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Conditions for the existence of higher symmetries of evolutionary equations on the lattice. (English) Zbl 0896.34057

A class of nonlinear differential-difference equations is given by

${u}_{n·t}\left(t\right)={f}_{n}\left({u}_{n-1}\left(t\right),{u}_{n}\left(t\right),{u}_{n+1}\left(t\right)\right),$

where ${u}_{n}\left(t\right)$ is a complex-dependent field expressed in terms of its dependent variables, $t$ varying over the complex numbers while $n$ is varying over the integers. Such an equation is a functional-differential relation that correlates the “time” evolution of a function calculated at a point $n$ to its values in its nearest neighbouring points $\left(n+1,n-1\right)$. It is considered as an infinite system of differential equations for the infinite number of functions ${u}_{n}$. Assuming that ${f}_{n}$ and ${u}_{n}$ are periodic functions in $n$ of period $k$ then the differential-difference equation becomes a (finite) system of differential equations. Differential-difference equations are important in applications as models for biological chains and as discretization of field theories. So, both as themselves and as approximation of continuous problems, they play an important role in many fields of mathematics, physics, biology and engineering.

Not many tools are available to solve such kind of problems. Apart from a few exceptional cases, solutions of nonlinear differential-difference equations are obtained only by numerical calculations or by going to the continuous limit when the lattice spacing vanishes and the system is approximated by a continuous nonlinear partial differential equation. Exceptional cases are those equations that, in one way or another, are either linearizable or integrable via the solution of an associated spectral problem on the lattice. In such cases a denumerable set of exact solutions corresponding to the symmetries of the nonlinear differential-difference equations is obtained. Such symmetries are either depending just on the dependent field and independent variable (point symmetries) or depending on the dependent field in various positions of the lattice (generalized symmetries). Any differential-difference equations can have point symmetries but the existence of generalized symmetries is usually associated only to the integrable ones.

A set of five conditions is constructed, necessary for the existence of generalized symmetries for the class of differential-difference equations described above depending only on the nearest neighbouring interactions. These conditions are applied to prove the existence of a new integrable equation belonging to this class.

##### MSC:
 34K05 General theory of functional-differential equations 35G10 Initial value problems for linear higher-order PDE 35K25 Higher order parabolic equations, general 39B32 Functional equations for complex functions