We consider the Lugiato-Lefever model of optical fibers in the periodic context. Spectrally stable periodic steady states were constructed recently in the studies of Delcey and Haragus [Philos. Trans. R. Soc., A **376**, 20170188 (2018)]; [Rev. Roumaine Math. Pures Appl. (to be published)]; and Hakkaev *et al.* (e-print arXiv:1806.04821). The spectrum of the linearization around such solitons consists of simple eigenvalues 0, −2*α* < 0, while the rest of it is a subset of the vertical line $ { \mu : R \mu = \u2212 \alpha } $. Assuming such a property abstractly, we show that the linearized operator generates a *C*_{0} semigroup and, more importantly, the semigroup obeys (optimal) exponential decay estimates. Our approach is based on the Gearhart-Prüss theorem, where the required resolvent estimates may be of independent interest. These results are applied to the proof of asymptotic stability with phase of the steady states.

## I. INTRODUCTION

The Lugiato-Lefever model has been object of intense investigations in the last decade, as it arises as a relevant model in optical fibers. More specifically, it is an envelope model which was derived from Maxwell’s equation,^{10} to describe the mechanism of pattern formation in the optical field of a cavity filled with a Kerr medium, which is then subjected to a radiation field. Related to this, high frequency optical combs generated by micro-resonators have been considered in Refs. 1, 9, 10, 13, 11, and 16, among others. There are also deep experimental studies,^{5} which confirm the relevance of this and related models.

There are numerous papers dealing with the model derivation, as well as further reductions to dimensionless variables.^{2,10,11} The model equation, considered in Refs. 3, 4, 14, and 15, is given as follows:

In this paper, we follow slightly different but equivalent formulation; see, for example, the derivation in Refs. 13 and 12. This is in the context of the whispering gallery mode resonators. More precisely, we will be studying

where *u* is the field envelope (complex-valued) function, *t* is the normalized time, and *x* is the azimuthal coordinate, while *α* > 0 is the detuning/damping parameter and the normalized pumping strength parameter is *h* > 0. The main interest will be in stationary/time independent solutions and their stability properties. These are of great interest to the practitioners, especially the stable ones and they are often referred to as frequency combs. We remark that mathematically, this model could be done in the periodic (and also in the whole line context), with a definite preference to the periodic case, as the dependence of the period *T* is an important aspect for the application.

To set the ideas, we are interested in solutions in the form *u*(*t*, *x*) = *φ*(*x*) and their stability. They satisfy the following time-independent equation:

In several recent studies, several different types of such solutions were shown to exist. Many of them turn out to be spectrally unstable, but some of them are stable, which makes them interesting from the point of view of their practical applications. Our results herein address the dynamics close to these spectrally stable solutions.

The main result of this article is about the asymptotic stability of such solutions. In order to state the main result, let us state its linearization around the solution *φ*. Set *u* = *φ* + $v$, *φ* = *φ*_{1} + *iφ*_{2}, $v$ = $ v 1 $ + $iv$_{2}. After ignoring *O*(|$v$|^{2}) terms, we obtain the following system:

In the eigenvalue ansatz, $ v 1 ( t , x ) v 2 ( t , x ) \u2192 e \lambda t v 1 ( x ) v 2 ( x ) $, and after rearranging terms, we obtain the eigenvalue problem

where

Introduce the linearized operator $H\u2009\u2254JL\u2212\alpha $. In the recent papers,^{3,4,7} spectrally stable solutions of (1.2) were constructed.^{15,16} The common feature of these smooth solutions *φ* was that the linearized operator $H$ has its spectrum to the left of the imaginary axes. In fact, $\sigma ( H \u2009\u2009 ) \u2229iR= { 0 } $, 0 is a simple eigenvalue and the rest of the spectrum is inside $ { \mu : R \mu \u2264 \u2212 \alpha } $. We now take this as an abstract assumption on *φ*, of which the solitons exhibited in Refs. 3, 4, and 7 are an example of; see (1.6) for a detailed description. We aim at showing that such $H$ generate a semigroup with exponential decay and consequently, the solitons are asymptotically stable.

The following is the main result of this paper.

*Let φ ∈ H*

^{1}

*[−T, T] be a solution of*(1.3)

*. Assume that the linearized operator*$H$

*satisfies*$H\u2009\u2009 \phi 1 \u2032 \phi 2 \u2032 =0$

*and*

*where 0, −2α are simple eigenvalues. Then,*$H$

*generates a C*

_{0}

*semigroup. In addition, when projected away from the zero eigenvalue,*$ e t H Q 0 $

*has exponential decay; see*

*(3.15)*

*.*

*Finally, φ it is an asymptotically stable soliton. More precisely, for every β: 0 < β < α, there is ϵ = ϵ*

_{β}

*and a constant C = C*

_{β}

*so that whenever*$ \Vert u 0 \u2212 \phi \Vert H 1 <\u03f5$

*, then the solution u exists globally and it satisfies*

*where |σ′(t)| ≤ ϵ*

^{3/2}

*e*

^{−βt}

*,*$\Vert v ( t , \u22c5 ) \Vert H 1 \u2264C\u03f5 e \u2212 \beta t $

*. In particular, there is σ*

_{∞}

*≔*lim

_{t→∞}

*σ(t) so that*

*Note that*$| \sigma \u221e |\u2264 \u222b 0 \u221e | \sigma \u2032 ( t ) |dt\u2264 C \beta \u03f5 3 2 .$

*Remark:*

The exponent $ 3 2 $ above is not sharp and it can be replaced by any *γ* < 2. Informally, if the initial data *u*_{0} are order *O*(*ϵ*) away from the soliton *φ*, then the asymptotic phase *σ*_{∞} is roughly order *O*(*ϵ*^{2}) small.

The plan of the paper is as follows. In Sec. II, we start by defining some preliminary notions and definitions and we give some well-known results. Section III contains the main new contribution of the article, namely, the exponential estimates for the linearized Lugiato-Lefever semigroup. These are shown through a powerful abstract tool, namely, the Gearhart-Prüss theorem, which requires a uniform estimate of the correspondent resolvent, close to the imaginary axes. In Sec. IV, we apply the exponential estimates to derive the nonlinear asymptotic stability with phase of the solitons, as a solution to the full Lugiato-Lefever (LL) equation.

## II. PRELIMINARIES

We use the following Fourier series representations for *f* ∈ *L*^{2}[−*π*, *π*]:

so that

### A. Semigroup generation and the Gearhart-Prüss theorem

We use the standard definition of *C*_{0} semigroups on a Banach space *X*, which is that there is a one parameter family of bounded operators $T: 0,\u221e \u2192B ( X ) $ so that *T*(0) = *Id*, *T*(*t* + *s*) = *T*(*t*)*T*(*s*) and for each fixed *φ* ∈ *X*, lim_{t→0+}∥*T*(*t*)*φ* − *φ*∥_{X} = 0. Its generator $A$ is always a closed operator, with a domain $D ( A \u2009 ) $, defined through the condition $ lim t \u2192 0 + T ( t ) x \u2212 x t $ existing in the ∥⋅∥_{X}. Clearly, it is not trivial or automatic to say that any given closed operator generates a *C*_{0} semigroup, but we shall use the following simple sufficient condition, in the case of Hilbert spaces, which is adaptation and consequence of Theorem X. 48, p. 240 and the lemma on p. 244, Ref. 17.

*Lemma 1.*

*Suppose that a closed operator H*_{0} *on a Hilbert space X has the property* $R \u27e8 H 0 x , x \u27e9 \u22640$ *(i.e,. −H*_{0} *is accretive), and in addition, there exists δ > 0 so that δ − H*_{0} *is invertible. Then, H*_{0} *generates a contraction semigroup. If H is another operator so that H − H*_{0} *is a bounded operator, then H generates a C*_{0} *semigroup as well.*

#### 1. $H$ generates a *C*_{0} semigroup

Let us now use this criterion in order to show that $H$ generates a *C*_{0} semigroup on the Hilbert space *L*^{2}[−*T*, *T*]. Without loss of generality, we may assume that *T* = *π*. Next, let $ L 0 $ denote the differential part of the self-adjoint operator $L$. In other words,

For $ H 0 \u2254J L 0 \u2212\alpha $, we have by integration by parts

since *α* > 0. Also, using the Fourier variables and the calculation,

we can invert $1\u2212 H 0 $ as follows:

which is a bounded operator, basically because the matrix in (2.1) is bounded in *k*. It follows that $ H 0 $ generates a contraction semigroup. Clearly $H\u2212 H 0 =J \u2212 ( 6 \phi 1 2 + 2 \phi 2 2 ) \u2003 \u2212 4 \phi 1 \phi 2 \u2212 4 \phi 1 \phi 2 \u2003 \u2212 ( 2 \phi 1 2 + 6 \phi 2 2 ) ,$is a bounded operator on *L*^{2} × *L*^{2} since $ \phi \u2192 \u2208 H 1 \u2282 L \u221e $. Thus, Lemma 1 applies and we have that $H$ generates a *C*_{0} semigroup.

Our next task will be to show that $ e t H $ has exponential decay bounds, at least when projected away from its zero eigenvalue. This will require much more effort, but the main tool is by now the classical result, namely, the Gearhart-Prüss theorem.

#### 2. Gearhart-Prüss theorem

*Let*$A$

*generate a C*

_{0}

*semigroup on a Hilbert space so that*$\sigma ( A ) \u2282 { \lambda : R \lambda < 0 } $

*. If*

*then there exists δ > 0 so that*$\Vert e t A f \Vert H \u2264 C \delta e \u2212 \delta t \Vert f \Vert H $

*.*

As we have mentioned above, $H$ does not exactly fit the Gearhart-Prüss statement, as it has spectrum on the imaginary axes, namely, a simple eigenvalue at zero. This is a frequent occurrence in dynamical system theory, as symmetries of the system manifest themselves each as eigenvalues at zero. Luckily, most of this spectrum is finite dimensional, so one could still apply Theorem 2 by projecting away on a finite co-dimension subspace. In order to do that, we need to look into the relevant object that lets us do that, namely, the Riesz projections.

### B. Riesz projections

In this section, we identify the concrete form of the Riesz projection onto the eigenspace corresponding to the zero eigenvalue. More precisely, we are looking to write explicitly the projection onto $Ker [ H \u2009 ] =span [ \phi 1 \u2032 \phi 2 \u2032 ] $. Define, for $\u03f5< \alpha 2 $,

and $ Q 0 \u2254Id\u2212 P 0 $. Denote the corresponding subspaces $ P 0 = P 0 [ L 2 \xd7 L 2 ] , Q 0 = P 0 [ L 2 \xd7 L 2 ] $.

The operator $ P 0 $ is a projection onto one dimensional subspace, so it is a rank-one operator. It is naturally given by

for some vector *e*_{*}.

We now aim to identify *e*_{*}. To that end, recall that by our assumption on $H$, (1.6), $ { \u2212 \alpha , \alpha } \u2208\sigma ( J L \u2009 ) $. Taking adjoints, we see that $ { \u2212 \alpha , \alpha } \u2208\sigma ( L J \u2009 ) $. We will show that properly normalized eigenvector *e*_{*} for the eigenvalue −*α* will do the job. Indeed, assume $ ( L J + \alpha ) e * =0$. For any $\mu \u22600,\mu \u2208\sigma ( H \u2009\u2009 ) $, consider the corresponding eigenfunction, $H\u2009 f \mu =\mu f \mu $. Compute

Thus, *e*_{*} ⊥ *Q*_{0}. On the other hand, *e*_{*} ≠ 0, so it follows that $ \u27e8 \phi 1 \u2032 \phi 2 \u2032 , e * \u27e9 \u22600$, otherwise *e*_{*} will be perpendicular to the whole Hilbert space, and hence *e*_{*} = 0. Thus, we pick a normalization of *e*_{*} so that $ \u27e8 \phi 1 \u2032 \phi 2 \u2032 , e * \u27e9 =1$. This determines *e*_{*} uniquely and this is the formula that we use henceforth. In fact, it is easy to see that $\eta \u2254J e * $ is the unique (when properly normalized) eigenvector of $H\u2009\eta =\u2212\alpha \eta $. So, we have proved the following lemma.

*Lemma 2.*

*The projection* $ P 0 $ *onto the eigenspace spanned by* $ \phi \u2032 \u2192 $ *is given by the formula* $ P 0 \u2009f= \u27e8 \u2009 f , e * \u27e9 \phi \u2192 \u2032 $*, where* $J e * $*is the unique eigenfunction of* $H$ *at −α, normalized with* $ \u27e8 \phi \u2192 \u2032 , e * \u27e9 =1$*.*

Next, we present a lemma, which provides an useful decomposition, if one is close to the soliton, with a modified radiation term, which is kept in the subspace *Q*_{0}.

*Lemma 3.*

*There exists ϵ*

_{0}

*> 0 so that whenever*$ u \u2192 \u2208 H 1 ( R ) \xd7 H 1 ( R ) :\Vert u \u2192 \u2212 \phi 1 \phi 2 \Vert H 1 < \u03f5 0 $

*, then there exists a pair (σ*, $v$)

*∈*

*R**× Q*

_{0}

*so that*$\sigma =\sigma ( u \u2192 ) , v \u2192 =v ( u \u2192 ) $

*,*

This is a well-known result, which is by now classical in the asymptotic stability literature. For a proof, in a slightly more general framework, we refer the reader to Lemma 2.2 in Ref. 8. Henceforth, we also adopt the notation $ \phi \u2192 \sigma ( x ) \u2254 \phi 1 \phi 2 ( x \u2212 \sigma ) $.

## III. EXPONENTIAL DECAY ESTIMATES FOR $ e t H $

We start by the observation that the operator $H$ generates a *C*_{0} semigroup on *L*^{2}[−*T*, *T*], as established in Sec. II A 1. We now show appropriate exponential decay estimates for the semigroup $ e t H $ generated by $H$. We use the Gearhart-Prüss theorem, which calls for uniform resolvent bounds, with respect to the spectral parameter on the imaginary axes. In fact, the difficult part of this estimate is to look for sufficiently large *μ*, while the control on compact intervals is a softer issue, which essentially reduces to knowing that the spectrum does not appear on *i***R**, an assumption in Theorem 1. See the proof of Corollary 1.

We have a more general result, which may be of independent interest. Note that estimates of similar flavor (but for the case of the whole line, with all essential spectrum) have been done in the past; see, for example, Ref. 6 and the Ph.D. thesis of the first author.^{18} Note that the argument here seems more straightforward as it completely avoids the Birman-Schwinger technology (and the associated compactness considerations). Instead, we precisely identify the problematic spectral parameters and we keep the necessary inversions in the Neumann realm.

*Proposition 1.*

*where V*

_{1}, W, V_{2}are real-valued and belong to H^{1}[−T, T]. Note that L is a self-adjoint operator, when taken with the domain H^{2}[−T, T] × H^{2}[−T, T].*Then, for every δ > 0, there exists N*

_{δ}

*so that whenever |μ| > N*

_{δ}

*,*$\delta +i\mu \u2208\rho ( M \u2009 ) $

*. More importantly, we have the estimate*

*Remark:*

*V*

_{1},

*W*,

*V*

_{2}∈

*H*

^{1}[−

*T*,

*T*] may be weakened to the following:

*Proof.*

*T*=

*π*. We need to invert $M\u2212 ( \delta + i \mu ) =J ( L + ( \delta + i \mu ) J ) $, so we need to provide bounds for the inverse of $L+ ( \delta + i \mu ) J$. Without loss of generality

*μ*> 0, since the other case is symmetric to this one. Thus, we set up the equation

*V*

_{1},

*W*,

*V*

_{2}, we see that on the

*k*

^{th}Fourier mode, we have the formula (in a Fourier multiplier form)

*μ*>> 1, 0 <

*δ*<< 1, this Fourier multiplier has lots of decay in

*μ*except when $| ( k 2 + 1 ) 2 + \delta 2 \u2212 \mu 2 |\u2264\mu $. We claim that if

*μ*> 100, there is at most one integer

*k*

_{0}=

*k*

_{0}(

*μ*) so that

*k*≠ ±

*k*

_{0}(

*μ*), we have

Indeed, the inequality $| ( k 2 + 1 ) 2 + \delta 2 \u2212 \mu 2 |\u2264\mu $ leads to |*k*^{2} − *μ*| ≤ 1, which then implies $|k\u2212 \mu |\u2264 1 \mu $. If now *μ* > 100, clearly, this inequality identifies at most one integer *k*_{0}(*μ*), namely, $ k 0 ( \mu ) = \mu $, the closest integer to $ \mu $. Furthermore, if *k* ≠ *k*_{0}(*μ*), by considering the two cases *k* ≥ *k*_{0}(*μ*) + 1 or *k* ≤ *k*_{0}(*μ*) − 1 separately, we can show (3.4). Thus, we have shown the following

*Lemma 4.*

*For every μ*> 100,

*there exists at most one integer k*

_{0}=

*k*

_{0}(

*μ*) > 0

*(which is necessarily*$ k 0 ( \mu ) = \mu $

*) so that*

*and for all k*≠ ±

*k*

_{0}(

*μ*),

*there is*

*(3.4)*

*. Note that if k*

_{0}(

*μ*)

*does not exist, then*

*(3.4)*

*holds for all k*.

*k*

_{0}(

*μ*). We will, with a slight abuse of notations, not distinguish between an

*L*

^{2}[−

*π*,

*π*] function and its (sequence of) Fourier coefficients. In order to introduce convenient notations, let $ z ^ ( k ) \u2254 f ^ ( k ) \u011d ( k ) $, $z= \u2211 k = \u2212 \u221e \u221e z ^ ( k ) e i k x ,$ and

*k*),

*k*≠

*k*

_{0}(

*μ*), we obtain from (3.1)

*k*≠ ±

*k*

_{0}(

*μ*), in the following schematic form:

*k*),

*k*=

*k*

_{0}(

*μ*), we obtain from (3.1)

*V*

_{1},

*V*

_{2},

*W*∈

*H*

^{1}, we have the following decay for its Fourier coefficients $| V ^ 1 ( n ) |+| V ^ 2 ( n ) |+|\u0174 ( n ) |\u2264C ( 1 + | n | ) \u2212 1 $, whence

*k*=

*k*

_{0}(

*μ*), we use the trivial bound for the determinant, coming from its imaginary part, namely,

*O*(1) terms involving $ \u222b \u2212 \pi \pi W ( x ) dx$ in the imaginary parts cancel out. It follows from (3.3) that

*O*(1) represents an

*B*(

*L*

^{2}) operator with norm no bigger than

*Cδ*

^{−1}and $\Vert O ( \mu \u2212 1 / 2 ) \Vert C \u2192 C \u2264C \delta \u2212 1 \mu \u2212 1 / 2 $. A similar calculation shows that the same form can be deduced for the other problematic Fourier mode $ z ^ $(−

*k*

_{0}), namely,

*μ*large enough (depending on various implicit constants involving the potentials

*V*

_{1},

*V*

_{2},

*W*),

*Id*+

*O*(

*μ*

^{−1/2}) will be invertible and therefore

Applying the Proposition 1 to the operator $H$ and taking into account the Gearhart-Prüss result, we obtain

*Corollary 1.*

*Let*$H$

*satisfy the assumption*

*(1.6)*

*. Then, the semigroup*$ e t H $

*has a growth rate*$ \omega 0 ( H \u2009\u2009 ) \u2264\u2212\alpha $

*. More precisely, for every δ > 0, there is C*

_{δ}

*so that*

*Proof.*

*δ*> 0, and for all large enough

*μ*, $ sup | \mu | > N \delta \Vert ( J L \u2212 ( \delta + i \mu ) ) \u2212 1 \Vert B ( L 2 ) <\u221e$. On the other hand, by the assumption (1.6), there is $ sup | \mu | < N \delta \Vert ( J L \u2212 ( \delta + i \mu ) ) \u2212 1 \Vert B ( L 2 ) \u2264 C \delta $. This is because $z\u2192 ( J L \u2212 z ) \u2212 1 $ is an analytic function on $\rho ( H \u2009\u2009 ) $, and as such, it is bounded on any compact subset of $\rho ( H \u2009\u2009 ) $. Thus,

*δ*

_{0}> 0,

*δ*> 0, there is

*C*

_{δ},

We now provide an extension of the exponential decay estimates to the case of Sobolev spaces *H*^{s}[−*T*, *T*].

*Lemma 5.*

*Let $H$ satisfy the assumption*(1.6)

*and s ∈*(0, 2).

*Then, for every δ > 0*,

*there exists C*

_{δ}

*so that*

*Proof.*

*s*= 0 has been handled already; see Corollary 1. The general case follows by an interpolation argument as follows. Recall that $L$ is a self-adjoint operator with a domain $D ( L \u2009 ) = H 2 [ \u2212 T , T ] \xd7 H 2 [ \u2212 T , T ] $. Moreover, it is clear that $L$ is bounded from below. This can be seen by taking

*γ*>> 1 and considering the quadratic form for $L+\gamma $. Estimating in a straightforward manner

*γ*, one may define $ ( L + \gamma ) s 2 $, with domain $D ( ( L + \gamma ) s 2 ) = H s \xd7 H s $ and so that

*a*= 0 is obvious from (3.15) and the fact that the operators $ ( L + \gamma ) i \mu ,\mu \u22600$ are unitary on

*L*

^{2}×

*L*

^{2}by Stone’s theorem.

*a*= 1, we have by Stone’s theorem $\Vert ( L + \gamma ) 1 + i \mu e t H Q 0 \u2009 f \u2192 \Vert L 2 \xd7 L 2 =\Vert ( L + \gamma ) e t H Q 0 \u2009 f \u2192 \Vert L 2 \xd7 L 2 $. Then, we have

## IV. ASYMPTOTIC STABILITY: PROOF OF THEOREM 1

We first set up the perturbation equation. We follow the standard construction of the perturbation ansatz; see, for example, Ref. 8.

### A. The equation for the perturbation

For a solution *u* = *u*_{1} + *iu*_{2} of (1.2), and under the *a priori* assumption for smallness, that is $\Vert u \u2192 ( t ) \u2212 \phi \u2192 \Vert H x 1 <<1$, we use the ansatz guaranteed by Lemma 3,

We obtain the system

where $ L \sigma $ is the self-adjoint operator displayed in (1.5) (with $ \phi \u2192 $ replaced by $ \phi \u2192 \sigma $) and non-linearity given by

Multiplying by $J$, we arrive at

Note that the system (4.2) is still not very good for our purposes, for example, *L*_{σ} is still a time dependent object at this point.

We have reduced matters to

Recall now that Lemma 3 provides us with a decomposition, subordinated to the functional calculus of the operator $H=JL\u2212\alpha $. More precisely, $ v \u2192 $ belongs to the subspace *Q*_{0}, that is, $ Q 0 v \u2192 = v \u2192 $, while $ P 0 v \u2192 =0$. At the same time, $ P 0 \phi \u2032 \u2192 = \phi \u2032 \u2192 $, while $ Q 0 \phi \u2032 \u2192 =0$. Applying the two projections to (4.3), we obtain the system of a coupled Partial Differential Equation (PDE) and ordinary differential equation (ODE),

Here, we have used that $ P 0 H=H P 0 , Q 0 H=H Q 0 $. Our goal is to show exponential time decay and smallness for both $\Vert v \u2192 ( t , \u22c5 ) \Vert H 1 $ and |*σ*′(*t*)| and just smallness for |*σ*(*t*)|. The last would follow from smallness/integrability for |*σ*′(*t*)| since *σ*(0) = 0, as we start close to the soliton.

Let us informally analyze the terms appearing in (4.4). We have put the higher order/small terms on the right-hand side of (4.4). Indeed, $ N j ( v \u2192 ) ,j=1,2$, are quadratic (or higher) in $ v \u2192 $, while the terms $ ( L \sigma \u2212 L ) v \u2192 $ are in the form *O*(*σ*$v$)—that is, they are linear in $v$, but they contain a small factor, proportional to *σ*. Finally, *σ*′(*t*) is multiplied by $ \u27e8 \phi \u2032 \u2192 \sigma , e * \u27e9 $ on the left, but note that this is close to $ \u27e8 \phi \u2032 \u2192 , e * \u27e9 =1$, up to order *O*(*σ*), so $ \u27e8 \phi \u2032 \u2192 \sigma , e * \u27e9 =1+O ( \sigma ) $. This explains the decay/smallness properties of *σ*′(*t*).

For the $ v \u2192 $ equation, note that it is driven by $H$ on the stable subspace *Q*_{0} and as such, the semigroup operator has exponential decay, according to Corollary 1; see also (3.15). The first two terms on the right are similar to the one discussed for *σ*′(*t*), while the last term $ Q 0 [ \phi \u2032 \u2192 \sigma ] $ is close to $ Q 0 [ \phi \u2032 \u2192 ] =0$. This informal analysis convinces us that the nonlinear problem is well behaved. In Sec. IV B, we present the details of the formal proof.

### B. Nonlinear stability of the wave *φ*

We start by noting that the ODE/PDE system is supplied by an initial condition, $ v \u2192 ( 0 ) = v \u2192 0 \u2208 Q 0 \u2282 H 1 $, which is small enough. This corresponds to the fact that we start close to the solitary wave: $\Vert u 0 \u2212 \phi \Vert H 1 <<1$. Also, we set *σ*(0) = 0. Write

Before we set the nonlinear persistence argument, let us derive some estimates on the nonlinear terms. For the nonlinear term, as it consists of quadratic and cubic terms of $ v \u2192 $,

where we have used that *H*^{1} is an algebra. Next,

Next, $ Q 0 [ \phi \u2032 \u2192 \sigma ] = \phi \u2032 \u2192 \sigma \u2212 \u27e8 \phi \u2032 \u2192 \sigma , e * \u27e9 \phi \u2032 \u2192 = \phi \u2032 \u2192 \sigma \u2212 \phi \u2032 \u2192 \u2212 \u27e8 \phi \u2032 \u2192 \sigma \u2212 \phi \u2032 \u2192 , e * \u27e9 \phi \u2032 \u2192 ,$ whence

Finally, recalling $ \u27e8 \phi \u2032 \u2192 , e * \u27e9 =1$, we have $ \u27e8 \phi \u2032 \u2192 \sigma , e * \u27e9 =1+ \u27e8 \phi \u2032 \u2192 \sigma \u2212 \phi \u2032 \u2192 , e * \u27e9 $ and

Note that while we are assured that |*σ*(*s*)| is appropriately small, $ \u27e8 \phi \u2032 \u2192 \sigma , e * \u27e9 =1+O ( | \sigma ( s ) | ) $ and its reciprocal is well defined and in fact $ 1 \u27e8 \phi \u2032 \u2192 \sigma , e * \u27e9 =1+O ( \sigma ( s ) ) $.

In view of these preparatory estimates, we are now ready to set up the non-linear persistence argument as follows. Fix 0 < *β* < *α*. Then, fix *σ* ∈ (*β*, *α*), say $\sigma = \alpha + \beta 2 $. We have then 0 < *β* < *σ* < *α*. Applying the estimate (3.17), we obtain for the free solution of (4.5) the bound

We will show that there exists *ϵ* : 0 < *ϵ* << 1 so that whenever the initial data $ v \u2192 0 $ satisfy $ C 0 \Vert v \u2192 0 \Vert H 1 \u2264 \u03f5 2 $, then the systems (4.5) and (4.6) have a global solution, which in addition satisfies the bounds

It remains to determine whether such an *ϵ* exists. For any initial data, the systems (4.5) and (4.6) certainly have a solution in some time interval, which may be very small, dependent on the data. Denote

Clearly, the problem is to show that for all small enough *ϵ*, *τ*_{*} = *∞*. Also, for all *t* ∈ (0, *t*_{*}), we have $|\sigma ( t ) |\u2264 \u222b 0 t | \sigma \u2032 ( s ) |ds\u2264 \u03f5 3 2 \beta $.

We are now ready for the bootstrapping step. Taking absolute values in (4.6), we obtain

where *Y* = *H*^{1} × *H*^{1}. Using the *a priori* bounds on $ \Vert v \u2192 ( s ) \Vert Y $ and |*σ*′(*s*)| guaranteed to us by *s* ∈ (0, *t*_{*}), we have

for small enough *ϵ*.

As a consequence, we obtain the following inequality for $I ( t ) \u2254 e \sigma t \Vert v \u2192 ( t ) \Vert Y $:

Denoting $M ( t ) \u2254 \u222b 0 t I ( s ) ds$, we have

whence, assuming $\u03f5:\sigma \u2212\beta \u2212C \u03f5 3 / 2 > \sigma \u2212 \beta 2 $ and taking into account *M*(0) = 0, we have

Plugging this back into (4.9) and after some elementary manipulations, we have the bound

for small enough *ϵ*. Thus, the nonlinear stability is proved in full.

## ACKNOWLEDGMENTS

Stanislavova is partially supported by NSF-DMS under Grant No. 1516245. Stefanov is partially supported by NSF-DMS under Grant No. 1614734. The authors are indebted to the anonymous referee, whose thoughtful suggestions significantly improved the manuscript.