随机 Kuramoto-Sivashinsky 方程行波解的非线性稳定

数学物理学报 ›› 2025, Vol. 45 ›› Issue (3): 790-806.

随机 Kuramoto-Sivashinsky 方程行波解的非线性稳定

刘羽¹(),陈光淦^1,^*(),李树勇²()

¹四川师范大学数学科学学院, 可视化计算与虚拟现实四川省重点实验室成都 610068
²绵阳师范学院数理学院四川绵阳 621000

收稿日期:2024-07-22 修回日期:2025-01-26 出版日期:2025-06-26 发布日期:2025-06-20
通讯作者: *陈光淦, E-mail: chenguanggan@hotmail.com
作者简介:刘羽, E-mail: liuyu96@hotmail.com; 李树勇, E-mail: shuyongli@sicnu.edu.cn
基金资助:
国家自然科学基金(12171343);四川省科技计划(2022JDTD0019)

Nonlinear Stability of Traveling Waves for Stochastic Kuramoto-Sivashinsky Equation

Liu Yu¹(),Chen Guanggan^1,^*(),Li Shuyong²()

¹School of Mathematical Science and V.C. $\&$ V.R.Key Lab, Sichuan Normal University, Chengdu 610068
²College of Mathematics and Physics, Mianyang Teachers' College, Sichuan Mianyang 621000

Received:2024-07-22 Revised:2025-01-26 Online:2025-06-26 Published:2025-06-20
Supported by:
NSFC(12171343);Sichuan Science and Technology Program(2022JDTD0019)

摘要/Abstract

摘要：

文章主要研究了随机 Kuramoto-Sivashinsky 方程的行波解的非线性稳定性. 利用随机相位变换法和分裂时间变量, 验证了当随机系统的噪声强度足够小并且其初始值足够接近所对应确定系统的行波时, 该随机系统所对应的确定系统的行波解保持非线性稳定性.

关键词: 随机 Kuramoto-Sivashinsky 方程, 行波, 相移, 非线性稳定

Abstract:

This work is concerned with the nonlinear stability of traveling wave for the stochastic Kuramoto-Sivashinsky equation. By stochastic phase shift method and splitting time argument, we prove that the traveling wave solution of the deterministic system retain the nonlinear stability when the noise intensity of the stochastic system is small enough and its initial value is sufficiently close to the traveling wave of the corresponding deterministic system.

Key words: stochastic Kuramoto-Sivashinsky equation, traveling wave, phase shift, nonlinear stability

中图分类号:

O175.2

刘羽, 陈光淦, 李树勇. 随机 Kuramoto-Sivashinsky 方程行波解的非线性稳定[J]. 数学物理学报, 2025, 45(3): 790-806.

Liu Yu, Chen Guanggan, Li Shuyong. Nonlinear Stability of Traveling Waves for Stochastic Kuramoto-Sivashinsky Equation[J]. Acta mathematica scientia,Series A, 2025, 45(3): 790-806.

参考文献 25

[1]	Kuramoto Y. Instability and turbulence of wavefronts in reaction-diffusion systems. Prog Theor Phys, 1980, 63(6): 1885-1903
[2]	Sivashinsky G. Nonlinear analysis of hyrodynamic instability in laminar flames. Acta Astronaut, 1977, 4(11/12): 1117-1206
[3]	Duan J, Ervin V J. On the stochastic Kuramoto-Sivashinsky equation. Nonlinear Anal, 2001, 44: 205-216
[4]	Rose G A P, Suvinthra M, Balachandran K. Large deviations for stochastic Kuramoto-Sivashinsky equation with multiplicative noise. Nonlinear Anal: Model and control, 2021, 26(4): 642-660
[5]	Wang G, Wang X, Wang Y. Long time behavior for nonlocal stochastic Kuramoto-Sivashinsky equations. Stat Probabil Lett, 2014, 87: 54-60
[6]	Wang G, Wang X, Xu G. Long time stability of nonlocal stochastic Kuramoto-Sivashinsky equations with jump noises. Stat Probabil Lett, 2017, 127: 23-32
[7]	Wu W, Cui S, Duan J. Global well-posedness of the stochastic generalized Kuramoto-Sivashinsky equation with multiplicative noise. Acta Math Appl Sin-E, 2018, 34: 566-584 doi: 10.1007/s10255-018-0769-3
[8]	Yang D. Random attractors for the stochastic Kuramoto-Sivashinsky equation. Stoch Anal Appl, 2006, 24: 1285-1303
[9]	Tadmor E. The well-posedness of the Kuramoto-Sivashinsky equation. SIAM J Math Anal, 1986, 17: 884-893
[10]	Nickel J. Travelling wave solutions to the Kuramoto-Sivashinsky equation. Chaos Soliton Fract, 2007, 33: 1376-1382
[11]	Zayed E M E, Nofal T A, Gepreel K A. The travelling wave solutions for non-linear initial-value problems using the homotopy perturbation method. Appl Anal, 2009, 88: 617-634
[12]	Ge J H, Hua C C, Feng Z S. A method for constructing traveling wave solutions to nonlinear evolution equations. Acta Appl Math, 2012, 118: 185-201
[13]	Barker B, Johnson M A, Noble P, Rodrigues M, Zumbrun K. Stability of periodic Kuramoto-Sivashinsky waves. Appl Math Lett, 2012, 25: 842-829
[14]	Hooper A P, Grimshaw R. Travelling wave solutions of the Kuramoto-Sivashinsky equation. Wave Motion, 1988, 5: 899-919
[15]	Zumbrun K. Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves. Q Appl Math, 2011, 69: 177-202
[16]	周永辉. 一类具有时滞的非局部反应扩散方程非单调临界行波解的全局稳定性. 数学物理学报, 2020, 40A(4): 983—992
	Zhou Y H. Global stability of the nonmonotone critical traveling waves for reaction diffusion equations. Acta Math Sci, 2020, 40A(4): 983-992
[17]	Shargatov V A, Chugainova A P, Kolomiytsev G V. Global stability of traveling wave solutions of generalized Korteveg-de Vries-Burgers equation with non-constant dissipation parameter. J Comput Appl Math, 2022, 412: 114354
[18]	Cornwell P, Jones C K R T. On the existence and stability of fast traveling waves in a doubly diffusive FitzHugh-Nagumo system. SIAM J Appl Dyn Syst, 2018, 17(1): 754-787
[19]	Lang E. A multiscale analysis of traveling waves in stochastic neural fields. SIAM J Appl Dyn Syst, 2016, 15: 1581-1614
[20]	Hamster C H S, Hupkes H J. Stability of traveling waves for reaction-diffusion equations with multiplicative noise. SIAM J Appl Dyn Syst, 2019, 18: 205-278 doi: 10.1137/17M1159518
[21]	Hamster C H S, Hupkes H J. Traveling waves for reaction-diffusion equations forced by translation invariant noise. Physica D, 2020, 401: 132233
[22]	Lorenzi L, Lunardi A, Metafune G, Pallara D. Analytic Semigroups and Reaction-Diffusion Problems. Internet Seminar, 2005
[23]	Howard P, Zumbrun K. Stability of undercompressive shock profiles. J Differ Equations, 2006, 225: 308-360
[24]	Yosida K. Functional Analysis. Berlin:Springer, 1980
[25]	Temam R, Wang X M. Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivashinsky equation in the general case. Differ Integral Equ, 1994, 7: 1095-1108