## Hylomorphic solitons in the nonlinear Klein-Gordon equation.(English)Zbl 1194.35096

The article is devoted to the solitary waves of the nonlinear Klein-Gordon equation $\psi_{tt} - \Delta \psi + f'(|\psi|) \psi = 0.$ In this paper, solitary waves are referred to as solitons if they are stable and have a particle-like behaviour. This is not a conventional definition as the word “solitons” is typically used in the context of integrable equations. The authors add another unconventional definition of “hylomorphic solitons” from the Greek words “hyle” (for matter or a set of particles) and “morphe” (for a form). The meaning of “hylomorphic solitons” is thought to be “solitons that give a suitable form to condensed matter”. The mathematical definition of the hylomorphic solitons is based on the ratio between the energy and the charge of the nonlinear Klein-Gordon equation.
The three aims of the article are formulated as follows. 5mm
1.
To give the definition of hylomorphic solitons.
2.
To describe a new variational approach to the proof of the existence of hylomorphic solitons in the context of the nonlinear Klein-Gordon equations.
3.
To classify the nonlinearities which give hylomorphic solitons.
In particular, four classes of nonlinearities are obtained and the necessary conditions for the nonlinear term to belong to a given class are derived. Numerical simulations show the different behaviour of hylomorphic solitons for these classes quantitatively.

### MSC:

 35C08 Soliton solutions 35Q51 Soliton equations 35L71 Second-order semilinear hyperbolic equations
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