Theorem can be proved by showing that the characteristic algebra L y of the system of hyperbolic type equations obtained from lattice (4.9) by imposing appropriate boundary conditions is of finite dimension (see [24]). To this end we show that the subalgebra Ly ⊂ L y generated by the operators {Y j } N j=1 and R has a finite basis (see Remark 1 above) R, {Y i } N i=1 , {Y i+1,i } N −1 i=1 , {Y i+2,i+1,i } N −2 i=1 , .…”

“…In our opinion, the presence of a sufficient number of reductions that are Darboux integrable (see [13]- [21]) can also be a sign of the integrability of a multidimensional model. In our recent study (see [22]- [24]), we tested this idea by applying it to a nonlinear chain of the form (1.1).…”

Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D

“…In the studies [23,24,26,39] a classification algorithm was developed for integrable equations with three independent variables based on the idea of the Darboux integrable reductions and characteristic Lie algebras. Efficiency of the method was illustrated by applying to the equations from the class (2) (see [23]).…”