随机 Kuramoto-Sivashinsky 方程行波解的非线性稳定

Abstract

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

CLC Number:

O175.2

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.

Trendmd

References

[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
[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
[16]	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
[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

Nonlinear Stability of Traveling Waves for Stochastic Kuramoto-Sivashinsky Equation

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Wu Wenbin, Ren Xue, Zhang Ran. Traveling Waves for a Discrete Diffusive Vaccination Model with Delay [J]. Acta mathematica scientia,Series A, 2025, 45(3): 858-874.
[2]	Liu Jia, Bao Xiongxiong. Asymptotic Stability of Pyramidal Traveling Front for Nonlocal Delayed Diffusion Equation [J]. Acta mathematica scientia,Series A, 2025, 45(1): 44-53.
[3]	Yang Yongli, Yang Yunrui. Traveling Wave Solutions to a Cholera Epidemic System with Spatio-Temporal Delay and Nonlocal Dispersal [J]. Acta mathematica scientia,Series A, 2025, 45(1): 110-135.
[4]	Zhang Guangxin, Yang Yunrui, Song Xue. Periodic Traveling Wave Solutions of Delayed SEIR Systems with Nonlocal Effects [J]. Acta mathematica scientia,Series A, 2024, 44(1): 140-159.
[5]	Liao Yuankang. Nonlinear Stability of Viscous Shock Waves for One-dimensional Isentropic Compressible Navier-Stokes Equations with Density-Dependent Viscosity [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1149-1169.
[6]	Yu Yang. Traveling Wave of a Nonlocal Dispersal SIR Model [J]. Acta mathematica scientia,Series A, 2022, 42(5): 1409-1415.
[7]	Kai Wang,Hongyong Zhao. Traveling Wave of a Reaction-Diffusion Dengue Epidemic Model with Time Delays [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1209-1226.
[8]	Leta Temesgen Desta,Wenjun Liu,Jian Ding. Dynamics of Traveling Wave Solutions to Fully Nonlinear Heavy Ion-Acoustic Degenerate Relativistic Quantum Plasmas [J]. Acta mathematica scientia,Series A, 2021, 41(2): 496-506.
[9]	Dong Deng,Yan Li. Traveling Waves in a Nonlocal Dispersal SIR Epidemic Model with Treatment [J]. Acta mathematica scientia,Series A, 2020, 40(1): 72-102.
[10]	Xue Zhang,Yuhuai Sun. Dynamical Analysis and Traveling Wave Solutions for Generalized (3+1)-Dimensional Kadomtsev-Petviashvili Equation [J]. Acta mathematica scientia,Series A, 2019, 39(3): 501-509.
[11]	Zou Xia, Wu Shiliang. Traveling Waves in a Nonlocal Dispersal SIR Epidemic Model with Delay and Nonlinear Incidence [J]. Acta mathematica scientia,Series A, 2018, 38(3): 496-513.
[12]	Guo Tingguang, Xu Zhiting. Existence of Traveling Waves in a Spatial Infectious Disease Model with Vaccination [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1129-1147.
[13]	Lou Cuijuan, Yang Yin. Existence of Traveling Wave Solutions for a Classical Chemotaxis Model [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1044-1058.
[14]	Kuang Jie,Wang Zejun. The Large Time Behavior of Inhomogeneous Burgers Equation with Periodic Initial Data [J]. Acta mathematica scientia,Series A, 2015, 35(1): 1-14.
[15]	ZHANG Wei-Guo, LIU Qiang, LI Zheng-Ming, LI Xiang. BOUNDED TRAVELING WAVE SOLUTIONS OF VARIANT BOUSSINESQ EQUATION WITH A DISSIPATION TERM AND DISSIPATION EFFECT [J]. Acta mathematica scientia,Series A, 2014, 34(3): 941-959.