230 research outputs found
Stability of solution of Kuramoto-Sivashinsky-Korteweg-de Vries system
Abstract-A model consisting of a mixed Kuramoto-Sivashinsky-Korteweg-de Vries equation, linearly coupled to an extra linear dissipative equation has been proposed in [1] in order to describe the surface waves on multilayered liquid films and stability criteria are discussed using wave mode analysis. In this paper, we study the linear stability of solutions to the model from the viewpoint of energy estimate
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
On the backward behavior of some dissipative evolution equations
We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows
up in the energy space, backward in time, provided the solution does not belong
to the global attractor. This is a phenomenon contrast to the backward behavior
of the periodic 2D Navier-Stokes equations studied by
Constantin-Foias-Kukavica-Majda [18], but analogous to the backward behavior of
the Kuramoto-Sivashinsky equation discovered by Kukavica-Malcok [50]. Also we
study the backward behavior of solutions to the damped driven nonlinear
Schrodinger equation, the complex Ginzburg-Landau equation, and the
hyperviscous Navier-Stokes equations. In addition, we provide some physical
interpretation of various backward behaviors of several perturbations of the
KdV equation by studying explicit cnoidal wave solutions. Furthermore, we
discuss the connection between the backward behavior and the energy spectra of
the solutions. The study of backward behavior of dissipative evolution
equations is motivated by the investigation of the Bardos-Tartar conjecture
stated in [5].Comment: 34 page
Stabilized Kuramoto-Sivashinsky system
A model consisting of a mixed Kuramoto - Sivashinsky - KdV equation, linearly
coupled to an extra linear dissipative equation, is proposed. The model applies
to the description of surface waves on multilayered liquid films. The extra
equation makes its possible to stabilize the zero solution in the model,
opening way to the existence of stable solitary pulses (SPs). Treating the
dissipation and instability-generating gain in the model as small
perturbations, we demonstrate that balance between them selects two
steady-state solitons from their continuous family existing in the absence of
the dissipation and gain. The may be stable, provided that the zero solution is
stable. The prediction is completely confirmed by direct simulations. If the
integration domain is not very large, some pulses are stable even when the zero
background is unstable. Stable bound states of two and three pulses are found
too. The work was supported, in a part, by a joint grant from the Israeli
Minsitry of Science and Technology and Japan Society for Promotion of Science.Comment: A text file in the latex format and 20 eps files with figures.
Physical Review E, in pres
Large dispersion, averaging and attractors: three 1D paradigms
The effect of rapid oscillations, related to large dispersion terms, on the
dynamics of dissipative evolution equations is studied for the model examples
of the 1D complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. Three
different scenarios of this effect are demonstrated. According to the first
scenario, the dissipation mechanism is not affected and the diameter of the
global attractor remains uniformly bounded with respect to the very large
dispersion coefficient. However, the limit equation, as the dispersion
parameter tends to infinity, becomes a gradient system. Therefore, adding the
large dispersion term actually suppresses the non-trivial dynamics. According
to the second scenario, neither the dissipation mechanism, nor the dynamics are
essentially affected by the large dispersion and the limit dynamics remains
complicated (chaotic). Finally, it is demonstrated in the third scenario that
the dissipation mechanism is completely destroyed by the large dispersion, and
that the diameter of the global attractor grows together with the growth of the
dispersion parameter
Whitham's Modulation Equations and Stability of Periodic Wave Solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky Equation
International audienceWe study the spectral stability of periodic wave trains of the Korteweg-de Vries-Kuramoto-Sivashinsky equation which are, among many other applications, often used to describe the evolution of a thin liquid film flowing down an inclined ramp. More precisely, we show that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to side-band perturbations. Here, we use a direct Bloch expansion method and spectral perturbation analysis instead of Evans function computations. We first establish, in our context, the now usual connection between first order expansion of eigenvalues bifurcating from the origin (both eigenvalue 0 and Floquet parameter 0) and the first order Whitham's modulation system: the hyperbolicity of such a system provides a necessary condition of spectral stability. Under a condition of strict hyperbolicity, we show that eigenvalues are indeed analytic in the neighborhood of the origin and that their expansion up to second order is connected to a viscous correction of the Whitham's equations. This, in turn, provides new stability criteria. Finally, we study the Korteweg-de Vries limit: in this case the domain of validity of the previous expansion shrinks to nothing and a new modulation theory is needed. The new modulation system consists in the Korteweg-de Vries modulation equations supplemented with a source term: relaxation limit in such a system provides in turn some stability criteria
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