{"title":"存在离散谱的五阶修正 KdV 方程求解的长时渐近线","authors":"Nan Liu, Mingjuan Chen, Boling Guo","doi":"10.1111/sapm.12742","DOIUrl":null,"url":null,"abstract":"<p>We investigate the Cauchy problem of an integrable focusing fifth-order modified Korteweg–de Vries (KdV) equation, which contains the fifth-order dispersion and relevant higher order nonlinear terms. The long-time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a <span></span><math>\n <semantics>\n <mover>\n <mi>∂</mi>\n <mo>¯</mo>\n </mover>\n <annotation>$\\bar{\\partial }$</annotation>\n </semantics></math> generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth-order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 3","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Long-time asymptotics of solution for the fifth-order modified KdV equation in the presence of discrete spectrum\",\"authors\":\"Nan Liu, Mingjuan Chen, Boling Guo\",\"doi\":\"10.1111/sapm.12742\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We investigate the Cauchy problem of an integrable focusing fifth-order modified Korteweg–de Vries (KdV) equation, which contains the fifth-order dispersion and relevant higher order nonlinear terms. The long-time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a <span></span><math>\\n <semantics>\\n <mover>\\n <mi>∂</mi>\\n <mo>¯</mo>\\n </mover>\\n <annotation>$\\\\bar{\\\\partial }$</annotation>\\n </semantics></math> generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth-order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights.</p>\",\"PeriodicalId\":51174,\"journal\":{\"name\":\"Studies in Applied Mathematics\",\"volume\":\"153 3\",\"pages\":\"\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2024-07-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Studies in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1111/sapm.12742\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studies in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/sapm.12742","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Long-time asymptotics of solution for the fifth-order modified KdV equation in the presence of discrete spectrum
We investigate the Cauchy problem of an integrable focusing fifth-order modified Korteweg–de Vries (KdV) equation, which contains the fifth-order dispersion and relevant higher order nonlinear terms. The long-time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth-order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights.
期刊介绍:
Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.