Stability of Breathers for a Periodic Klein–Gordon Equation
The existence of breather-type solutions, i.e., solutions that are periodic in time and exponentially localized in space, is a very unusual feature for continuum, nonlinear wave-type equations. Following an earlier work establishing a theorem for the existence of such structures, we bring to bear a...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-09-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/26/9/756 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850261046053306368 |
|---|---|
| author | Martina Chirilus-Bruckner Jesús Cuevas-Maraver Panayotis G. Kevrekidis |
| author_facet | Martina Chirilus-Bruckner Jesús Cuevas-Maraver Panayotis G. Kevrekidis |
| author_sort | Martina Chirilus-Bruckner |
| collection | DOAJ |
| description | The existence of breather-type solutions, i.e., solutions that are periodic in time and exponentially localized in space, is a very unusual feature for continuum, nonlinear wave-type equations. Following an earlier work establishing a theorem for the existence of such structures, we bring to bear a combination of analysis-inspired numerical tools that permit the construction of such waveforms to a desired numerical accuracy. In addition, this enables us to explore their numerical stability. Our computations show that for the spatially heterogeneous form of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> model considered herein, the breather solutions are generically unstable. Their instability seems to generically favor the motion of the relevant structures. We expect that these results may inspire further studies towards the identification of stable continuous breathers in spatially heterogeneous, continuum nonlinear wave equation models. |
| format | Article |
| id | doaj-art-2e50234441d64bfa94027cba51acf362 |
| institution | OA Journals |
| issn | 1099-4300 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj-art-2e50234441d64bfa94027cba51acf3622025-08-20T01:55:31ZengMDPI AGEntropy1099-43002024-09-0126975610.3390/e26090756Stability of Breathers for a Periodic Klein–Gordon EquationMartina Chirilus-Bruckner0Jesús Cuevas-Maraver1Panayotis G. Kevrekidis2Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The NetherlandsGrupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla, Escuela Politécnica Superior, C/Virgen de África, 7, 41011 Sevilla, SpainDepartment of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USAThe existence of breather-type solutions, i.e., solutions that are periodic in time and exponentially localized in space, is a very unusual feature for continuum, nonlinear wave-type equations. Following an earlier work establishing a theorem for the existence of such structures, we bring to bear a combination of analysis-inspired numerical tools that permit the construction of such waveforms to a desired numerical accuracy. In addition, this enables us to explore their numerical stability. Our computations show that for the spatially heterogeneous form of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> model considered herein, the breather solutions are generically unstable. Their instability seems to generically favor the motion of the relevant structures. We expect that these results may inspire further studies towards the identification of stable continuous breathers in spatially heterogeneous, continuum nonlinear wave equation models.https://www.mdpi.com/1099-4300/26/9/756nonlinear Klein-Gordon PDEspectral stabilitybreathersheterogeneous mediacenter manifold reduction |
| spellingShingle | Martina Chirilus-Bruckner Jesús Cuevas-Maraver Panayotis G. Kevrekidis Stability of Breathers for a Periodic Klein–Gordon Equation Entropy nonlinear Klein-Gordon PDE spectral stability breathers heterogeneous media center manifold reduction |
| title | Stability of Breathers for a Periodic Klein–Gordon Equation |
| title_full | Stability of Breathers for a Periodic Klein–Gordon Equation |
| title_fullStr | Stability of Breathers for a Periodic Klein–Gordon Equation |
| title_full_unstemmed | Stability of Breathers for a Periodic Klein–Gordon Equation |
| title_short | Stability of Breathers for a Periodic Klein–Gordon Equation |
| title_sort | stability of breathers for a periodic klein gordon equation |
| topic | nonlinear Klein-Gordon PDE spectral stability breathers heterogeneous media center manifold reduction |
| url | https://www.mdpi.com/1099-4300/26/9/756 |
| work_keys_str_mv | AT martinachirilusbruckner stabilityofbreathersforaperiodickleingordonequation AT jesuscuevasmaraver stabilityofbreathersforaperiodickleingordonequation AT panayotisgkevrekidis stabilityofbreathersforaperiodickleingordonequation |