Discontinuous solutions for the short-pulse master mode-locking equation

1.   Introduction

In recent years, the ultrafast science has experimentally crossed the threshold from the routine and direct laser production of few-cycle optical pulses in the $5-8$ fs regime ^[29,36,37], to subfemtosecond pulses in the range of hundreds of attoseconds ^[35,54], leading to the hitherto unexplored field of attosecond physics ^[24,39,58].

Such short pulse durations and broad spectral content have opened up vast new possibilities for exploring the fundamental nature of atomic and molecular physics at the fastest time scales, including molecular vibrations, chemical reactions, and light-matter interactions. Indeed, even single-electron transition events can now be captured ^[35] and an absolute measure of time potentially established ^[23,25].

Yet the achievement of attosecond pulses in experiments also highlights the difficulties in measurement and interpretation, especially as the optical field information is interwoven with the physics of the interaction. Thus, theoretical and experimental methods are needed to help guide the understanding of attosecond science.

A first method is based on the master mode-locking (MML) equation. It was developed by Haus ^[33,34] and has dominated mode-locking theory for many years.

The Haus theory is fundamentally based upon a center-frequency expansion of the electric field and a derivation of an envelope approximation of the nonlinear Schrödinger equation (NLS) type. Inherent in the model is the assumption that the envelope is slow in comparison to the underlying fast carrier.

In the case of the pulses which contain only a few cycles of the carrier, this approximation fails to hold, even if higher-order terms are incorporated into the NLS-based description. Regardless, the NLS-based approach has been shown to work quantitatively beyond its expected breakdown, into the tens of femtoseconds regime, and has been used extensively for modeling supercontinuum generation ^[28].

However, when pushed to the extreme of a few femtosecond pulses, the NLS description becomes suspect, and a theory not founded upon a center-frequency expansion is required.

As an alternative to the Haus model, Farnum and Kutz ^[30,31,32] develop a model to describe the pulses like those described in ^{[29,35,36,37,54]} without relying on center-frequency expansion.

The Farnum and Kutz model is based on a short-pulse master mode-locking (SPMML) theory, which derives from the Maxwell equations ^{[1,2,4,5,38,50,51,55,59]}. In such theories, it is assumed that the propagation occurs for a broadband pulse so that the center-frequency expansion is circumvented.

From a mathematical point of view, the Farnum and Kutz model is formulated in terms of the following equation:

also known as SPMML equation, where $u(t, x)$ is the electric field amplitude, $t$ is the time, $x$ is distance propagated in the laser cavity ^[55], $b$ is the linear attenuation, $c$ is a cubic gain term giving rise to intensity discrimination (saturable absorption) ^[33,34] and $\gamma$ is a real parameter ^[38,55].

If $b = c = 0$, (1.1) reads

It was introduced by Kozlov and Sazonov ^[38] 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 ^[55] as a model equation describing the propagation of ultra-short light pulses in silica optical fibers. Hence, with respect to Kozlov, Sazonov, Schäfer and Wayne, Farnum and Kutz take into consideration the linear attenuation and the saturable absorption. Moreover, in ^[42,43,44] the authors show that (1.2) is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons.

If we take $c = 0$ in (1.1), we have the short pulse equation with linear attenuation, which is deduced in ^[63].

It also is interesting to remind that Eq (1.2) was proposed earlier in ^[48] in the context of plasma physic and that the similar equations describe the dynamics of radiating gases ^[41,57]. In ^[62], the authors deduce (1.2) to describe the short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. Moreover, ^[3,13,52,53] show that (1.2) is a particular Rabelo equation which describes pseudospherical surfaces.

In literature, we have the following generalization of (1.2):

It was derived by Costanzino, Manukian and Jones ^[22] in the context of the nonlinear Maxwell equations with high-frequency dispersion. Kozlov and Sazonov ^[38] show that (1.3) is an more general equation than (1.2) to describe the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media.

Recently, wellposedness results for the Cauchy problem of (1.2) are proven in the context of energy spaces (see ^[26,49,60]). A similar result is proven in ^[11,16] in the context of the entropy solution, while, in ^[7,12,18], the wellposedness of the homogeneous initial boundary value problem is studied. Finally, the convergence of a finite difference scheme is studied in ^[21].

Moreover, mathematical properties of (1.3) are studied in many different contexts, including the local and global wellposedness in energy spaces ^[22,49] and stability of solitary waves ^[22,46]. Observe that, letting $\beta\to0$ in (1.3), we obtain (1.2). Hence, following ^[9,45,56], in ^[15,16], the convergence of the solution of (1.3) to the unique entropy solution of (1.2) is proven.

In this papaer, we assume that $c > 0$ and we write $c = \kappa^2$. Therefore, (1.1) reads

We are interested in the Cauchy problem for this equation, thus we augment (1.4) with the initial condition

on which assume that

Following ^[10,11,16], on the function

we assume that

Integrating (1.4) in $(0, x)$, we gain the integro-differential formulation of (1.4) and (1.6).

that is equivalent to

One of the main issues in the analysis of (1.10) is that the equation is not preserving the $L^1$ norm, as a consequence the nonlocal source term $P$ and the solution $u$ are a priori only locally bounded. Indeed, from (1.9) and (1.10) is clear that we cannot have any $L^\infty$ bound without an $L^1$ bound. Since we are interested in the bounded solutions of (1.4), some assumptions on the decay at infinity of the initial condition $u_0$ are needed (see (1.8)).

The unique useful conserved quantities are

In the sense that if $u(t, \cdot)$ has zero mean at time $t = 0$, then it will have zero mean at any time $t > 0$. In addition, the $L^2$ norm of $u(t, \cdot)$ is constant with respect to $t$. Therefore, we require that initial condition $u_0$ belongs to $L^2\cap L^\infty$ and has zero mean.

Due to the regularizing effect of the $P$ equation in (1.10), we have that

Following ^[10,11,16], we give the following definition of solution.

Definition 1. We say that $u\in L^{\infty}((0, T)\times \mathbb{R})$ is an entropy solution of the initial value problem (1.10) and (1.5) if

$i$) $u$ is a distributional solution of (1.10);

$ii$) for every convex function $\eta\in C^2(\mathbb{R})$ the entropy inequality

holds in the sense of distributions in $(0, \infty)\times \mathbb{R}$.

The main result of this paper is the following theorem.

Theorem 1.1. Assume (1.6) and (1.8). The initial value problem (1.10) and (1.5) possesses an unique entropy solution $u$ in the sense of Definition1. Moreover, if $u$ and $v$ are two entropy solutions (1.10) and (1.5) in the sense of Definition 1, the following inequality holds

for almost every $0 < t < T$, $R > 0$, and some suitable constant $C(T) > 0$.

The paper is organized as follows. In Section 2, we prove several a priori estimates on a vanishing viscosity approximation of (1.10). Those play a key role in the proof of our main result, that is given in Section 3.

2.   Vanishing viscosity approximation

Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (1.10).

Fix $\varepsilon > 0$ a small number and let $u_\varepsilon = u_\varepsilon(t, x)$ be the unique classical solution of the following mixed problem ^[14,20]:

where $u_{ \varepsilon, 0}$ is a $C^{\infty}$ approximation of $u_0$ such that

where $C_0$ is a constant independent on $\varepsilon$.

Let us prove some a priori estimates on $u_\varepsilon$ and $P_\varepsilon$, denoting with $C_0$ the constants which depend only on the initial data, and $C(T)$ the constants which depend also on $T$.

Arguing as in [8,Lemma $2.1$], or [10,Lemma $2.1$], we have the following result.

Lemma 2.1. Let us suppose that

Then, the following statements are equivalent

Lemma 2.2. For each $t\in[0, \infty)$, (2.3) and (2.4) hold. In particular, fixed $T\ge t\ge 0$, we have that

Proof. Arguing as in [8,Lemma $2.2$], we have (2.3), and (2.4). Lemma 2.1 says that (2.5) also hold. Therefore, we get

Fixed $T > 0$, the Gronwall Lemma and (2.2) give

that is (2.6).

Lemma 2.3. Fix $T > 0$. There exists a constant $C(T) > 0$, independent on $\varepsilon$, such that

Proof. Let $0\le t\le T$. Multiplying (2.1) by $4 u_\varepsilon^3$, an integration on $\mathbb{R}$ gives

Hence,

Due to (2.6) and the Young inequality,

Consequently, by (2.8),

It follows from (2.2), (2.5) and an integration on $(0, t)$ that

which give (2.7).

Lemma 2.4. Fix $T > 0$. For each $t\in[0, T]$, we have that

Moreover,

Proof. We begin by observing that, integrating the second equation of (2.1) on $(0, x)$, we have

Consequently, by (2.3),

Differentiating (2.13) with respect to $t$, we obtain

Integrating the first equation on $(0, x)$, we have that

It follows from (2.3) and the regularity of $u_\varepsilon$ that

Consequently, by (2.3), (2.14), (2.15) and (2.16),

that is

Due to (2.6), (2.7) and the Hölder inequality,

(2.9) follows from (2.17) and (2.18).

We prove (2.10). By (2.3) and (2.12), we have

Consequently, differentiating (2.19) with respect to $t$, we obtain

(2.3) and the the regularity of $u_\varepsilon$ give

Therefore, by (2.15), (2.20) and (2.21),

Due to (2.6), (2.7) and the Hölder inequality,

(2.22) and (2.23) give (2.10).

Finally, we prove (2.11). Thanks to (2.9) and (2.10), we can consider (2.17) and (2.22) which give (2.11).

Following [15,Lemma $2.4$], we prove the following result.

Lemma 2.5. Fix $T > 0$. There exists a constant $C(T) > 0$, independent on $\varepsilon$, such that

In particular, for every $0\le t\le T$, we have

Proof. Let $0\le t \le T$. We begin by observing that, thanks to (2.9), we can consider the following function

Integrating the second equation of (2.1), by (2.3), we get

Differentiating (2.27) with respect to $t$, we obtain that

Integrating the first equation of (2.1) on $(-\infty, x)$, from (2.26), (2.27) and (2.28), we have

Multiplying (2.29) by $2 P_\varepsilon$, an integration on $\mathbb{R}$ give

Observe that, by (2.11) and (2.26),

Again by (2.11) and (2.26),

Moreover, by (2.1) and (2.3),

Therefore, it follows from (2.30), (2.31), (2.32) and (2.33) that

By (2.29), we have

Multiplying (2.35) by $2\kappa^2 u_\varepsilon^3$, an integration on $\mathbb{R}$ gives

Since

we have that

Using (2.36) in (2.34), we get

Observe that

Consequently,

Using (2.38) in (2.37), we have

where

Multiplying the first equation of (2.1) by $- \frac{6\kappa^2}{\gamma} P_\varepsilon u_\varepsilon^2$, an integration on $\mathbb{R}$ give

Observe that by (2.1) and (2.3),

Hence, by (2.41),

Substituting (2.42) in (2.39), we get

Due to (2.6), (2.7) and the Young inequality and the Hölder inequality,

where $D, \, E$ are two positive constants which will be specified later. Consequently, by (2.43),

Observe that by (2.40),

Thanks to (2.6), (2.7) and the Young inequality,

It follows from (2.44), (2.45) and (2.46) that

The Gronwall Lemma, (2.2), (2.6), (2.7), (2.40), (2.46) and the Young inequality give

We prove (2.9). Thanks to (2.1), (2.3), (2.6) and the Hölder inequality,

Therefore, by (2.47),

Hence,

Choosing

we have

which gives (2.24).

Finally (2.25) follows from (2.7), (2.24), (2.47) and (2.48).

Following [10,Lemma $3.1$], or [19,Lemma $3.1$], we prove the following result.

Lemma 2.6. Let $T > 0$. There exists a constant $C(T) > 0$, independent on $\varepsilon$, such that

for every $0\le t\le T$.

Proof. Let $0\le t\le T$. We begin by observing that, by (2.9), we have that

Therefore,

The proof of (2.49) splits into two parts. In the first part, we consider $b\ge0$. Instead, in the second one, we consider $b\le0$.

Case $b\ge0$. We assume that

Therefore, by the first equation of (2.1), (2.51) and (2.52), we have

A supersolution of (2.1) satisfies the following ordinary differential equation:

We consider the map

where $A$ is a positive constant, which will be specified later. Observe that

Choosing

we have that

for every $t\in(0, T)$. Then, $Z_1(t)$ is a supersolution of (2.54). (2.56), the comparison principle for parabolic equations and the comparison principle for ordinary differential equations yield

Observe that, by the first equation of (2.1), (2.51) and (2.52), we have

Therefore, a subsolution of (2.1) satisfies the following ordinary differential equation:

We consider the map

where $B$ is a positive constant, which will be specified later. Observe that

Choosing

we have that

for every $t\in(0, T)$. Then, $Z_2(t)$ is a subsolution of (2.61). (2.52), the comparison principle for parabolic equations and the comparison principle for ordinary differential equations yield

It follows from (2.58) and (2.64) that

which give (2.49).

Case $b\le0$. We assume that

Thanks to (2.66), arguing as in previous case, we get

A supersolution of (2.1) satisfies the following ordinary differential equation:

We consider the map

where $D, \, E$ are two positive constants, which will be specified later. Observe that

We search $D, \, E$ such that,

From the first inequality of (2.71), we obtain that

Choosing

it follows from the second inequality of (2.71) that

that is

From (2.72) and (2.74), we get

Choosing

from (2.69) and (2.73), we have that

Moreover, by (2.70), (2.73) and (2.76),

for every $0\le t\le T$. Then, $Z_3(t)$ is a supersolution of (2.68). (2.77), the comparison principle for parabolic equations and the comparison principle for ordinary differential equations yield

Arguing as in previous case, we have that

Therefore, a subsolution of (2.1) satisfies the following ordinary differential equation:

We consider the map

where $F, \, G$ are two positive constants, which will be specified later. Observe that

We search $F, \, G$ such that

Choosing

by (2.83), we have

Arguing as before, we gain

Choosing

then, by (2.81) and (2.84),

Moreover, by (2.82), (2.84) and (2.86), we have

for every $0\le t\le T$. Then, $Z_4(t)$ is a subsolution of (2.80). (2.87), the comparison principle for parabolic equations and the comparison principle for ordinary differential equations yield

It follows from (2.78) and (2.88) that

Hence,

which gives (2.49).

3.   Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1.

Let us begin by proving the existence of a distributional solution to (1.10) satisfying (1.13).

Lemma 3.1. Let $T > 0$. There exists a function $u\in L^{\infty}((0, T)\times \mathbb{R})$ that is a distributional solution of (2.1) and satisfies (1.13) for every convex entropy $\eta\in C^2(\mathbb{R})$.

We construct a solution by passing to the limit in a sequence $\left\{{u_{ \varepsilon}}\right\}_{ \varepsilon > 0}$ of viscosity approximations (2.1). We use the compensated compactness method ^[61].

Lemma 3.2. Let $T > 0$. There exists a subsequence $\{ u_{\varepsilon_k}\}_{k\in \mathbb{N}}$ of $\{ u_\varepsilon\}_{ \varepsilon > 0}$ and a limit function $u\in L^{\infty}((0, T)\times \mathbb{R})$ such that

Moreover, we have that

where

Proof. Let $\eta: \mathbb{R}\to \mathbb{R}$ be any convex $C^2$ entropy function, and $q: \mathbb{R}\to \mathbb{R}$ be the corresponding entropy flux defined by $q'(u) = 3au^2\eta'(u)$. By multiplying the first equation in (2.1) with $\eta'(u_\varepsilon)$ and using the chain rule, we get

where $\mathcal{L}_{1, \varepsilon}$, $\mathcal{L}_{2, \varepsilon}$, $\mathcal{L}_{3, \varepsilon}$, $\mathcal{L}_{4, \varepsilon}$ and $\mathcal{L}_{5, \varepsilon}$ are distributions. Let us show that

Since

by (2.6) and Lemma 2.6,

We claim that

Again by (2.6) and Lemma 2.6,

We have that

Let $K$ be a compact subset of $(0, T)\times \mathbb{R}$. Using (2.24) and Lemma 2.6,

We show

Let $K$ be a compact subset of $(0, T)\times \mathbb{R}$. By Lemma 2.6,

We claim that

Let $K$ be a compact subset of $(0, T)\times \mathbb{R}$. Again by Lemma 2.6,

Therefore, Murat's lemma ^[47] implies that

The $L^{\infty}$ bound stated in Lemma 2.6, (3.4) and the Tartar's compensated compactness method ^[61] give the existence of a subsequence $\{ u_{\varepsilon_k}\}_{k\in \mathbb{N}}$ and a limit function $u\in L^{\infty}((0, T)\times \mathbb{R}), \, T > 0,$ such that (3.1) holds.

Finally, (3.2) follows from (3.1), the Hölder inequality and the identity

Now, we prove Theorem 1.1.

Proof of Theorem 1.1. Lemma 3.2 gives the existence of an entropy solution $u$ for (1.9), or equivalently (1.10).

We prove that $u(t, x)$ is unique and (1.14) holds. Fix $T > 0$. Let $u(t, x)$ and $v(t, x)$ be two entropy solution of (1.9), or (1.10) such that

Consequently, by (3.5), we have that

where

We define

Thanks to (3.6), following ^[6,17,27,40], we can prove that

holds in sense of distributions in $(0, \infty)\times \mathbb{R}$, and

for $0 < t < T,$ where

Observe that

Instead, thanks to (3.6),

Since

due to (3.8),

Considered the following function,

It follows from (3.9), (3.11), (3.12) and (3.14) that

The Gronwall inequality and (3.15) give

that is (1.14).

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Conflict of interest

The authors declare no conflict of interest.