Multi-pulse solitary waves in a fourth-order nonlinear Schrödinger equation
aa r X i v : . [ n li n . PS ] S e p Multi-pulse solitary waves in a fourth-order nonlinear Schr¨odingerequation
Ross Parker a , Alejandro Aceves a a Department of Mathematics, Southern Methodist University, Dallas, Texas 75275
Abstract
In the present work, we consider the existence and spectral stability of multi-pulse solitarywave solutions to a nonlinear Schr¨odinger equation with both fourth and second order dis-persion terms. We first give a criterion for the existence of a single solitary wave solutionin terms of the coefficients of the dispersion terms, and then show that a discrete family ofmulti-pulse solutions exists which is characterized by the distances between the individualpulses. We then reduce the spectral stability problem for these multi-pulses to computingthe determinant of a matrix which is, to leading order, block diagonal. Under an additionalassumption, which can be verified numerically, we show that all multi-pulses are spectrallyunstable. For double pulses, numerical computations are presented which are in good agree-ment with our analytical results.
Keywords: nonlinear Schr¨odinger equation, solitary waves, multi-pulse solutions, nonlinearoptics
1. Introduction
It has been nearly 50 years since the discovery by Zakharov and Shabat [17] of theintegrability of the nonlinear Schr¨odinger equation (NLS) and the corresponding solitonsolutions, and 40 years since the first experimental demonstration by Mollenauer, Stolenand Gordon of optical solitons propagating in fibers [7]. At its most fundamental level,the NLS soliton represents the balance of chromatic second order dispersion and the Kerrself-focusing nonlinearity. The robustness of the soliton opened up new directions in boththeoretical and experimental fronts that continue to this day. As better fibers were built,and technological advances led to the invention of photonic crystal fibers, that enabled theengineering of the dispersion resulting in new discoveries such as supercontinuum generationrealized in regimes far from the NLS equation. In this vein, recent experimental work insilicon photonic crystal waveguides has produced for the first time what is now known aspure quartic solitons (PQS) [2]; the term reflects that for this waveguide the leading orderdispersion term is fourth order. In [13], spectral stability of PQS was shown numerically,as well as evolution into PQS from Gaussian initial conditions, and this is extended to amore general model in [14]. Our results presented here provide a more rigorous study of this
Email addresses: [email protected] (Ross Parker), [email protected] (Alejandro Aceves)
Preprint submitted to Elsevier September 7, 2020 eneral model, including for the first time the existence and spectral stability of multi-pulsesolutions.After a brief background, we first present results on the existence and stability of the pri-mary soliton solution in the more general case where both second and fourth order dispersionare accounted for. This includes the particular case of PQS. Under a standard assumption,which can be verified numerically, the primary pulse solution is orbitally stable. The secondpart of the paper proves the existence of multi-pulses. These are solutions which resemblemultiple, well-separated copies of the primary soliton; neighboring pulses in the multi-pulsecan be either in phase or out of phase. We then look at the spectral stability of thesemulti-pulses. Under a mild assumption, which can be verified numerically, all of these pulsetrains are unstable. Numerical examples are then presented, followed by a brief discussionof conclusions and directions for future work. The final section contains the proofs for thespectral stability results.
2. Background
The fourth-order generalization of the nonlinear Schr¨odinger equation (NLS) iu t + β u xxxx − β u xx + γ | u | u = 0 (1)was recently investigated in [14] in a study of the properties of solitary wave solutions undera combination of second and fourth order dispersion. (We use the independent variables( t, x ) in place of ( z, τ ), which is used in [2, 13, 14] and is common in the optics literature).Ordinary NLS solitons are solutions when β < β = 0. Pure quartic solitons (PQS)occur when β = 0 and β <
0. In that case, u ( x, t ) satisfies the equation iu t + β u xxxx + γ | u | u = 0 . (2)Unlike ordinary NLS solitons, PQS have oscillatory, exponentially decaying tails. Therehas been much recent interest in PQS due to the their discovery in experimental media byBlanco-Redondo et al. in 2016 [2]. The existence and spectral stability of PQS solutions wasshown numerically in [13], and the existence of solitary wave solutions to the more generalequation (1) in terms of the parameters β , β , and ω is discussed in [14].Real-valued, standing wave solutions, i.e. solutions of the form e iωt u ( x ), satisfy the ODE β u xxxx − β u xx + γu − ωu = 0 , (3)which is a rescaling of [3, (7)]. For PQS, equation (3) can be written in parameter free formby using the rescaling u ( x ; ω ) = r ωγ ˜ u (cid:18) ω | β | (cid:19) / x ! to obtain the equation −
124 ˜ u xxxx + ˜ u − ˜ u = 0 . (4)2e observe that the power or photon number of PQS scales as ω / compared to the ω / scal-ing of classical NLS solitons. A more general rescaling [14, Section VI] transforms equation(3) into the one-parameter equation˜ u xxxx + 2 σ ˜ u xx + ˜ u − | ˜ u | ˜ u = 0 , (5)where σ = s ω | β | β is a non-dimensional parameter characterizing the relative strengths of the quadratic andquartic dispersion terms.For ordinary NLS, an analytic solution can be obtained by the inverse scattering trans-form [17]. For β < β <
0, an analytic solution has been obtained by Karlsson andH¨o¨ok [6] when ω = 24 β / | β | .
3. Mathematical setup
Our analysis follows Grillakis, Shatah, and Strauss [4]. Equation (1) can be written inHamiltonian form as ∂u∂t = J E ′ ( u ( t )) , (6)where J = − i and the energy E is given by E ( u ) = 12 Z ∞−∞ (cid:18) β | u xx | + β | u x | + γ | u | (cid:19) dx. (7)The energy E is invariant under the complex rotation group T ( θ ), given by T ( θ ) u = e iθ u .The corresponding conserved quantity, often called the charge [4, Section 6.C], is given by Q ( u ) = − Z ∞∞ | u | dx. (8)We make the following hypothesis regarding the well-posedness of (6), which is the same as[4, Assumption 1]. Hypothesis 1.
For each initial condition u , there exists T > depending only on K , where k u k ≤ K , such that the PDE (6) has a solution u ( t ) on [0 , T ] with u (0) = u . Standing waves are solutions of the form T ( ωt ) u , where u is independent of t . A standingwave solution satisfies the standing wave equation E ′ ( u ) − w Q ′ ( u ) = 0 [4, 2.15]. Since Q ′ ( u ) = − u , this equation has the form E ′ ( u ) + wu = 0, which can be written as β u xxxx − β u xx + γ | u | u − ωu = 0 . (9)The following theorem gives criteria for the existence of real-valued solitary wave solutionsto (9) in terms of the parameters β , β and ω . For the remainder of this paper, we will onlyconsider β <
0, since that is the physically relevant regime.3 heorem 1.
Let β < , and define ω c = 32 β | β | . (10) Then for either (i) ω > ω c , or (ii) < ω < ω c and β < , there exists a real-valued,symmetric, exponentially localized solution φ ( x ; ω ) ∈ H ( R ) ∩ C ( R ) of the standing waveequation (9) .Proof. The existence result follows directly from [5], which uses the mountain pass lemma andthe concentration-compactness principle. Exponential localization follows from the stablemanifold theorem.
Remark 1.
For ω > ω c , it follows from [5] that there exists a countably infinite family ofdistinct solitary-wave solutions, which are the multi-pulse solutions we will construct below.In addition, for β = 0 , ω c = 0 , thus PQS exist for all ω > . We make the following standard smoothness assumption (see, for example, [4, Assumption2]) concerning the solutions φ ( x ; ω ) to (3). Hypothesis 2.
The map ω φ ( x ; ω ) from I to H ( R ) is C , where I is the interval forwhich the primary pulse solution φ ( x ; ω ) exists. Define the scalar d ( ω ) = E ( φ ( ω )) − ω Q ( φ ( ω )) . (11)By [4, (2.21)], d ′′ ( ω ) = hQ ′ ( φ ( x ; ω )) , ∂ ω φ ( x ; ω ) i = Z ∞−∞ φ ( x ; ω ) ∂ ω φ ( x ; ω ) dx, (12)where ∂ ω φ ( x ; ω ) is well-defined by Hypothesis 2. By [4, Theorem 3.5], the standing wave φ ( x ; ω ) is orbitally stable if d ′′ ( ω ) >
0. This quantity can be computed numerically, and wetake this stability criterion as a hypothesis.
Hypothesis 3.
For each ω such that a primary pulse solution φ ( x ; ω ) exists, d ′′ ( ω ) > . Let β < β ∈ R , and choose ω > φ ( x ) = φ ( x ; ω ) exists by Theorem 1. From this point forward, we will suppress the dependence on ω for simplicity of notation. The linearization of the PDE (1) about φ is the linear operator L ( φ ) : H ( R ) ⊂ L ( R ) L ( R ), given by L ( φ ) = (cid:18) L − ( φ ) −L + ( φ ) 0 (cid:19) , (13)where L − ( φ ) = − β ∂ xxxx + β ∂ xx + ω − γφ L + ( φ ) = − β ∂ xxxx + β ∂ xx + ω − γφ .
4t is straightforward to verify that L − ( φ ) φ = 0 L + ( φ ) ∂ x φ = 0 L + ( φ )( − ∂ ω φ ) = φ. (14)Furthermore, since L − ( φ ) is self-adjoint and φ ′ ⊥ ker L − ( φ ), there exists a function z suchthat L − ( φ ) z = φ ′ . For the classical NLS equation ( β = 0 , β = 0), z = β xφ . Thus L ( φ )has a kernel with (at least) algebraic multiplicity 4 and geometric multiplicity 2, i.e. L ( φ ) (cid:18) φ (cid:19) = 0 , L ( φ ) (cid:18) ∂ ω φ (cid:19) = (cid:18) φ (cid:19) L ( φ ) (cid:18) ∂ x φ (cid:19) = 0 , L ( φ ) (cid:18) z (cid:19) = (cid:18) ∂ x φ (cid:19) . (15)The spectrum of L ( φ ) can be divided into two disjoint sets: the essential spectrum isthe set of λ ∈ C for which L ( φ ) − λ I is not Fredholm, and the point spectrum is the set of λ ∈ C for which ker L ( φ ) − λ I is nontrivial. To find the essential spectrum, which dependsonly on the background state and is independent of the solution φ we are linearizing about, L ( φ ) is exponentially asymptotic to the linear operator L (0), given by L (0) = (cid:18) L −L (cid:19) , L = − β ∂ xxxx + β ∂ xx + ω, (16)thus the eigenvalue problem L (0) v = λv is equivalent to ( L + λ ) p = 0. By [15, Theorem3.1.13], the essential spectrum is given by the curves (cid:20) − β
24 ( ik ) + β ik ) + ω (cid:21) + λ = 0 k ∈ R , from which it follows that σ ess = (cid:26) ± i (cid:18) − β k − β k + ω (cid:19) : k ∈ R (cid:27) . If β < β ≤
0, the essential spectrum is given by σ ess = { ki : k ∈ R , | k | ≥ ω } , (17)which is purely imaginary, bounded away from the origin, and independent of β and β . Inparticular, this is the case for PQS. If β < β >
0, and ω > ω c , the essential spectrum isgiven by σ ess = { ki : k ∈ R , | k | ≥ ω − ω c } , (18)which is also purely imaginary and bounded away from the origin, but does depend on β and β via ω c .By the stability assumption in Hypothesis 3, no element of the spectrum of L ( φ ) can havea positive real part. Since the PDE (1) is Hamiltonian, all elements of the spectrum of L ( φ )5ust come in quartets ± α ± βi , thus the spectrum of L ( φ ) is contained in the imaginaryaxis. For PQS, there is an additional pair of imaginary eigenvalues located right before theessential spectrum boundary (approximately ± . ωi ), which corresponds to an internalmode of the solitary wave [13]. For β = 0, there can be multiple pairs of internal modeeigenvalues (an example of two pairs internal mode eigenvalues is shown in [14, Figure 9]).By Hypothesis 3, these internal mode eigenvalues must be purely imaginary. In a recent conference presentation [1], it was shown this by incorporating an intracavityprogrammable pulse-shaper in a mode-locked fiber laser, one can manipulate the net cavitydispersion by applying a phase to the pulse so that to leading order, the stationary pulsegenerated is modeled by the higher-order NLS equation − ( i ) k d k udx k + ωu − γu = 0 , (19)where k ≥ u ( x ; ω ) = q ωγ ˜ u ( ω /k x ).
4. Existence of multi-pulse solitary waves
A multi-pulse is a multi-modal solitary wave resembling multiple, well-separated copiesof the primary solitary wave. To prove the existence of multi-pulse solutions to (3), wewill reframe the problem using a spatial dynamics approach. From this perspective, theprimary solitary wave is a homoclinic orbit connecting the unstable and stable manifolds ofa saddle equilibrium. A multi-pulse is a multi-loop homoclinic orbit which remains close tothe primary homoclinic orbit. Letting U = ( u , u , u , u ) = ( u, ∂ x u, ∂ x u, β ∂ x u ), we rewriteequation (3) as the first order system U ′ = F ( U ) = u u β u ωu − γu . (20)This system has a conserved quantity H ( u , u , u , u ) = − u u − u + β u − γ u + 12 ωu , (21)which we obtain by multiplying (3) by u x and integrating once. F (0) = 0, and the charac-teristic polynomial of DF (0) is p ( t ) = t − β β t − β ω, which has a quartet of complex eigenvalues ± a ± bi when ω > ω c . For ω > ω c , U = 0 isa hyperbolic saddle equilibrium of (20) with two-dimensional stable and unstable manifolds6hich intersect to form a homoclinic orbit. The exponentially localized primary pulse solu-tion corresponding to this homoclinic orbit will have oscillatory tails, with the frequency ofoscillations approximately equal to b . We have the following result concerning the existenceof multi-pulse solutions, which follows immediately from [10, Theorem 3.6]. Theorem 2.
Assume Hypothesis , Hypothesis , and Hypothesis , and fix β < and ω > ω c . Let φ ( x ) be the real-valued, symmetric, exponentially localized primary pulse solutionto (3) from Theorem , and let U ( x ) = ( φ ( x ) , ∂ x φ ( x ) , ∂ x φ ( x ) , ∂ x φ ( x )) be the correspondinghomoclinic orbit solution to (20) . Let ± a ± bi be the eigenvalues of DF (0) , with a > and b > . Then for any(i) n ≥ (ii) Sequence of nonnegative integers { k , . . . , k n − } , with at least one of the k j ∈ { , } (iii) Sequence of phase parameters { θ , . . . , θ n } ∈ {− , } n , with θ = 1 there exists a nonnegative integer m such that for any integer m with m ≥ m , there existsa unique n − modal solution U n ( x ) to (20) which is defined piecewise via U n x + 2 i − X k =1 X k ! = ( θ i U ( x ) + ˜ U − i ( x ) x ∈ [ − X i − , θ i U ( x ) + ˜ U − i ( x ) x ∈ [0 , X i ] (22) for i = 1 , . . . , n , where X = X n = ∞ . Uniqueness is up to translation and multiplication by T ( θ ) . The distances between consecutive peaks are given by X i , where X i ≈ πb (2 m + k i ) + ˜ X, (23) and ˜ X is a constant. In addition, we have the estimates k ˜ U ± i k ∞ ≤ Ce − aX min ˜ U + i ( X i ) = θ i +1 U ( − X i ) + O ( e − aX min )˜ U − i +1 ( − X i ) = θ i U ( X i ) + O ( e − aX min ) , (24) where X min = min { X , . . . X n − } , which hold as well for all derivatives with respect to x .Proof. Since the spectrum of DF (0) is a quartet of eigenvalues ± a ± bi for ω > ω c , equation(20) has a conserved quantity (21), and the Melnikov integral M = R ∞−∞ φ x dx is positive,the result follows from [10, Theorem 3.6], with the straightforward modification that themulti-pulse is constructed from copies of U ( x ) and − U ( x ). The estimates (24) follow from[11, 12].
5. Spectrum of multi-pulse solitary waves
Let U ( x ) = ( φ ( x ) , ∂ x φ ( x ) , ∂ x φ ( x ) , β ∂ x φ ( x )) be the primary homoclinic orbit correspond-ing to the primary pulse φ ( x ), and let U n = ( φ n ( x ) , ∂ x φ n ( x ) , ∂ x φ n ( x ) , β ∂ x φ n ( x )) be a multi-loop homoclinic orbit solution to (20) constructed according to Theorem 2. The first com-ponent φ n of U n is a multi-pulse solitary wave solution to (3). As in [8, 11], we will locate theeigenvalues near the origin of the linearized operator L ( φ n ). Both L ( φ ) and L ( φ n ) have two7igenfunctions in the kernel. Since the kernel of L ( φ ) is 2-dimensional, we expect that L ( φ n )will have 4( n −
1) additional eigenvalues near 0. Since these arise from nonlinear interactionsbetween the tails of neighboring pulses in the multi-pulse structure, we call them interac-tion eigenvalues. Once again using a spatial dynamics approach, we rewrite the eigenvalueproblem L ( φ n ) v = λv as the first order system V ′ ( x ) = K ( φ n ) V ( x ) + λB V ( x ) , (25)where K ( φ n ) = (cid:18) K + ( φ n ) 00 K − ( φ n ) (cid:19) , B = (cid:18) B − B (cid:19) ,K − ( φ n ) = β ω − γφ n β , K + ( φ ) = β ω − γφ n β , B = . The associated variational equation V ′ ( x ) = K ( φ ) V ( x ) (26)has two linearly independent, exponentially decaying solutions ˜ Q ( x ) = ( U ′ ( x ) , T and Q ( x ) = (0 , U ( x )) T . The corresponding adjoint variational equation W ′ ( x ) = − K ( φ ) ∗ W ( x ) (27)has two linearly independent, exponentially decaying solutions ˜ Q ∗ ( x ) = (Ψ ′ ( x ) , T and Q ∗ ( x ) = (0 , Ψ( x )) T , whereΨ( x ) = (cid:18) − β ∂ x φ ( x ) + β ∂ x φ ( x ) , β ∂ x φ ( x ) − β φ ( x ) , − β ∂ x φ ( x ) , φ ( x ) (cid:19) . (28)The following theorem, which is analogous to [11, Theorem 2], reduces the problem oflocating the eigenvalues of L ( φ n ) in a ball around the origin in the complex plane to findingthe determinant of a 2 n × n matrix which is, to leading order, block diagonal. The proof isgiven in subsection 8.1. Theorem 3.
Assume Hypothesis , Hypothesis , and Hypothesis . Let U ( x ) be the pri-mary homoclinic orbit from Theorem , and let U n ( x ) be an n -pulse solution constructed ac-cording to Theorem with phase parameters { θ , . . . , θ n } and pulse distances X , . . . , X n − .Let ± a ± bi be the eigenvalues of DF (0) , with a > and b > . Then there exists δ > withthe following property. There exists a bounded, nonzero solution V ( x ) of (25) for | λ | < δ ifand only if E ( λ ) = det S ( λ ) = 0 , (29) where S ( λ ) is the n × n block matrix S ( λ ) = (cid:18) A + λ M I
00 ( a + b ) A − λ ˜ M I (cid:19) + R ( λ ) . (30)8 he tri-diagonal matrix A is defined by A = − a a a − a − a a a − a − a a . . . . . . a n − − a n − , where a i = θ i θ i +1 h Ψ( X i ) , U ( − X i ) i , and Ψ( x ) is defined by (28) . The constants M and ˜ M are given by M = Z ∞−∞ φ ( x ) ∂ ω φ ( x ) dx = d ′′ ( ω ) > , ˜ M = Z ∞−∞ ∂ x φ ( x ) z ( x ) dx, where z ( x ) is defined in section after (14) . The remainder term R ( λ ) is analytic in λ andhas uniform bound | R ( λ ) | ≤ C (cid:0) | λ | ( | λ | + e − αX min ) + e − (2 α + γ ) X min ) (cid:1) , where γ > . If ˜
M >
0, which is supported by numerical computation, then all multi-pulse solutionsare unstable by the following corollary.
Corollary 1.
Let U n ( x ) be a n-pulse constructed using Theorem . Then there are n − pairs of interaction eigenvalues λ , . . . λ n − and ˜ λ , . . . ˜ λ n − , given by λ i = r µ i M + O (cid:0) e − (2 α + γ ) X min (cid:1) i = 1 , . . . , n − λ i = r − ( a + b ) µ i ˜ M + O (cid:0) e − (2 α + γ ) X min (cid:1) i = 1 , . . . , n − , (31) where { µ , . . . , µ n − , } are the real, distinct eigenvalues of A . If ˜ M > , then one of each pair λ i , ˜ λ i is real and the other is purely imaginary, thus there are n − positive real eigenvalues. Finally, we compute the interaction eigenvalues of a 2-pulse solution U ( x ). Corollary 2.
Let U ( x ) be a 2-pulse constructed using Theorem with pulse distance X and phase parameters θ , θ . Then there are four interaction eigenvalues associated with U ( x ) , which are, to leading order, given by λ = ± r a M , ˜ λ = ± s − a + b ) a ˜ M , (32) where a = θ θ h Ψ( X ) , U ( − X ) i . If ˜ M > , then one pair is real and one pair is purelyimaginary. Remark 2.
In addition, the internal mode eigenvalues of the primary pulse will duplicateas pulses are added to the multi-pulse structure. For example, for the pure quartic solitarywave, the 2-pulse will have two pairs of internal mode eigenvalues. If ˜ M > , the 2-pulse isunstable, and these internal mode eigenvalues have no additional effect on stability. . Numerical Results To construct the primary pulse solution φ ( x ), we start with the known solitary wavesolution for NLS and gradually modify the parameters β and β , solving for the new solitarywave solution at each step using a Newton conjugate-gradient method [16, Chapter 7.2.4]implemented in MATLAB. To obtain the pure quartic solitary wave for β = − β , β ) = ( − ,
0) and ( β , β ) =(0 , − -20 -15 -10 -5 0 5 10 15 20 x -0.200.20.40.60.811.21.4 -20 -15 -10 -5 0 5 10 15 20 x -35-30-25-20-15-10-505 l og Figure 1: Pure quartic solitary wave solution φ ( x ) to (3) with β = 0, β = − ω = 1. (left panel). Plotof log φ ( x ) vs x (right panel) showing exponentially-decaying oscillatory tails. Spatial discretization is auniform grid with N = 1024 grid points, and we use periodic boundary conditions. To determine the spectrum of the linearization about the primary pulse, we constructthe linear operator L ( φ ) using Fourier spectral differentiation matrices and compute theeigenvalues using Matlab’s eigenvalue solver eig (Figure 2, left panel). We note that theessential spectrum is discrete, which is a consequence of the spatial discretization, as well asthe presence of a pair of internal mode eigenvalues on the imaginary axis.10 Re -1.5-1-0.500.511.5 I m Figure 2: Spectrum of pure quartic solitary wave (left panel), β = 0, β = − ω = 1. Kernel eigenvalues inblack, internal mode eigenvalues in red. Eigenvalues in blue correspond to the discrete essential spectrum.Eigenfunction corresponding to internal mode eigenvalue (right panel). For the primary solitary wave solutions, we can compute the stability criterion (12) from[4]. In all cases, M = d ′′ ( ω ) >
0, which suggests that primary pulse solution is orbitallystable. In addition, numerical computation suggests that ˜
M >
0. To construct doublepulses, we glue together two copies of the primary pulse at the pulse distances predictedby Theorem 2 and solve for the double pulse solution using the same Newton conjugate-gradient method we used above. The first eight double pulse solutions are shown in Figure 3.Arbitrary multi-pulses can similarly be constructed. -20 -15 -10 -5 0 5 10 15 20 x -0.200.20.40.60.811.21.4 -20 -15 -10 -5 0 5 10 15 20 x -1.5-1-0.500.511.5 Figure 3: First eight double pulse solutions φ ( x ) constructed from two pure quartic solitary wave solution φ ( x ) to (3) with β = 0, β = − ω = 1. In-phase double pulses (left panel), opposite phase double pulses(right panel). For the spectrum of L ( φ ), the linearization about the double pulse solution φ , there is(for both in phase and out of phase double pulses) a pair of purely imaginary interactioneigenvalues and a pair of real interaction eigenvalues, thus the double pulse solutions are allunstable (Figure 4), which verifies Corollary 2. There is also a duplication of the internal11ode eigenvalues (Figure 5); these appear to be purely imaginary, but they do not affectstability. -0.5 0 0.5 Re -1.5-1-0.500.511.5 I m -0.5 0 0.5 Re -1.5-1-0.500.511.5 I m Figure 4: Eigenvalues for first in-phase double pulse (left panel) and first out-of-phase double pulse (rightpanel). Interaction eigenvalues shown in red, and kernel eigenvalues in black. Eigenvalues in blue corre-spond to the discrete essential spectrum; as expected, these eigenvalues are on the imaginary axis and havemagnitude | λ | ≥ ω . Internal mode eigenvalues are not shown. β = 0, β = − ω = 1. -0.5 0 0.5 Re I m Figure 5: Close-up of spectrum showing pair of internal mode eigenvalues (red) and eigenvalues correspondingto the essential spectrum (blue) for third in-phase double pulse. β = 0, β = − ω = 1. Finally, we verify the formulas for the interaction eigenvalues from Corollary 2 by plottingthe log of the relative error between the leading order term in (32) and the eigenvaluescomputed by Matlab versus the pulse separation distance X (Figure 6).12 Pulse separation (X) -16-14-12-10-8-6-4 l og r e l a t i v e e rr o r +- Figure 6: Log of the relative error for the eigenvalues λ ± versus the pulse separation X for the first fivein-phase double pulses. β = 0, β = − ω = 1.
7. Conclusions and future directions
In this paper, we studied single and multi-pulse solitary wave solutions to a generalnonlinear Schr¨odinger equation with both second and fourth order dispersion terms. Wegave criteria for the existence of a primary soliton solution in terms of the parameters ofthe system, and provided numerical verification that the primary soliton is orbitally stable.We then constructed n -pulse solutions by splicing together multiple copies of the primarypulse, and reduced the problem of finding the small eigenvalues resulting from interactionbetween neighboring pulses to that of computing the determinant of a 2 n × n block matrix.Under a mild assumption, which can be verified numerically, we showed that all multi-pulse solutions are unstable. For future research, we could investigate solitons and multi-pulses in higher order NLS equations, as discussed in [1]. We expect that these resultswould hold for these higher order variants, and that all multi-pulse solutions would beunstable. We could also study generalizations to other nonlinearities. It would be interestingto perform numerical time-stepping starting with perturbed multi-pulses to investigate howthe instability evolves in time. Although all multi-pulse solutions to this fourth order modelare unstable, this equation represents an idealization of the experimental situation sinceenergy is always conserved. A more realistic model might incorporate gains and losses ofenergy in the laser cavity, and it is possible that stable multi-pulses could exist in such ascenario.
8. Proof of stability results
The proof is adapted from [8, Section 3.4] and the proof of [11, Theorem 2], and uses animplementation of the Lyapunov-Schmidt reduction known as Lin’s method. It follows from(15) that [ Y ( x )] ′ = K ( φ n ) Y ( x ) , [ Z ( x )] ′ = K ( φ n ) Z ( x ) + B Y ( x )[ ˜ Y ( x )] ′ = K ( φ n ) ˜ Y ( x ) , [ ˜ Z ( x )] ′ = K ( φ n ) ˜ Z ( x ) + B ˜ Y ( x ) , (33)13here Y ( x ) = (0 , U n ( x )) T , Z ( x ) = ( ∂ ω U n ( x ) , T ˜ Y ( x ) = ( ∂ x U n ( x ) , T , ˜ Z ( x ) = (0 , Z n ( x )) T , (34) Z n ( x ) = ( z n ( x ) , ∂ x z n ( x ) , ∂ x z n ( x ) , β ∂ x z n ( x )), and the first component z n ( x ) solves L − ( φ n ) z n = φ ′ n . The analysis is identical to that of [8], except the piecewise ansatz for the eigenfunctionalso involves Z ( x ) and ˜ Z ( x ). Writing the functions (33) in piecewise form as with (22), wetake the ansatz V ± i ( x ) ′ = d i ( Y ± i ( x ) + λZ ± i ( x )) + ˜ d i ( ˜ Y ± i ( x ) + λ ˜ Z ± i ( x )) + W ± i i = 1 , . . . , n, (35)where V − i ∈ C ([ − X i − , , C ) and V + i ∈ C ([0 , X i ] , C ). Substituting (35) into (25) andsimplifying using (33), the remainder functions W ± i ( x ) solve the equation W ± i ( x ) ′ = K ( φ n ) W ± i ( x ) + λ d i BZ ± i ( x ) + λ ˜ d i B ˜ Z ± i ( x ) i = 1 , . . . , n. (36)Following [8, 11], we obtain a unique piecewise solution W ± i ( x ) which generically has n jumpsat x = 0 in the direction of Q ∗ (0) ⊕ ˜ Q ∗ (0). Using the definitions of Q ∗ ( x ) and ˜ Q ∗ ( x ) togetherwith (24) and [8, (3.19)], these jumps are given by ξ i = θ i +1 h Ψ( X i ) , U ( − X i ) i ( d i +1 − d i ) + θ i − h Ψ( − X i − ) , U ( X i − ) i ( d i − d i − )+ λ θ i d i Z ∞−∞ h Ψ( y ) , B∂ ω U ( y ) i dy + O (( | λ | + e − αX min ) )˜ ξ i = θ i +1 h Ψ ′ ( X i ) , U ′ ( − X i ) i ( ˜ d i +1 − ˜ d i ) + θ i − h Ψ ′ ( − X i − ) , U ′ ( X i − ) i ( ˜ d i − ˜ d i − ) − λ θ i ˜ d i Z ∞−∞ h Ψ( y ) , BZ ( y ) i dy + O (( | λ | + e − αX min ) ) , (37)where Z ( x ) = ( z ( x ) , ∂ x z ( x ) , ∂ x z ( x ) , β ∂ x z ( x )). By symmetry,Ψ( − x ) = − R Ψ( x ) , U ( − x ) = RU ( x ) , (38)where R is the standard reversor operator R ( u , u , u , u ) = ( u , − u , u , − u ) , thus h Ψ( − X i − ) , U ( X i − ) i = −h Ψ( X i − ) , U ( − X i − ) ih Ψ ′ ( − X i − ) , U ′ ( X i − ) i = −h Ψ ′ ( X i − ) , U ′ ( − X i − ) i . (39)Finally, we relate h Ψ( X i ) , U ( − X i ) i and h Ψ ′ ( X i ) , U ′ ( − X i ) i . Since DF ( φ ) = K + ( φ ), Ψ( x )is the unique bounded solution to the adjoint equation W ′ ( x ) = − DF ( φ ) ∗ W ( x ). Thus by[11, Lemma 6.1], with Ψ ′ ( x ) in place of Ψ( x ), p in place of φ , and no parameter µ , h Ψ ′ ( x ) , U ( − x ) = h Ψ ′ ( − x ) , U ( x ) i = se − ax sin(2 bx + p ) + O ( e − (2 α + γ ) x ) (40) h Ψ ′ ( x ) , U ′ ( − x ) i = h Ψ ′ ( x ) , U ′ ( − x ) i = − se − ax ( b cos(2 bx + p ) − a sin(2 bx + p )) + O ( e − (2 α + γ ) x ) , (41)14here s > γ >
0. Differentiating h Ψ( − x ) , U ( x ) i with respect to x , since the operator ∂ x is skew symmetric, ddx h Ψ( x ) , U ( − x ) i = 2 h Ψ ′ ( x ) , U ( − x ) i , thus we can integrate (40) by parts to get h Ψ( x ) , U ( − x ) i = − a + b se − ax ( b cos(2 bx + p ) + a sin(2 bx + p )) + O ( e − (2 α + γ ) x ) . In the proof of [11, Theorem 3], the distances X i are chosen to solve se − aX i sin(2 bX i + p ) = O ( e − (2 α + γ ) X i ), thus for x = X i we have h Ψ ′ ( X i ) , U ′ ( − X i ) i = ( a + b ) h Ψ( X i ) , U ( − X i ) i + O ( e − (2 α + γ ) X i ) . (42)Using (42) and (39), multiplying by θ i , and using the definition of B , equations (37) simplifyto the jump conditions ξ i = θ i θ i +1 h Ψ( X i ) , U ( − X i ) i ( d i +1 − d i ) − θ i − θ i h Ψ( X i − ) , U ( − X i − ) i ( d i − d i − )+ λ d i Z ∞−∞ φ ( y ) ∂ ω φ ( y ) dy + O ( | λ | ( | λ | + e − αX min ) + e − (2 α + γ ) X min ))˜ ξ i = ( a + b ) θ i θ i +1 h Ψ( X i ) , U ( − X i ) i ( ˜ d i +1 − ˜ d i ) − ( a + b ) θ i − θ i h Ψ( X i − ) , U ( − X i − ) i ( ˜ d i − ˜ d i − ) − λ ˜ d i Z ∞−∞ ∂ y φ ( y ) z ( y ) dy + O ( | λ | ( | λ | + e − αX min ) + e − (2 α + γ ) X min )) , which we write in matrix form as in the statement of the theorem. For Corollary 1, let { µ , . . . , µ n − , } be the eigenvalues of A , which are real and distinctas in the proof of [9, Theorem 5]. Following the steps in that proof and using the rescaling in[11, Theorem 3], there are 2( n −
1) pairs of interaction eigenvalues, given by (31), which areeither real or purely imaginary by Hamiltonian symmetry. Since
M >
M > λ i , ˜ λ i is real and the other is purely imaginary.Corollary 2 is the specific case n = 2, where the nonzero eigenvalue of A can be computeddirectly. Acknowledgments.
This material is based upon work supported by the U.S. National ScienceFoundation under the RTG grant DMS-1840260 (R.P. and A.A.).
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