GLOBAL CONTINUA OF SOLUTIONS TO THE LUGIATO-LEFEVER MODEL FOR FREQUENCY COMBS OBTAINED BY TWO-MODE PUMPING ELIAS GASMI TOBIAS JAHNKE MICHAEL KIRN AND WOLFGANG REICHEL

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GLOBAL CONTINUA OF SOLUTIONS TO THE LUGIATO-LEFEVER MODEL

FOR FREQUENCY COMBS OBTAINED BY TWO-MODE PUMPING

ELIAS GASMI, TOBIAS JAHNKE, MICHAEL KIRN, AND WOLFGANG REICHEL

ABSTRACT. We consider Kerr frequency combs in a dual-pumped microresonator as time-

periodic and spatially 2π-periodic traveling wave solutions of a variant of the Lugiato-

Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation

given by iaτ= (ζ−i)a−daxx − |a|2a+if0+if1ei(k1x−ν1τ). The main new feature of the

problem is the speciﬁc form of the source term f0+f1ei(k1x−ν1τ)which describes the simul-

taneous pumping of two different modes with mode indices k0=0 and k1∈N. We prove

existence and uniqueness theorems for these traveling waves based on a-priori bounds and

ﬁxed point theorems. Moreover, by using the implicit function theorem and bifurcation

theory, we show how non-degenerate solutions from the 1-mode case, i.e. f1=0, can be

continued into the range f16=0. Our analytical ﬁndings apply both for anomalous (d>0)

and normal (d<0) dispersion, and they are illustrated by numerical simulations.

1. INTRODUCTION

Optical frequency comb devices are extremely promising in many applications such as,

e.g., optical frequency metrology [25], spectroscopy [20,27], ultrafast optical ranging [24],

and high capacity optical communications [14]. For many of these applications the Kerr

soliton combs are generated by using a monochromatic pump. However, recently new

pump schemes have been discussed, where more than one resonator mode is pumped, cf.

[23]. The pumping of two modes can have a number of important advantages. In partic-

ular, 1-solitons arising from a dual-pump scheme can be spectrally broader and spatially

more localized than 1-solitons arising from a monochromatic pump, cf. [7] for a compre-

hensive discussion of the theoretical advantages. Mathematically, Kerr comb dynamics

are described by the Lugiato-Lefever equation (LLE), a damped, driven and detuned non-

linear Schrödinger equation [9,12,16]. Our analysis relies on a variant of the LLE which

is modiﬁed for two-mode pumping, cf. [23] and [7] for a derivation. Using dimensionless,

normalized quantities this equation takes the form

(1) iaτ= (ζ−i)a−daxx −|a|2a+if0+if1ei(k1x−ν1τ),a2π-periodic in x.

Here, a(τ,x)represents the optical intracavity ﬁeld as a function of normalized time τ=

2tand angular position x∈[0, 2π]within the ring resonator. The constant κ>0 describes

the cavity decay rate and d=2

κd2quantiﬁes the dispersion in the system (where ωk=

Date: October 19, 2022.

2000 Mathematics Subject Classiﬁcation. Primary: 34C23, 34B15; Secondary: 35Q55, 34B60.

Key words and phrases. Nonlinear Schrödinger equation, bifurcation theory, continuation methods.

arXiv:2210.09779v1 [math.AP] 18 Oct 2022

2 ELIAS GASMI, TOBIAS JAHNKE, MICHAEL KIRN, AND WOLFGANG REICHEL

ω0+d1k+d2k2is the cavity dispersion relation between the resonant frequencies ωkand

the relative indices k∈Z). Here, the case d<0 amounts to normal and the case d>0

to anomalous dispersion. The resonant modes in the cavity are numbered by k∈Zwith

k0=0 being the ﬁrst and k1∈Nthe second pumped mode. With f0,f1we describe

the normalized power of the two input pumps and ωp0,ωp1denote the frequencies of

the two pumps. Since there are now two pumped modes there are also two normalized

detuning parameters denoted by ζ=2

κ(ω0−ωp0)and ζ1=2

κ(ωk1−ωp1). They describe

the offsets of the input pump frequencies ωp0and ωp1to the closest resonance frequency

ω0and ωk1of the microresonator. The particular form of the pump term i f0+if1ei(k1x−ν1τ)

with ν1=ζ−ζ1+dk2

1suggests to change into a moving coordinate frame and to study

solutions of (1) of the form a(τ,x) = u(s)with s=x−ωτ and ω=ν1

k1. These traveling

wave solutions propagate with speed ωin the resonator and their proﬁles usolve the

ordinary differential equation

(2) −du00 +iωu0+ (ζ−i)u−|u|2u+if0+if1eik1s=0, u2π-periodic.

In the case f1=0 equation (1) amounts to the case of pumping only one mode. This case

has been thoroughly studied, e.g. in [5,6,8,9,13,15,16,17,18,19,22]. In this paper we

are interested in the case f16=0. Since the speciﬁc form of the forcing term is not essential

for many of our results, we allow in the following for more general forcing terms

f(s) = f0+f1e(s)

with a 2π-periodic (not necessarily continuous) function e:R→Cand f0,f1∈R. Hence,

we consider the LLE

(3) −du00 +iωu0+ (ζ−i)u−|u|2u+if(s) = 0, u2π-periodic.

Our main results on the existence of solutions to (3) are stated in Section 2. In Section 3

we illustrate our main analytical results by numerical simulations. The proofs of the main

results are given in Section 4(a-priori bounds), Section 5(existence and uniqueness), and

Section 6(continuation results). The appendix contains a technical result and a considera-

tion of the case where in (2) the value k1is not an integer but close to an integer.

2. MAIN RESULTS

In the following we state our main results.

•Theorem 1provides existence of at least one solution of (3) for any choice of the

parameters and any choice of f.

•Theorem 6and Corollary 8describe how trivial (constant) solutions from the spe-

cial case f1=0 can be continued into non-trivial solutions for f16=0.

•Theorem 9and Corollary 10 show how a non-trivial solution from the case f1=0

can be continued to f16=0.

Our ﬁrst theorem, which ensures the existence of a solution of (3) in the general case where

f1does not need to vanish, is based on a-priori bounds and a variant of Schauder’s ﬁxed

point theorem known as Schaefer’s ﬁxed point theorem. A corresponding uniqueness

GLOBAL CONTINUA OF SOLUTIONS TO THE LUGIATO-LEFEVER MODEL 3

result, which applies whenever |ζ|  1 is sufﬁciently large or (essentially) kfk21 is

sufﬁciently small is given in Theorem 17 in Section 5together with more precise details.

We will use the following Sobolev spaces. For k∈Nthe space Hk(0, 2π)consists of all

square-integrable functions on (0, 2π)whose weak derivatives up to order kexist and are

square-integrable on (0, 2π). By Hk

per(0, 2π)we denote all locally square-integrable 2π-

periodic functions on Rwhose weak derivatives up to order kexist and are locally square-

integrable on R. In both spaces the norm is given by kuk=∑k

j=0k(d

ds )juk2

L2(0,2π)1/2.

Clearly Hk

per(0, 2π)is a proper subspace of Hk(0, 2π)since u∈Hk

per(0, 2π)implies that

ds )ju(0)=(d

ds )ju(2π)for j=0, . . . , k−1. Unless otherwise stated, all of the above

Hilbert spaces are spaces of complex valued functions over the ﬁeld R. In particular, for

v,w∈L2(0, 2π)we use the inner product hv,wi2:=Re R2π

0vw ds. The induced norm is

denoted by k ·k2.

Theorem 1. Equation (3)has at least one solution u ∈H2

per(0, 2π)for any choice of the parame-

ters d ∈R\ {0},ζ,ω∈Rand any choice of f ∈H2(0, 2π).

Next we address the question whether a known solution u0of (3) for f1=0 can be

continued into the regime f16=0. This continuation will be done differently depending

on whether u0is constant (trivial) or non-constant (non-trivial). Moreover, we ﬁrst con-

centrate on one-sided continuations for f1>0 (or f1<0). Two-sided continuations will

be discussed in Section 2.3.

2.1. One-sided continuation of trivial solutions. In the special case f1=0 there are

trivial (constant) solutions u0∈Cof (3) satisfying the algebraic equation

(4) (ζ−i)u0−|u0|2u0+if0=0.

From [13, Lemma 2.1] we know that for given f0∈Rthe curve of constant solutions can

be parameterized by

(5) ζ(t) = (1−t2)f2

0+t

√1−t2,u0(t) = (1−t2)f0−if0tp1−t2,t∈(−1, 1).

In Figure 1we show the curve of the squared L2-norm of all constant solutions of (3) for

f1=0 and f0=1, f0=2√2

√27 and f0=2. The curve may or may not have turning points

which are characterized by ζ0(t) = 0. This condition can be formulated independently of

tby the equivalent condition ζ2−4|u0|2ζ+1+3|u0|4=0. By a straightforward analysis

one can show that with f∗=2√2

√27 we have

•no turning point for |f0|<f∗(cf. Figure 1green curve),

•exactly one (degenerate) turning point for |f0|=f∗(cf. Figure 1red curve),

•exactly two turning points for |f0|>f∗(cf. Figure 1blue curve).

4 ELIAS GASMI, TOBIAS JAHNKE, MICHAEL KIRN, AND WOLFGANG REICHEL

FIGURE 1. Curve of squared L2-norm of all

constant solutions of (3) for f1=0 and f0=

1 (green), f0=2√2

√27 (red) and f0=2 (blue)

when ζ∈[−1, 5]. Turning points (if they ex-

ist) are marked with a cross.

Note that for |f0|>f∗, as

a consequence of the exis-

tence of two turning points,

three different constant solu-

tions exist for certain values

of ζ.

Starting from f1=0 we

use a kind of global implicit

function theorem to continue

a constant solution u0∈Cof

(3) with respect to f1. This

procedure is analyzed in The-

orem 6. The continuation

works if the constant solution

u0∈Cis non-degenerate in the following sense.

Deﬁnition 2. A solution u∈H2

per(0, 2π)of (3) for f1=0 is called non-degenerate if the

kernel of the linearized operator

Luϕ:=−dϕ00 +iωϕ0+ (ζ−i−2|u|2)ϕ−u2ϕ,ϕ∈H2

per(0, 2π)

consists only of span{u0}.

Remark 3. Note that Lu:H2

per(0, 2π)→L2(0, 2π)is a compact perturbation of the iso-

morphism −dd2

dx2+sign(d):H2

per(0, 2π)→L2(0, 2π)and hence an index-zero Fredholm

operator. Notice also that span{u0}always belongs to the kernel of Lu. Non-degeneracy

means that except for the obvious candidate u0(and its real multiples) there is no other

element of the kernel of Lu. Notice also that a constant solution u0is non-degenerate if the

linearized operator Lu0is injective, and, as a consequence, invertible in suitable spaces.

Lemma 4. A trivial solution u0∈Cof (3)for f1=0is non-degenerate if and only if

(a) Case ω6=0:

ζ2−4|u0|2ζ+1+3|u0|46=0.

(b) Case ω=0:

(ζ+dm2)2−4|u0|2(ζ+dm2) + 1+3|u0|46=0for all m ∈N0.

Proof. Let ϕ∈H2

per(0, 2π)be in the kernel of the linearized operator, i.e.,

−dϕ00 +iωϕ0+ (ζ−i−2|u0|2)ϕ−u2

0ϕ=0.

This implies that the Fourier coefﬁcients ϕmof the Fourier series ϕ=∑m∈Zϕmeims have

the property that

(dm2−ωm+ζ−i−2|u0|2)ϕm−u2

0ϕ−m=0

for all m∈Z. If we also write down the complex conjugate of this equation

−u02ϕm+ (dm2+ωm+ζ+i−2|u0|2)ϕ−m=0

GLOBAL CONTINUA OF SOLUTIONS TO THE LUGIATO-LEFEVER MODEL 5

then we see that non-degeneracy of u0is equivalent to the non-vanishing of the determi-

nant for this two-by-two system in the variables ϕm,ϕ−mfor all m∈N0. Computing the

determinant we obtain the condition

(6) (ζ+dm2)2−4|u0|2(ζ+dm2) + 1+3|u0|4−ω2m2−2iωm6=0 for all m∈N0.

In the case ω6=0 this is trivially satisﬁed for all m6=0 (because then the imaginary part

is non-zero) and for m=0 by assumption (a) of the lemma. In the case ω=0 condition

(6) can only be guaranteed by assumption (b). 

Remark 5. Trivial solutions of (3) for f1=0 are determined by (4). For ω6=0 all trivial so-

lutions u0of (3) for f1=0 are non-degenerate except those at the turning points described

above. In the case ω=0 all trivial solutions u0of (3) for f1=0 are non-degenerate except

those at the (potential) bifurcation points and the turning points. This is true (up to addi-

tional conditions ensuring transversality and simplicity of kernels) because the necessary

condition for bifurcation w.r.t. ζfrom the curve of trivial solutions is fulﬁlled if and only

if the expression in (b) vanishes for at least one m∈N, cf. [6],[13].

Theorem 6. Let d ∈R\{0},ζ,ω,f0∈Rand e ∈H2(0, 2π)be ﬁxed. Let furthermore u0∈C

be a constant non-degenerate solution of (3)for f1=0. Then the maximal continuum*C+⊂

[0, ∞)×H2

per(0, 2π)of solutions (f1,u)of (3)with (0, u0)∈ C+has the following properties:

(i) locally near (0, u0)the set C+is the graph of a smooth curve f17→ (f1,u(f1)),

(ii) C+∩[0, M]×H2

per(0, 2π)is bounded for any M >0.

Moreover, if pr1(C+)denotes the projection of C+onto the f1-parameter component, then at least

one of the following properties hold:

(a) pr1(C+) = [0, ∞),

(b) ∃u+

06=u0:(0, u+

0)∈ C+.

A maximal continuum C−⊂(−∞, 0]×H2

per(0, 2π)with corresponding properties also exists.

Remark 7. If property (a) of Theorem 6holds, then C+is unbounded in the direction of

the parameter f1∈[0, ∞)and hence this is an existence result for all f1∈[0, ∞). Property

(b) means that the continuum C+returns to the f1=0 line at a point u+

06=u0.

Corollary 8. Property (a) in Theorem 6holds in any of the following three cases,

(i) sign(d)ζ<−C(d,f0)21d<0−271+πf2

0|ω|

|d|+π2f4

|d|C(d,f0)6,

(ii) sign(d)ζ>3C(d,f0)2+ω2

4|d|,

(iii) √3C(d,f0)<1,

*A continuum is a closed and connected set.

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GLOBALCONTINUAOFSOLUTIONSTOTHELUGIATO-LEFEVERMODELFORFREQUENCYCOMBSOBTAINEDBYTWO-MODEPUMPINGELIASGASMI,TOBIASJAHNKE,MICHAELKIRN,ANDWOLFGANGREICHELABSTRACT.WeconsiderKerrfrequencycombsinadual-pumpedmicroresonatorastime-periodicandspatially2p-periodictravelingwavesolutionsofavariantoftheLugiato-Lefeve...

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GLOBAL CONTINUA OF SOLUTIONS TO THE LUGIATO-LEFEVER MODEL FOR FREQUENCY COMBS OBTAINED BY TWO-MODE PUMPING ELIAS GASMI TOBIAS JAHNKE MICHAEL KIRN AND WOLFGANG REICHEL

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