The steady states of the Kuramoto--Sivashinsky equation
The Kuramoto--Sivashinsky equation is one of the simple nonlinear partial diffential equations. It represents a system in which the transport of energy through nonlinear mode coupling produces a balance between long wavelength instability and short wavelength dissipation. The steady states and their stabilities give a foundation for understanding the dynamics of the system. They are distinguished according to their energy levels and fall into different branches that are identified by their harmonic content. Various types of solution include laminar states, N-cell states, long wave distorted N-cell states, the giant states, and steady traveling solutions. The linear stability around the steady states gives a guide to the origin of additional steady states through bifurcation and to the identification of each steady state. The index rule is also useful. Conjectured from the numerical observation, the parameter dependence of the steady traveling waves is estimated in the limit of large values of the parameter. In addition, a singular perturbation calculation of traveling waves near bifurcation is carried out. The dynamics of a system can be guided by the systematic study of the bifurcation and stability. For example, if all the stationary and periodic solutions are unstable for some parameter range, chaotic behavior of the system is expected, since any orbit governed by the equation has to wander around forever in the phase space. The orbits will then lie on strange attractors. 18 refs., 9 figs.
- Research Organization:
- General Atomics Co., San Diego, CA (USA)
- DOE Contract Number:
- FG03-85ER13402
- OSTI ID:
- 7230864
- Report Number(s):
- GA-A-19031; ON: DE88011819
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
990230* -- Mathematics & Mathematical Models-- (1987-1989)
DIFFERENTIAL EQUATIONS
DYNAMICS
EIGENVALUES
ENERGY TRANSPORT
EQUATIONS
FOURIER ANALYSIS
INSTABILITY
MATHEMATICAL SPACE
MECHANICS
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
ORBITS
PARTIAL DIFFERENTIAL EQUATIONS
PERTURBATION THEORY
PHASE SPACE
SPACE
TRAVELLING WAVES