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Wave Dispersion in the Linearised Fractional Korteweg - de Vries equation

In this paper we discuss some properties of linear fractional dispersive waves. In particular, we compare the dispersion relations emerging from the kinematic wave equation and from the linearised Korteweg - de Vries equation with the corresponding time-fractionalized versions. For this purpose, we evaluate the expressions for the phase velocity and for the group velocity, highlighting the differences not only analytically, but also by means of illuminating plots.


Introduction
Linear dispersive waves are defined as physical phenomena for which the relation that connects the wave number k with the angular frequency ω is non-trivial. This leads to different dependences in the behaviours of the phase velocity v p and of the group velocity v g as we vary the wave number.
In general, the relation between ω and k, known as the dispersion relation, takes the form where D is a suitable real function of ω and k. Such a relation is, in general, satisfied by certain ω, k ∈ C.
Let us assume that (1) can be solved explicitly in terms of a real variable (k or ω) by means of complex valued branches: where , m are two positive integers called mode indices. These branches provide the so-called Normal Mode Solutions for our physical system For sake of simplicity, in the following we will denote a normal mode simply by ϕ (k) and ϕ m (ω) so dropping the dependence on the space-time coordinates x, t, respectively.
The normal mode solutions represent a sort of pseudo-monochromatic modes since generally they are not sinusoidal in both space and time.
Now, for sake of brevity, we will omit the mode labels. Then we define, for the two cases (4) and (5) respectively, the phase velocity as In this paper, we will consider the relation (6) for the phase velocity, depending on k.
Furthermore, we define for both cases the group velocity as Despite the fact that the theory of linear dispersive waves is a very well established and developed field of mathematical physics, the effects of fractional extensions of such linear systems on the dispersion of waves can still represent an interesting, and utterly non-trivial, research topic. The aim of this paper is to present some examples of dispersion relations related to fractional properties of mechanical systems.
Particularly, in Section 2 we introduce the problem of dispersion for the simple case of the kinematic wave equation. Then, in Section 3 we deal with waves satisfying the linearised Korteweg -de Vries (KdV) equation.

The kinematic wave equation
We first introduce the problem of dispersion in fractional viscoelasticity showing the case of the kinematic wave equation.
The well-known kinematic wave equation usually found in literature is where the velocity of the waves c 0 is set to one in the following for sake of simplicity.
This equation leads to a dispersion equation from which one can easily infer v p = v g . Therefore, this is a clear example of a nondispersive scenario.
We can appreciate a different behavior replacing the time derivative with the fractional derivative of order α. Applying this change, our wave equation, will take the form where D α t is the well known fractional Caputo derivative of order α (see [4]), defined, for a certain function of time f (t), where, in general n ∈ N such that n − 1 < α < n. In this case, we consider values 0 < α < 1, so n = 1. Thus, we can write (9) in the Fourier domain by means of the relations and the dispersion relation becomes Thus, the angular frequency presents both a real and an imaginary part. Indeed, for k > 0, At this point, we can easily evaluate the velocities, respectively the complex phase velocity and complex the group velocity It is then important to remark that one can immediately infer that a value of α = 1 introduces dispersion effects.

Numerical Results.
Comparing the plots of certain relevant quantities can therefore be useful to understand the phenomenon. Firstly, it could be helpful to separate the real value and the imaginary value of the expressions (18) and (19). Indeed, one immediately finds that and From Figure 1 and Figure 2 we can qualitatively estimate the differences respectevely for α = 0.75 and α = 0.5. Interestingly, one finds that the real part vanishes for certain values of α = 1/2(m + 1), where m is an integer number (e.g. α = 1/4, α = 1/2).
In fact, while in Figure 1 we can appreciate the difference between real and imaginary part of both phase and group velocities, in Figure 2 is an example when the wave disappears.

The Korteweg -de Vries equation
Now, we discuss a similar situation for the KdV equation. It is a non-linear equation with several applications, such as in the study of waves on shallow water surfaces (see [1]) or solitons descriptions (see [7]). The most general form of KdV equation is where λ, µ and c 0 are real numbers, as argued in [2]. However, in this paper, we will deal the linearised KdV equation, that we recover setting λ = 0, namely fixing also c 0 = µ = 1 for sake of simplicity. In this way the linearised KdV equation is presented as the kinematic wave equation, with a dispersive perturbation term of the third order in space.
We can now focus our attention on the waves described by the related dispersion relation Thanks to the latter equation it is not difficult to compute the phase velocity v p (k) = 1 − k 2 (27) and the group velocity It is worth remarking that, in this case, we have dispersive effects even for the unmodified wave equation. Now, following a procedure akin to the one discussed in the previous section, we get The resulting dispersion relation reads and, again, the angular frequency can virtually be a complex number with non-vanishing imaginary part.
Once more, from this expression of ω(k) we get, and for the group velocity which can be split as follows and  3.1. Numerical Results. We can conclude that for α = 1, so in the ordinary case, phase velocity and group velocity are real-valued, as well shown by the Figure 3, and totally imaginary-valued for other values of α, as can be stated looking at Figure 4. In particular, it is remarkable that the argument of the cosine functions in (33) and (35) is equivalent to the corresponding one for the kinematic wave, emerging in (20) and (22). So, for α = 1/2(m + 1), with m ∈ N, velocities do not present real part.
For other real values of α, we find a mixed behavior, as we can see from Figure 5. Furthermore, we can note that there is a point where Re v p (k) = Re v g (k) and a point where Im v p (k) = Im v g (k).

Conclusion
In conclusion, it seems that the procedure of fractionalizing a linear wave equation leads to major modifications of the corresponding dispersion relation.
This analysis can surely be extended further by considering fractional derivative with respect to the space spatial coordinate, however this discussion is left for future investigations.