一类 mCH-CH 方程的持久性和传播速度

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 609-620.

一类 mCH-CH 方程的持久性和传播速度

李耀红^1,²(),田守富^2,^*()

1.中国矿业大学数学学院江苏徐州 221116
2.宿州学院数学与统计学院安徽宿州 234000

收稿日期:2022-10-26 修回日期:2023-12-19 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *田守富, Email: sftian@cumt.edu.cn
作者简介:李耀红, Email:liz.zhanghy@163.com
基金资助:
安徽省教育厅自然科学研究项目(KJ2021ZD0136);安徽省教育厅自然科学研究项目(KJ2021A1102)

Persistence Property and Propagation Speed for the mCH-CH Equation

Li Yaohong^1,²(),Tian Shoufu^2,^*()

1. School of Mathematics, China University of Mining and Technology, Jiangsu Xuzhou 221116
2. School of Mathematics and Statistics, Suzhou University, Anhui Suzhou 234000

Received:2022-10-26 Revised:2023-12-19 Online:2024-06-26 Published:2024-05-17
Supported by:
NSF of Anhui Provincial Education Department(KJ2021ZD0136);NSF of Anhui Provincial Education Department(KJ2021A1102)

摘要/Abstract

摘要：

研究了一类具有平方和立方非线性项的 mCH-CH 方程初值问题, 该方程是利用双哈密顿对偶方法对 Gardner 方程约化得到的一类重要的可积方程. 首先, 通过构建权函数, 利用能量方法, 结合 Gronwall 不等式, 获得了该方程具有指数或代数衰减初值时强解的持久性. 其次, 证明了方程初始值 $ m_{0}, u_{0} $ 有紧支集时, 解 $ m(x, t) $ 有紧支集, 非平凡解 $ u $ 不再具有紧支集, 但在无穷远处有指数衰减性质.

关键词: mCH-CH 方程, 持久性, 传播速度

Abstract:

Initial value problem of the mCH-CH equation with cubic and quadratic nonlinearities is studied, which is an important integrable equations obtained by applying the bi-Hamiltonian duality approach to reduce the Gardner equation. Firstly, by constructing the weight function and using the energy method and Gronwall's inequality, the persistence property of the strong solution at infinity was obtained when the initial data decays exponentially or algebraically. Secondly, we prove that the strong solution $ m(x, t) $ has compact support when the initial data $ m_{0}, u_{0} $ has compact support, and the nontrivial solution $ u(x, t) $ no longer has compact support, but has exponential decay property at infinity.

Key words: mCH-CH equation, Persistence property, Propagation speed

中图分类号:

O175.29

李耀红, 田守富. 一类 mCH-CH 方程的持久性和传播速度[J]. 数学物理学报, 2024, 44(3): 609-620.

Li Yaohong, Tian Shoufu. Persistence Property and Propagation Speed for the mCH-CH Equation[J]. Acta mathematica scientia,Series A, 2024, 44(3): 609-620.

参考文献 25

[1]	Fokas A S. The Korteweg-de Vries equation and beyond. Acta Appl Math, 1995, 39: 295-305
[2]	Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71(11): 1661-1664 doi: 10.1103/PhysRevLett.71.1661 pmid: 10054466
[3]	Fokas A. On a class of physically important integrable equations. Phys D, 1995, 87: 145-150
[4]	Fuchssteiner B. Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation. Phys D, 1996, 95: 229-243
[5]	Olver P, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E, 1996, 53: 1900-1906 pmid: 9964452
[6]	Qiao Z. A new integrable equation with cuspons and W/M-shape-peaks solitons. J Math Phys, 2006, 47: 112701
[7]	Fisher M, Schiff J. The Camassa-Holm equation: Conserved quantities and the initial value problem. Phys Lett A, 1999, 259(5): 371-376
[8]	Himonas A, Mantzavinos D. The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation. Nonlinear Anal, 2014, 95: 499-529
[9]	Chang X, Szmigielski J. Lax integrability and the peakon problem for the modified Camassa-Holm equation. Commun Math Phys, 2018, 358: 295-341
[10]	Kang J, Liu X, Qu C. On an integrable multi-component Camassa-Holm system arising from Mbius geometry. Proc R Soc A, 2021, 477(2251): 20210164
[11]	Yang S, Qiao Z. Qualitative analysis for a two-component peakon system with cubic nonlinearity. J Math Phys, 2022, 63: 121504
[12]	Chen M, Liu Y, Qu C, Zhang H. Oscillation-Induced blow-up to the modified Camassa-Holm equation with linear dispersion. Adv Math, 2015, 272: 225-251
[13]	Gui G, Liu Y, Olver P, Qu C. Wave-Breaking and peakons for a modified Camassa-Holm equation. Commun Math Phys, 2013, 319: 731-759
[14]	Constantin A, Escher J. Well-posedness, global existence, and blow up phenomena for a periodic quasi-linear hyperbolic equation. Comm Pure Appl Math, 1998, 51(5): 475-504
[15]	Liu X, Qiao Z, Yin Z. On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity. Commun Pure Appl Anal, 2014, 13: 1283-1304
[16]	Xu R, Yang Y. Local well-posedness and decay for some generalized shallow water equations. J Differ Equations, 2023, 367: 689-728
[17]	Chen M, Guo F, Liu Y, Qu C. Analysis on the blow-up of solutions to a class of integrable peakon equations. J Funct Anal, 2016, 270(6): 2343-2374
[18]	Xia B, Qiao Z, Li J. An integrable system with peakon, complex peakon, weak kink, and kink-peakon interactional solutions. Commun Nonlinear Sci Numer Simulat, 2018, 63: 292-306
[19]	Qin G, Yan Z, Guo B. The Cauchy problem and multi-peakons for the mCH-Novikov-CH equation with quadratic and cubic nonlinearities. J Dyn Differ Equ, 2023, 35: 3295-3354
[20]	Himonas A A, Misiolek G, Ponce G, Zhou Y. Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Comm Math Phys, 2007, 271: 511-522
[21]	Cui W, Han L. Infinite propagation speed and asymptotic behavior for a generalized Camassa-Holm equation with cubic nonlinearity. Appl Math Lett, 2018, 77: 13-20
[22]	Tian S. Infinite propagation speed of a weakly dissipative modified two-component Dullin-Gottwald-Holm system. Appl Math Lett, 2019, 89: 1-7
[23]	Zhu M, Jiang Z, Qiao Z. Persistence property and infinite propagation speed for the b-family of Fokas- Olver-Rosenau-Qiao (bFORQ) model. Appl Math Lett, 2022, 124: 10765
[24]	Zhu M, Jiang Z, Qiao Z. Analytical properties for the fifth-order b-family Novikov model. J Evol Equ, 2022, 22: Article number 19
[25]	田守富. 一个弱耗散修正的二分量Dullin-Gottwald-Holm 系统解的行为研究. 数学物理学报, 2020, 40A(5): 1204-1223
	Tian S F. On the behavior of the solution of a weakly dissipative modified two-component Dullin-Gottwald- Holm system. Acta Math Sci, 2020, 40A(5): 1204-1223