{"title":"Novel solitary patterns in a class of Klein–Gordon equations","authors":"Philip Rosenau , Slava Krylov","doi":"10.1016/j.physd.2025.134640","DOIUrl":null,"url":null,"abstract":"<div><div>We study the emergence, stability and evolution of solitons and compactons in a class of Klein–Gordon equations <span><span><span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>n</mi></mrow></msup><mo>−</mo><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow></msup><mo>,</mo><mspace></mspace><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo><</mo><mi>n</mi><mo>,</mo></mrow></math></span></span></span>endowed with both trivial and non-trivial stable equilibria, and demonstrate that similarly to the classical <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span> cases, solitons are linearly unstable, but the instability weakens as <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow></msub><mi>↑</mi></mrow></math></span>, and vanishes at <span><math><mrow><msubsup><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow><mrow><mi>c</mi><mi>r</mi><mi>i</mi><mi>t</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>n</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mn>2</mn><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span>, where solitons disappear and kink forms.</div><div>As the growing unstable soliton approaches the non-trivial equilibrium, it morphs into a ’mesaton’, a robust box shaped sharp pulse with a flat-top plateau, which expands at a sonic speed. In the <span><math><msubsup><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow><mrow><mi>c</mi><mi>r</mi><mi>i</mi><mi>t</mi></mrow></msubsup></math></span> vicinity, where instability is suppressed, whereas the internal modes have hardly changed, solitons persist for a very long time but then, rather than turn into mesaton, convert into a breather-like formation.</div><div>Linear damping tempers the conversion and slows mesaton’s propagation. When <span><math><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo><</mo><mi>n</mi><mo><</mo><mn>0</mn></mrow></math></span>, compactons emerge and being unstable morph either into a mesaton or into a breather-like formation.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134640"},"PeriodicalIF":2.7000,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167278925001198","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We study the emergence, stability and evolution of solitons and compactons in a class of Klein–Gordon equations endowed with both trivial and non-trivial stable equilibria, and demonstrate that similarly to the classical cases, solitons are linearly unstable, but the instability weakens as , and vanishes at , where solitons disappear and kink forms.
As the growing unstable soliton approaches the non-trivial equilibrium, it morphs into a ’mesaton’, a robust box shaped sharp pulse with a flat-top plateau, which expands at a sonic speed. In the vicinity, where instability is suppressed, whereas the internal modes have hardly changed, solitons persist for a very long time but then, rather than turn into mesaton, convert into a breather-like formation.
Linear damping tempers the conversion and slows mesaton’s propagation. When , compactons emerge and being unstable morph either into a mesaton or into a breather-like formation.
期刊介绍:
Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.