Fourth-order Evolution Equations Research Articles

The modulational theory of nonlinear gravity-wave in fluid with varying depth and nonuniform current

By means of WKB expansions, new fourth order evolution equations are derived for two-dimensional Stokes waves over the bottom with arbitrary depth. The effects of slowly varying depthh=h(e2x) and currentU=U(e2x,e2t,e4z) on the evolution of a packet of Stokes waves are considered as well. In addition, numerical simulation is performed for the evolution of single envelope by finite-difference method.

A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water

The stability of a train of nonlinear surface gravity waves in deep water in the presence of wind blowing over water is considered. An evolution equation is derived for the wave envelope that is correct to fourth order in the wave steepness. The importance of the fourth-order term in the evolution equation was pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)]. From this evolution equation the expressions for the maximum growth rate of the instability and the frequency at marginal stability are derived, and graphs are plotted for those two expressions against the wave steepness.

The potential form of the fourth-order evolution equation for deep-water gravity–capillary waves

The fourth-order evolution equation for the wave envelope of a train of deep-water gravity–capillary waves is derived in terms of the complex velocity potential Ψ(x,t). Despite a difference between Ψ(x,t) and the complex amplitude a(x,t) to this order, the evolution equations are curiously almost identical.

The fourth-order evolution equation for deep-water gravity-capillary waves

The stability of a train of nonlinear gravity-capillary waves on the surface of an ideal fluid of infinite depth is considered. An evolution equation is derived for the wave envelope, which is correct to fourth order in the wave steepness. The derivation is made from the Zakharov equation under the assumption of a narrow band of waves, and including the full form of the interaction coefficient for gravity-capillary waves. It is assumed that conditions are away from subharmonic resonant wavelengths. Just as was found by K. B. Dysthe ( Proc. R. Soc. Lond . A 369 (1979)) for pure gravity waves, the main difference from the third-order evolution equation is, as far as stability is concerned, the introduction of a mean flow response. There is a band of waves that remains stable to fourth order. In general the mean flow effects for pure capillary waves are of opposite sign to those of pure gravity waves. The second-order corrections to first-order stability properties are shown to depend on the interaction between the mean flow and the envelope frequency-dispersion term in the governing equation. The results are shown to be in agreement with some recent computations of the full problem for sufficiently small values of the wave steepness.

Unconditionally stable schemes for higher order inpainting

Higher order equations, when applied to image inpainting, have certain advantages over second order equations, such as continuation of both edge and intensity information over larger distances. Discretizing a fourth order evolution equation with a brute force method may restrict the time steps to a size up to orderx 4 wherex denotes the step size of the spatial grid. In this work we present efficient semi-implicit schemes that are guaranteed to be unconditionally stable. We explain the main idea of these schemes and present applications in image processing for inpainting with the Cahn-Hilliard equation, TV-H −1 inpainting, and inpainting with LCIS (low curvature image simplifiers).