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Well-Posedness Results for the Continuum Spectrum Pulse Equation

The continuum spectrum pulse equation is a third order nonlocal nonlinear evolutive equation related to the dynamics of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides. In this paper, the well-posedness of the classical solutions to the Cauchy problem associated with this equation is proven.

The constants a, b, g, q, α, κ, β, γ, in (1), take into account the frequency dispersion of the effective linear refractive index and the nonlinear polarization response, the excitation efficiency of the vibrations, the frequency and the decay time (see [7,8,14]).
Moreover, (1) generalizes the following system: whose the well-posedness is studied in [15]. From a mathematical point of view, the presence of the term 3gu 2 ∂ x u − a∂ 3 x u in the first equation of (1) makes the analysis of such system more subtle than the one of (6). Observe that, taking b = α = β = 0, (1) becomes the modified Korteweg-de Vries equation (see [16][17][18][19][20]) In [8,9,[21][22][23][24], it is proven that (7) is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. In [6,18], the Cauchy problem for (7) is studied, while, in [16,19], the convergence of the solution of (7) to the unique entropy solution of the following scalar conservation law is proven. On the other hand, taking α = β = 0 in (1), we have the following equations that were deduced by Kozlov and Sazonov [12] for the description of the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media and by Schäfer and Wayne [25] for the description of the propagation of ultra-short light pulses in silica optical fibers. Moreover, (9) is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons (see [22][23][24][26][27][28]), a particular Rabelo equation which describes pseudospherical surfaces (see [29][30][31][32]), and a model for the descriptions of the short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response (see [33]).
Finally, (9) was deduced in [34] in the context of plasma physic and that similar equations describe the dynamics of radiating gases [35,36], in [37][38][39][40] in the context of ultrafast pulse propagation in a mode-locked laser cavity in the few-femtosecond pulse regime and in [41] in the context of Maxwell equations.
Observe that, taking α = β = 0, (1) reads It was derived by Costanzino, Manukian and Jones [53] in the context of the nonlinear Maxwell equations with high-frequency dispersion. Kozlov and Sazonov [12] show that (10) is an more general equation than (9) to describe the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media.
Mathematical properties of (10) are studied in many different contexts, including the local and global well-posedness in energy spaces [43,53] and stability of solitary waves [53,54], while, in [6], the well-posedness of the classical solutions is proven.
The main result of this paper is the following theorem.
The proof of Theorem 1 is based on the Aubin-Lions Lemma (see [58][59][60]). The paper is organized as follows. In Section 2, we prove several a priori estimates on a vanishing viscosity approximation of (1). Those play a key role in the proof of our main result, that is given in Section 3. Appendix A is an appendix, where we prove the posedness of the classical solutions of (1), under the assumption

Vanishing Viscosity Approximation
Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (1).
Fix a small number 0 < ε < 1 and let u ε = u ε (t, x) be the unique classical solution of the following mixed problem [19,61,62]: where u ε,0 is a C ∞ approximation of u 0 such that Let us prove some a priori estimates on u ε , P ε and v ε . We denote with C 0 the constants which depend only on the initial data, and with C(T), the constants which depend also on T.
In particular, we have that Proof. We begin by proving (18). Thanks to the regularity of u ε and the first equation of (16), we have that which gives (18). Finally, we prove (19). Integrating the second equation of (16) on (−∞, x), by (16), we have that Equation (19) follows from (18) and (20).
There exists a constant C 0 > 0, independent on ε, such that Proof. Let 0 ≤ t ≤ T. Thanks to the third equation of (16), we have that Therefore, by (24), which gives (26).
In particular, we have for every 0 ≤ t ≤ T. Moreover, for every 0 ≤ t ≤ T.
for every 0 ≤ t ≤ T.
In particular, we have that for every 0 ≤ t ≤ T.
Proof. Let 0 ≤ t ≤ T. Consider two real constants D, E which will be specified later Multiplying the first equation of (16) by we have that Observe that Consequently, an integration on R of (58) gives Thanks to the second equation of (16) and (18), we have that Therefore, by (59), Observe that Consequently, by (60), We search D, E such that Since D = E − 10 is the unique solution of (62), it follows from (61) that Due to (41), (42), (43), (55), Lemma 3 and the Young inequality, . Therefore, defining by (63) and (64), we have .

Proof of Theorem 1
This section is devoted to the proof of Theorem 1. We begin by proving the following lemma.
We are ready for the proof of Theorem 1.

Conclusions
In this paper we studied the Cauchy problem for the Spectrum Pulse equation. It is a third order nonlocal nonlinear evolutive equation related to the dynamics of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides. Our existence analysis is based on on passing to the limit in a fourth order perturbation of the equation. If the initial datum belongs to H 2 (R) ∩ L 1 (R) and has zero mean we use the Aubin-Lions Lemma while if it belongs to H 3 (R) ∩ L 1 (R) and has zero mean we use the Sobolev Immersion Theorem. Finally, we directly prove a stability estimate that implies the uniqueness of the solution.

Conflicts of Interest:
The authors declare no conflict of interest.
In this appendix, we consider the Cauchy problem (1), where, on the initial datum, we assume while on the function P(x), defined in (3), we assume (4). Moreover, we assume (5). The main result of this appendix is the following theorem.
To prove Theorem A1, we consider the approximation (16), where u ε,0 is a C ∞ approximation of u 0 such that where C 0 is a positive constant, independent on ε.
It follows from (A10) that .
Using the Sobolev Immersion Theorem, we begin by proving the following result.