Traveling waves in nonlinear media with dispersion, dissipation, and reaction
Koçak Type Equations
The competition of dispersion, diffusion, convection, reaction and nonlinearity generally exhibits blow-up behaviour because of the complexity of nonlinear phenomena. However, fruitful interactions in nonlinear wave phenomena, such as soliton, kink, breather and optical waves, can rarely be obtained by the mix of such different entities with suitable parameters. In [1], a third-order dispersion-dissipation-reaction model, called the KdV-Burgers-Fisher (KdVBF) equation, is newly proposed as
ut + ε u ux - ν uxx + μ uxxx = r u(1 - u),(1)
where ε, ν, μ and r are real parameters for convection, diffusion, dispersion and reaction terms, respectively. Once can easily see that equation (1) can be reduced to the KdV equation when ν = r = 0, the Burgers’ equation when μ = r = 0, the KdV-Burgers equation when r = 0, the Fisher’s or the KPP (Kolmogorov-Petrovskii-Piskunov) equation when ε = μ = 0, the Burgers-Fisher equation when μ = 0 and the dispersive-Fisher equation,
ut + μ uxxx = r u(1 - u),(2)
when ε = ν = 0. Closed-form solutions of the equation (1) and (2) can be found in [1] and [2], respectively.
In order to specify such combined nonlinear phenomena including dispersion and reaction, we would like to call the equations (1) and (2) as the Koçak type equations. Recent analytical and numerical studies on the corresponding models, which are intriguing, can be seen here.
Moreover, in [1], the fifth-order dispersion-dissipation-reaction model, let us call the fifth-order Koçak's equation (Koçak-5 equation), is proposed as
ut + ε u2 ux - ν uxx + μ uxxx - γ uxxxxx = r u(1 - u),(3)
where ε, ν, μ, r and γ are real parameters. In addition to above well-known models, the fifth-order KdV equation and the modified Kawahara equation can be obtained by taking r = 0 and ν = r = 0 in (3), respectively.
Additionally, in [3], the generalized KdV-Fisher, let us call the generalized Koçak's equation (gKoçak equation), is proposed as
ut + ε un ux + μ uxxx = r u(1 - un),(4)
where ε, μ, and r are real parameters.
The first numerical attempt on the equations (1), (2) and (4) can be found in [4].
All above mentioned celebrated models have been deeply studied in the literature by using various analytical and numerical methods. However, the combination of the KdV, Burgers’ and Fisher’s equation has not been studied in details. It is believed that the fractional, higher-odd-order, generalised, and spatial system versions of the corresponding models will be analytically and numerically studied in physics, engineering, epidemiology, physiology, ecology and even in sociology [5].
For fruitful discussions on higher-order dispersive-Fisher equations, we refer to Prof. Victor A. Galaktionov's detailed paper [6].
Feel free to get in touch if you have any questions or requests regarding the corresponding models.