{"title":"双分量Novikov方程的多峰系统","authors":"Xiang-Ke Chang , Jacek Szmigielski","doi":"10.1016/j.physd.2025.134781","DOIUrl":null,"url":null,"abstract":"<div><div>The Novikov equation is a fascinating integrable modification of the Camassa–Holm equation with remarkable properties. This paper concentrates on a two-component Novikov equation featuring a non-self-adjoint 4 × 4 Lax operator. We explore the associated forward and inverse spectral maps, as well as global existence and long-time asymptotics concerning the peakon sector. On one hand, we execute an isospectral deformation to the long-time regime to compute the relevant eigenvalues using a Moser-inspired technique. To achieve this, we first demonstrate the global existence of the peakon flows, then establish a connection between the long-time asymptotics of positions and momenta and the non-zero eigenvalues. We introduce a trio of Weyl functions and show that they are matrix-valued Stieltjes transforms of discrete positive measures, again employing the deformation technique. This effectively linearizes the peakon flows on the spectral data side. Conversely, we tackle the inverse problem using Hermite–Padé approximation techniques that involve tensor products and an added symmetry condition. The thorough analysis guarantees existence and uniqueness, yielding global multipeakon formulas.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"481 ","pages":"Article 134781"},"PeriodicalIF":2.7000,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the multipeakon system of a two-component Novikov equation\",\"authors\":\"Xiang-Ke Chang , Jacek Szmigielski\",\"doi\":\"10.1016/j.physd.2025.134781\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The Novikov equation is a fascinating integrable modification of the Camassa–Holm equation with remarkable properties. This paper concentrates on a two-component Novikov equation featuring a non-self-adjoint 4 × 4 Lax operator. We explore the associated forward and inverse spectral maps, as well as global existence and long-time asymptotics concerning the peakon sector. On one hand, we execute an isospectral deformation to the long-time regime to compute the relevant eigenvalues using a Moser-inspired technique. To achieve this, we first demonstrate the global existence of the peakon flows, then establish a connection between the long-time asymptotics of positions and momenta and the non-zero eigenvalues. We introduce a trio of Weyl functions and show that they are matrix-valued Stieltjes transforms of discrete positive measures, again employing the deformation technique. This effectively linearizes the peakon flows on the spectral data side. Conversely, we tackle the inverse problem using Hermite–Padé approximation techniques that involve tensor products and an added symmetry condition. The thorough analysis guarantees existence and uniqueness, yielding global multipeakon formulas.</div></div>\",\"PeriodicalId\":20050,\"journal\":{\"name\":\"Physica D: Nonlinear Phenomena\",\"volume\":\"481 \",\"pages\":\"Article 134781\"},\"PeriodicalIF\":2.7000,\"publicationDate\":\"2025-06-19\",\"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/S0167278925002581\",\"RegionNum\":3,\"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":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167278925002581","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
On the multipeakon system of a two-component Novikov equation
The Novikov equation is a fascinating integrable modification of the Camassa–Holm equation with remarkable properties. This paper concentrates on a two-component Novikov equation featuring a non-self-adjoint 4 × 4 Lax operator. We explore the associated forward and inverse spectral maps, as well as global existence and long-time asymptotics concerning the peakon sector. On one hand, we execute an isospectral deformation to the long-time regime to compute the relevant eigenvalues using a Moser-inspired technique. To achieve this, we first demonstrate the global existence of the peakon flows, then establish a connection between the long-time asymptotics of positions and momenta and the non-zero eigenvalues. We introduce a trio of Weyl functions and show that they are matrix-valued Stieltjes transforms of discrete positive measures, again employing the deformation technique. This effectively linearizes the peakon flows on the spectral data side. Conversely, we tackle the inverse problem using Hermite–Padé approximation techniques that involve tensor products and an added symmetry condition. The thorough analysis guarantees existence and uniqueness, yielding global multipeakon formulas.
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.