Existence of Peregrine Solitons in fractional reaction-diffusion equations

In this article, we will analyze the existence of Peregrine type solutions for the fractional diffusion reaction equation by applying Splitting-type methods. These functions that have two main characteristics, they are direct sum of functions of periodic type and functions that tend to zero at infinity. Global existence results are obtained for each particular characteristic, for then finally combining both results.


Introduction
We consider the non autonomous system ∂ t u + σ(−∆) β u = F (t,u), (1.1) where u(x,t) ∈ Z for x ∈ R n , t > 0, σ ≥ 0 and 0 < β ≤ 1, F : R × Z → Z a continuous map and Z a Banach space. We consider the initial problem u(x,0) = u 0 (x). The aim of this paper is to prove the existence of Peregrine type of solutions for the fractional reaction diffusion equation, using recent numerical splitting techniques ( [5], [12], and [4]) introduced for other purposes. Peregrine solitons were studied in ( [20]), and has multiple applications (See for example, [3], [11], [15], [14] and [24]). "Peregrine solitons" are functions with two main characteristics: These are direct sum of periodic functions and functions that tend to zero when the spatial variable tends to infinity.
Fractional reaction-diffusion equations are frequently used on many different topics of applied mathematics such as biological models, population dynamics models, nuclear reactor models, just to name a few (see [2], [7], [8] and references therein).
The fractional model captures the faster spreading rates and power law invasion profiles observed in many applications compared to the classical model (β = 1). The main reason for this behavior is given by the fractional Laplacian, that is described by standard theories of fractional calculus (for a complete survey see [19]). There are many different equivalent definitions of the fractional Laplacian and its behavior is well understood (see [6], [13], [16], [18], [25], [21] and [17]).
The non-autonomous nonlinear reaction diffusion equation dynamics were studied by [22] and others, analyzing the stability and evolution of the problem.
The paper is organized as follows: In Section 2 we set notations and preliminary results and in Section 3 we present main results, focusing first on each characteristic of the direct sum separately, for finally joining both results to reach the existence of Peregrine Solitons.

Notations and Preliminaries.
We are interested in continuous functions to vectorial values, that is to say, whose evaluations take values in Banach Spaces.
Let Z be a Banach space, we define C u (R d ,Z) as the set of uniformly continuous and bounded functions on R d with values in Z. Taking the norm the Bochner integral is defined in the following way, This defines an element of C u (R d ,Z) and the linear operator u → g * u is continuous (see [10]).
The following results show that the operator −(−∆) β defines a continuous contraction semigroup in the Banach space C u (R d ,Z). The following lemma is a consequence of Lévy-Khintchine formula for infinitely divisible distributions and the properties of the Fourier transform. Lemma 1. Let 0 < β ≤ 1 and g β ∈ C 0 (R d ) such thatĝ β (ξ) = e −|ξ| 2β , it holds g β is positive, invariant under rotations of R d , integrable and Proof. The first statement follows from Theorem 14.14 of [23], the remaining claims are immediate from the definition ofĝ β .
Based on the previous lemma, we study Green's function associated to the linear operator ∂ t + σ(−∆) β . Proposition 2. Let σ > 0 and 0 < β ≤ 1, the function G σ,β given by ii. G σ,β (.,t) ∈ L 1 (R d ) and Proof. The first and second statements are a consequence of the definition ofĝ β . The third and fourth statements are immediate applying Fourier transform.
In the following proposition, we have that the linear operator −σ(−∆) β defines a contraction continuous semigroup in the set C u (R d ,Z).
In this paper, we consider integral solutions of the problem (1.1). We say that Since our method to build solutions of (2.1) is based on the application of the Lie-Trotter method, it is necessary to study the non-linear problem associated with F . We remark that some regularity condition is necessary for convergence, as it is shown in the counterexample given in [9]. Let F : R + × Z → Z be a continuous map, we say that is locally Lipschitz in the second variable if, given R, In this case, for any z 0 ∈ Z there exists a unique (maximal) solution of the Cauchy problem It is easy to see that there exists a nonincreasing function T : Also, one of the following alternatives holds: We can see that F : ) is continuous and locally Lipschitz in the second variable. Therefore, we can consider problem The following result relates the solutions of (2.2) with the problem (2.1) in the case of having constant initial data.
Proof. Since u 0 is a constant function, from the uniqueness of the problem (2.2), we have u(t) is a constant function for any t > 0 where the solution is defined. Therefore, which proves our assertion. Theorem 6. There exists a function T * : C u (R d ,Z) → R + such that for u 0 ∈ C u (R d ,Z), exists a unique u ∈ C([0,T * (u 0 )),C u (R d ,Z)) mild solution of (1.1) with u(0) = u 0 . Moreover, one of the following alternatives holds:

Periodic solutions
In this section, we will analyze the existence of solutions for the fractional reaction diffusion equation by applying Splitting methods to functions that have two main characteristics: these are direct sum of functions of periodic type and functions that tend to zero at infinity. This type of solution is also studied in the non-linear Schroedinger equation, under the name of "Peregrine solitons" [20]. Well posedness results are obtained for each particular characteristic, to then combine both results. In addition, we will observe that the evolution of the periodic part is independent of the part that tends to zero at infinity. For instance, suppose that the non-linearity is of polynomial type (as in the Fitzhugh-Nagumo equation, see [1]), in this case we use F (u) = u 2 . If u(t) = v(t) + w(t), where v(t) is a periodic function and w(t) is a function that tends to zero when the spatial variable tends to infinity, then we have that where, v 2 is periodic and 2vw + w 2 tends to zero. In this specific case we can appreciate the absorption of the part that tends to zero, in the crossed terms. As v 2 = F (v), we expect that the periodic part of the initial data evolve independently from the part that tends to zero for the non linear equation. In this section we obtain general results to which this example refers.
We consider the space C 0 (R d ,Z) of functions which converge to 0 when |x| → ∞. It is easy to prove the following result.
Lemma 9. Let X be a Banach space and let X 1 ,X 2 ⊂ X be closed subspaces such that X 1 X 2 = {0}, the following statement are equivalent i. X 1 ⊕ X 2 is closed.
ii. The projector P : X 1 ⊕ X 2 → X 1 is continuous.
Proof. Since X 1 ⊕ X 2 is a Banach space, the linear map φ : X 1 × X 2 → X 1 ⊕ X 2 given by φ(x 1 ,x 2 ) = x 1 + x 2 is bijective, and from the closed graph theorem we have φ and φ −1 are continuous operator. We can write P = π 1 φ −1 and then P is continuous. On the other hand, X 1 ⊕ X 2 = P −1 X 1 , since P continuous and X 1 a closed subspace, X 1 ⊕ X 2 is closed.
To obtain the existence of solutions in the space X Γ,Z , we first study each case separately. We analyze the existence of solutions for the case of Γ periodic functions using the translation function. Given is a convolution operator, it is easy to see that T γ S(t) = S(t)T γ . Using that T γ F (t,u) = F (t,T γ u) we obtain Therefore, T γ u is the solution of (2.1) with initial data T γ u 0 . Proposition 12. If u 0 ∈ C u (R d /Γ,Z), then the solution u of the equation (2.1) verifies u(t) ∈ C u (R d /Γ,Z) for 0 ≤ t < T * (u 0 ).
Proof. Since T γ u 0 = u 0 for any γ ∈ Γ, T γ u,u are solutions with the same initial data. From uniqueness, we have T γ u = u. Therefore, u(t) ∈ C u (R d /Γ,Z).
We now analyze the existence of solutions of functions that tend to zero when the spatial variable tends to infinity.
Suppose that T * (v 0 ) < T * (u 0 ). Let T ∈ (0,T * (u 0 )), M = max 0≤t≤T u(t) . We define T = {t ∈ [0,T ] : u(t) / ∈X Γ,Z }, that is, the times for which we have that u(t) is not a direct sum. Suppose that T = ∅. Then there exists t 1 = inf T . We analize if the infimum can be equal to zero or greater than zero.
The case in which t 1 = 0 is not possible because we have already seen that u(t) ∈ X Γ,Z , in the interval [0,T * (v 0 )). In the same way, if t 1 > 0 and additionally t 1 < T * (v 0 ) we have that u(t) ∈ X Γ,Z . We analize the remaining case, t 1 > 0 and T > t 1 > T * (v 0 ).
We observe that, by theorem 6 we obtain that lim t→T * (v0) v(t) = +∞ but on the other hand, by lemma 10 we have that v(t) ≤ P u(t) ≤ P M that is, the norm v(t) is bounded for t ∈ [0,T * (v 0 )) ⊂ [0,T ], which is a contradiction.