Numerical solution of the generalized, dissipative KdV–RLW–Rosenau equation with a compact method

Abstract

The nonlinear dynamics of the one–dimensional, generalized Korteweg–de Vries–regularized–long wave–Rosenau (KdV–RLW–Rosenau) equation with second– and fourth–order dissipative terms subject to initial Gaussian conditions is analyzed numerically by means of three–point, fourth–order accurate, compact finite differences for the discretization of the spatial derivatives and a trapezoidal method for time integration. By means of a Fourier analysis and global integration techniques, it is shown that the signs of both the fourth–order dissipative and the mixed fifth–order derivative terms must be negative. It is also shown that an increase of either the linear drift or the nonlinear convection coefficients results in an increase of the steepness, amplitude and speed of the right–propagating wave, whereas the speed and amplitude of the wave decrease as the power of the nonlinearity is increased, if the amplitude of the initial Gaussian condition is equal to or less than one. It is also shown that the wave amplitude and speed decrease and the curvature of the wave’s trajectory increases as the coefficients of the second– and fourth–order dissipative terms are increased, while an increase of the RLW coefficient was found to decrease both the damping and the phase velocity, and generate oscillations behind the wave. For some values of the coefficients of both the fourth–order dissipative and the Rosenau terms, it has been found that localized dispersion shock waves may form in the leading part of the right–propagating wave, and that the formation of a train of solitary waves that result from the breakup of the initial Gaussian conditions only occurs in the absence of both Rosenau’s, Kortweg–de Vries’s and second– and fourth–order dissipative terms, and for some values of the amplitude and width of the initial condition and the RLW coefficient. It is also shown that negative values of the KdV term result in steeper, larger amplitude and faster waves and a train of oscillations behind the wave, whereas positive values of that coefficient may result in negative phase and group velocities, no wave breakup and oscillations ahead of the right–propagating wave.