Fourth-order Nonlinear Evolution Equation Research Articles

Symmetry analysis of the Infeld-Rowlands equation.

A fourth-order nonlinear evolution equation in 2+1 dimensions, arising in the study of soliton stability, is analyzed. Its symmetry group is shown to be infinite dimensional and is used to obtain particular solutions. The equation is shown not to have the Painlev\'e property.

Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water

Fourth-order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves as first pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)] and later elaborated by Janssen [J. Fluid Mech. 126, 1 (1983)], are derived for a deep-water surface gravity wave packet in the presence of a second wave packet. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. Stability analysis is made for a uniform Stokes wave train in the presence of a second wave train. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness. Significant deviations are noticed from the results obtained from the third-order evolution equations which consist of two coupled nonlinear Schrödinger equations.

Properties of nonlinear waves in dissipative-dispersive media with instability

Wave processes in dissipative-dispersive media with instability described by a fourth-order nonlinear evolution equation are considered. Analytic solutions in the form of solitary and cnoidal waves are obtained. The existence of a critical value of the dispersion coefficient beyond which an initial disturbance (in particular, “white noise”) is transformed into a structure is demonstrated by numerical modeling.

Global behavior of the solutions of certain fourth-order nonlinear equations

Two classes of fourth-order nonlinear evolution equations are considered. For the first class of equations, including the known Cahn-Hilliard equation, it is proved that there exists a global minimal B-attractor; it is compact and connected. For the second class, one of the representatives of which is the Sivashinsky equation, a theorem regarding the blowup of the solutions in finite time is proved. In addition, for the Kuramoto-Sivashinsky equation, in the one-dimensional case, the existence of a global minimal B-attractor from W2 1 in the class of even functions is proved. This attractor is compact and connected. In the multidimensional case (n=2, 3) a conditional theorem is proved regarding the existence of a compact attractor.