Abstract
In this chapter we construct a functional abstract framework which allows to define solitary waves, solitons and hylomorphic solitons (Sects. 2.1.1 and 2.1.3). Then, we will give some abstract existence theorems (Sect. 2.2). These theorems are based on two general minimization principles related to the concentration compactness techniques (see Sects. 2.2.3 and 2.2.4). These results are able to cover all the situations considered in the rest of this book and in most of the present literature on this subject. In the last two Sects. 2.3.1 and 2.3.2, we will discuss the meaning, the structure and possible interpretations of hylomorphic solitons.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Actually, in many concrete models, this is the generic case; this is the reason why \(\Gamma \) is called soliton manifold even if it might happen that it is not a manifold.
References
V. Benci, D. Fortunato, A minimization method and applications to the study of solitons. Nonlinear Anal. 75, 4398–4421 (2012)
V. Benci, D. Fortunato, Solitons in Schrödinger-Maxwell equations. J. Fixed Point Theory Appl. 15, (2014). arXiv:1303.1415
V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Annales de l’institut Henri Poincaré (C) Analyse non linéaire 20, 419–475 (2003)
S. Cuccagna, On asymptotic stability in 3D of kinks for the φ 4 model. Trans. Am. Math. Soc. 360(5), 2581–2614 (1986)
S. Cuccagna, T. Mizumachi, On asymptotically stability in energy space of ground states for nonlinear Schrödinger equations. Commun. Math. Phys. 284(1), 51–77 (2008)
A. Komech, B. Vainberg, On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations. Arch. Ration. Mech. Anal. 134(3), 227–248 (1996)
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(2), 109–145 (1984)
P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(4), 223–283 (1984)
K. Tintarev, K.H. Fieseler, Concentration Compactness (Imperial College Press, London, 2007)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Benci, V., Fortunato, D. (2014). Solitary Waves and Solitons: Abstract Theory. In: Variational Methods in Nonlinear Field Equations. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06914-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-06914-2_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06913-5
Online ISBN: 978-3-319-06914-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)