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Perturbation Theory for PDEs

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Perturbation theory for PDEs is a part of the qualitative theory of differential equations. One of the most effective methods of perturbationtheory is the normal form theory which consists of using coordinate transformations in order to describe the qualitative features of a given orgeneric equation. Classical normal form theory for ordinary differential equations has been used all along the last century in many different domains,leading to important results in pure mathematics, celestial mechanics, plasma physics, biology, solid state physics, chemistry and many otherfields.

The development of effective methods to understand the dynamics of partial differential equations is relevant in pure mathematics as well as in allthe fields in which partial differential equations play an important role. Fluidodynamics, oceanography, meteorology, quantum mechanics, andelectromagnetic theory are just a few examples of potential applications. More precisely, the normal...

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Abbreviations

Perturbation theory:

The study of a dynamical systems which is a perturbation of a system whose dynamics is known. Typically the unperturbed system is linear or integrable.

Normal form:

The normal form method consists of constructing a coordinate transformation which changes the equations of a dynamical system into new equations which are as simple as possible. In Hamiltonian systems the theory is particularly effective and typically leads to a very precise description of the dynamics.

Hamiltonian PDE:

A Hamiltonian PDE is a partial differential equation (abbreviated PDE) which is equivalent to the Hamilton equation of a suitable Hamiltonian function. Classical examples are the nonlinear wave equation , the Nonlinear Schrödinger equation , and the Kortweg–de Vries equation .

Resonance vs. Non-Resonance:

A frequency vector \( { \{\omega_k\}_{k=1}^n } \) is said to be non-resonant if its components are independent over the relative integers. On the contrary, if there exists a non-vanishing \( { K\in{\mathbb{Z}}^n } \) such that \( { \omega\cdot K=0 } \) the frequency vector is said to be resonant. Such a property plays a fundamental role in normal form theory. Non-resonance typically implies stability.

Actions:

The action of a harmonic oscillator is its energy divided by its frequency. It is usually denoted by I. The typical issue of normal form theory is that in nonresonant systems the actions remain approximatively unchanged for very long times. In resonant systems there are linear combinations of the actions with such properties.

Sobolev space:

Space of functions which have weak derivatives enjoying suitable integrability properties. Here we will use the spaces H s, \( { s\in {\mathbb{N}} } \) of the functions which are square integrable together with their first s weak derivatives.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italia

    Dario Bambusi

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  1. Dario Bambusi
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Bambusi, D. (2009). Perturbation Theory for PDEs. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_401

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