DISCONTINUOUS SOLUTIONS FOR THE GENERALIZED SHORT PULSE EQUATION

. The generalized short pulse equation is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. This is a nonlinear evolution equation. In this paper, we prove the wellposedness of the Cauchy problem associated with this equation within a class of discontinuous solutions.


1.
Introduction. In past years, intense ultrashort light pulses comprising merely a few-optical cycles became routinely available; for a review of the various techniques of their production and measurement as well as relevant theoretical methods used to model their unique features (see [7]). These intense ultrashort optical pulses have various applications in the field of light-matter interactions, high-order harmonic generation, extreme nonlinear optics (see [52]), and attosecond physics (see [43]). Several theoretical approaches have been considered thus far to describe the physics of few-cycle-pulse optical solitons; chiefly we have three classes of governing models: the first one is the full quantum approach (see [36,39,40,46]), the second one is the refinements of envelope nonlinear Schrödinger type equations, in the framework of the slowly-varying envelope approximation (SVEA) (see [6,49,51]) and the third one is the non-SVEA models (see [3,4,27]).
A non-SVEA model is represented by the following equation (see [2,4,5,29,30,31]): known as the modified Korteweg-de Vries equation [16,22,26,42,47]. In [26], the Cauchy problem for (1) is studied, while, in [16,42], the convergence of the solution 738 GIUSEPPE MARIA COCLITE AND LORENZO DI RUVO of (1) to the unique entropy solution of the following scalar conservation law is proven. One additional non-SVEA model is (see [1,4,27,38,41,44,45,50]): It was introduced by Kozlov and Sazonov [27] as a model equation describing the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, and by Schäfer and and Wayne [41] as a model equation describing the propagation of ultra-short light pulses in silica optical fibers. An interpretation of (3) in the context of Maxwell equations is given in [38]. Recently, wellposedness results for the Cauchy problem of (3) are proven in the context of energy spaces (see [24,37,45]). A similar result is proven in [14,19] in the context of the entropy solution, while, in [15,20], the wellposedness of the homogeneous initial boundary value problem is studied. Finally, the convergence of a finite difference scheme is studied in [9].
The papers [29,30,32] show that the following evolution equation known as the generalized short pulse equation, is the most general of all approximate non-SVEA models for FCPs, and in fact contains all of them. Observe that, if d = 0, (4) reads It was derived by Costanzino, Manukian and Jones [23] in the context of the nonlinear Maxwell equations with high-frequency dispersion. Kozlov and Sazonov [27] show that (5) is an more general equation than (3) to describe the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media. Mathematical properties of (5) are studied in many different contexts, including the local and global wellposedness in energy spaces [23,37] and stability of solitary waves [23,34].
In this paper, we assume d = 0, and investigate the wellposedness of classes of discontinuous functions for (4) with b = 0, that is Integrating (6) in x, we gain the integro-differential formulation of (6) that is equivalent to The presence of the cubic term in the first equation of (8) makes the analysis of such equation more subtle than the one of (3). We are interested in the Cauchy problem for this equation, thus we augment with the initial condition The assumption, which we make on (9), depends on the conserved quantities of (8).
In fact, one of the main issues during the analysis of (8) is that the equation does not preserve the L 1 norm, while, in in general, the unique conserved quantity is (see We consider only the case being the one c/d < 0 similar. On the initial condition u 0 we assume that Following [12,14,19], on the function we assume that in order to get the boundedness of u.
The following Cauchy problem can be rewritten as follows Due to the regularizing effect of the P equation in (16), we have that Following [12,14], we give the following definition of solution.
Definition 1.1. We say that u ∈ L ∞ ((0, T ) × R) is an entropy solution of the initial value problem (15) if i) u is a distributional solution of (16); ii) for every convex function η ∈ C 2 (R) the entropy inequality holds in the sense of distributions in (0, ∞) × R.
The main result of this section is the following theorem.
Theorem 1.2. Assume (11), (12) and (14). The initial value problem (15) possesses an unique entropy solution u in the sense of Definition 1.1. Moreover, if u and v are two entropy solutions of (6) in the sense of Definition 1.1, the following inequality holds for almost every 0 < t < T , R > 0, and some suitable constant C(T ) > 0.
This paper is organized into two sections. In Section 2, we prove Theorem 1.2.
2. Proof of Theorem 1.2. Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (16). Fix a small number ε > 0 and let u ε = u ε (t, x) be the unique classical solution of the following mixed problem [17,21]: where u ε,0 is a C ∞ c approximation of u 0 such that Clearly, (20) is equivalent to the integro-differential problem Let us prove some a priori estimates on u ε and P ε , denoting with C 0 the constants which depend only on the initial data, and C(T ) the constants which depend also on T .
that is In particular, we have that Proof. We begin by proving that By (20), we have that Consequently, and due to (11), we gain (26).
In a similar way, we can prove that We continue by proving (24).
Lemma 2.4. Assume (11). Let T > 0. There exists a constant C(T ) > 0, independent on ε, such that In particular, for every 0 ≤ t ≤ T , we have Proof. Let 0 ≤ t ≤ T . Multiplying the first equation of (20) by 4u 3 ε , an integration on R gives Hence, we have that Thanks to (32), we can consider the following function Integrating the second equation of (20), by (23), we get Differentiating (46) with respect to t, we obtain that Integrating the first equation of (20) on (−∞, x), from (45) and (47), we have Multiplying (48) by cP ε + dP 3 ε , an integration on R give
Proof. Due to (20) and (42), solves the equation the comparison principle for parabolic equations implies that In a similar way we can prove that Therefore, that is (67).