# Preliminary Considerations

• Maciej Blaszak
Part of the Texts and Monographs in Physics book series (TMP)

## Abstract

The main subject of our interest in this book is equations of the form
$${u_l} = K\left( u \right),$$
where K(u) denotes a vector field on a certain manifold M and u is a point of this manifold, which we shall refer to as a dynamical system. We do not impose any restrictions on the dimensionality of M. When M is finite dimensional, (1.1) is represented by a set of first order ordinary differential equations called a finite dimensional dynamical system. When M is infinite dimensional but with a countable number of degrees of freedom, (1.1) turns into a set of differential-difference equations known as a lattice dynamical system. Finally, when M is infinite dimensional and such that each point u = u(x) ∈ M is represented by a function of the x-variable, K = K(u,u x ,…) takes the form of a differential function of u, hence, (1.1) turns into a system of partial differential equations and we refer to it as a field dynamical system. We are going to consider all three types of dynamical systems (1.1) with K depending on u in a nonlinear way. We assume that the reader is familiar with the concept of differential manifolds. Fortunately, the majority of the interesting field dynamical systems can be considered in a topological linear space V which is reflexive, i.e. V** = V, like in the finite dimensional case, instead of an arbitrary manifold. Hence, we can avoid the problems connected with differential geometry on manifolds of infinite dimension. Actually, for most of our further considerations it is sufficient to assume that V consists of, in general complex-valued, ℂ∞—functions f of a real variable x ∈ ℝ such that f and all its derivatives vanish ‘rapidly’ at ±∞. ‘Rapidly’ means for example faster than any rational function. Typical examples of such a V are the Schwartz space S(ℝ) or L 1(ℝ) space, respectively. In the case of lattice functions, the continuous space variable x ∈ ℝ is replaced by a discrete integer variable n ∈ ℤ. In such a case, this means that to be an element of L 1(ℤ) the series $$\sum {_{n = - \infty }^{n = + \infty }} f\left( u \right)$$ f (u) must be absolutely convergent. Moreover, all differential formulas are introduced in a way which is required for our further considerations. For a more detailed discussion of differential calculus in a topological vector space we refer the reader to [191].

## Keywords

Poisson Bracket Recursion Operator Tensor Invariant Preliminary Consideration Finite Dimensional Case
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