# Multiple-scale analysis of dynamical systems on the lattice

@article{Levi2011MultiplescaleAO, title={Multiple-scale analysis of dynamical systems on the lattice}, author={Decio Levi and Piergiulio Tempesta}, journal={Journal of Mathematical Analysis and Applications}, year={2011}, volume={376}, pages={247-258} }

We propose a new approach to the multiple-scale analysis of difference equations, in the context of the finite operator calculus. We derive the transformation formulae that map any given dynamical system, continuous or discrete, into a rescaled discrete system, by generalizing a classical result due to Jordan. Under suitable analytical hypotheses on the function space we consider, the rescaled equations are of finite order. Our results are applied to the study of multiple-scale reductions of… Expand

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#### References

SHOWING 1-10 OF 41 REFERENCES

Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV

- Mathematics, Physics
- 2005

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an… Expand

Multiscale analysis of discrete nonlinear evolution equations

- Mathematics, Physics
- 1999

The method of multiscale analysis is constructed for discrete systems of evolution equations for which the problem is that of the far behaviour of an input boundary datum. Discrete slow space… Expand

FAST TRACK COMMUNICATION: Multiscale expansion of the lattice potential KdV equation on functions of an infinite slow-varyness order

- Mathematics, Physics
- 2007

We present a discrete multiscale expansion of the lattice potential Korteweg–de Vries (lpKdV) equation on functions of an infinite order of slow varyness. To do so, we introduce a formal expansion of… Expand

Umbral calculus, difference equations and the discrete Schrödinger equation

- Mathematics, Physics
- 2004

In this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The… Expand

Discretization of nonlinear evolution equations over associative function algebras

- Mathematics
- 2010

Abstract A general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces, defined over a regular lattice of points, in such a way that… Expand

Nonlinear evolution equations, rescalings, model PDEs and their integrability: II

- Mathematics
- 1987

For pt. I see ibid., vol.3, p.229-62, 1987. The authors continue their investigation of the model equations that govern the wave modulations induced by weakly nonlinear effects, in the context of… Expand

On a discrete version of the Korteweg-De Vries equation

- Mathematics
- 2005

In this short communication, we consider a discrete example of how to perform multiple scale expansions and by starting from the discrete nonlinear Schroodinger equation (DNLS) as well as the… Expand

Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice

- Mathematics, Physics
- 1995

`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and… Expand

Multiple-scale perturbation beyond the nonlinear Schroedinger equation. I

- Mathematics
- 1997

Abstract We consider the effect of weak nonlinearity on the propagation of one-dimensional strongly dispersive waves. In the standard quasi-monochromatic approximation, it is well known that the… Expand

Multi-scale expansions in the theory of systems integrable by the inverse scattering transform

- Mathematics
- 1986

Abstract It is shown that using multi-scale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the IST method into other systems of this type.