Modulational instability of periodic standing waves in the derivative NLS equation
MMODULATIONAL INSTABILITY OF PERIODIC STANDING WAVESIN THE DERIVATIVE NLS EQUATION
JINBING CHEN, DMITRY E. PELINOVSKY, AND JEREMY UPSAL
Abstract.
We consider the periodic standing waves in the derivative nonlinear Schr¨odinger(DNLS) equation arising in plasma physics. By using a newly developed algebraic methodwith two eigenvalues, we classify all periodic standing waves in terms of eight eigenvalues ofthe Kaup–Newell spectral problem located at the end points of the spectral bands outsidethe real line. The analytical work is complemented with the numerical approximation ofthe spectral bands, this enables us to fully characterize the modulational instability of theperiodic standing waves in the DNLS equation. Introduction
The derivative nonlinear Schr¨odinger (DNLS) equation arises in a long-wave, weakly non-linear limit from the one-dimensional compressible magnetohydrodynamic equations in thepresence of the Hall effect [34, 35]. This equation is a canonical model for Alfv´en wavespropagating along the constant magnetic field in cold plasmas. It was shown by D. Kaupand A. Newell in [28] that this equation has the same isospectral property as in the canonicalKorteweg-de Vries (KdV) equation considered by P. Lax in [30]. For future reference, wetake the DNLS equation in the following normalized form iu t + u xx + i ( | u | u ) x = 0 , (1.1)where i = √− u ( x, t ) : R × R (cid:55)→ C . The DNLS equation is the compatibility conditionfor the following Lax pair of linear equations: ϕ x = (cid:18) − iλ λu − λ ¯ u iλ (cid:19) ϕ, (1.2)and ϕ t = (cid:18) − iλ + iλ | u | λ u + λ ( iu x − | u | u ) − λ ¯ u + λ ( i ¯ u x + | u | ¯ u ) 2 iλ − iλ | u | (cid:19) ϕ, (1.3)where ¯ u denotes the complex-conjugate of u and ϕ ( x, t ) : R × R (cid:55)→ C . The x -derivativepart (1.2) of the Lax pair is referred to as the Kaup–Newell (KN) spectral problem. Date : September 14, 2020.
Key words and phrases. derivative nonlinear Schr¨odinger equation, periodic standing waves, Kaup-Newellspectral problem, algebraic method, eigenvalues, spectral stability, modulational stability. a r X i v : . [ n li n . S I] S e p JINBING CHEN, DMITRY E. PELINOVSKY, AND JEREMY UPSAL
When the DNLS equation is posed on the real line, the Cauchy problem is locally well-posed in H s ( R ) for s ≥ [39] and ill-posed in H s ( R ) for s < due to lack of the continuousdependence on initial data [1]. If functional-analytic methods are used, global well-posednessof the Cauchy problem can only be shown for u ∈ H s ( R ), s ≥ with small initial data in L ( R ) (see [22, 23, 32] and more recent works [19, 41, 42]). On the other hand, by usingtools of the inverse scattering transform, one can solve the Cauchy problem globally in asubspace of H ( R ) without restricting the L ( R ) norm of the initial data [24, 31, 37, 38].Travelling solitary waves of the DNLS equation are well known due to their importantapplications in plasma physics [28, 34, 35]. These solutions can be expressed as the standingwave u ( x, t ) = φ ω,ν ( x − νt ) e iωt , (1.4)where φ ω,ν is available in the polar form φ ω,ν = R ω,ν e i Θ ω,ν with R ω,ν ( x ) = (cid:18) ω − ν ) √ ω cosh( √ ω − ν x ) − ν (cid:19) / , Θ ω,ν ( x ) = ν x − (cid:90) x −∞ R ω,ν ( y ) dy. (1.5)The speed parameter ν is arbitrary, whereas the frequency parameter ω is restricted underthe constraint 4 ω − ν >
0. Orbital stability of the travelling waves in the energy space H ( R ) was proven for ν < ν ∈ ( −√ ω, √ ω ) [10] (see also recentworks [29, 33]).There are very few results available on the periodic standing wave solutions, which can beexpressed in the form (1.4) with | φ ω,ν ( x + L ) | = | φ ω,ν ( x ) | , x ∈ R , (1.6)for some fundamental period L >
0. Such solutions are generally quasi-periodic when theyare expressed in the polar form φ ω,ν = R ω,ν e i Θ ω,ν with R ω,ν and Θ (cid:48) ω,ν being periodic with thesame period L . The simplest periodic standing wave solutions to the DNLS equation (1.1)can be analyzed directly by separating the variables in the polar form [15]. Convergence ofperiodic waves to the solitary waves (1.5) was shown in [21]. Spectral stability of periodicwaves with non-vanishing φ ω,c was established with respect to perturbations of the sameperiod in [20]. The main purpose of this work is to classify all periodic standing waves of the DNLSequation and to characterize their spectral stability with respect to localized perturbations.We use an algebraic method which allows us to relate the periodic standing waves withsolutions of the complex finite-dimensional Hamiltonian systems.The algebraic method of nonlinearization of linear equations in the Lax pair to finite-dimensional Hamiltonian systems was developed by C.W. Cao and X.G. Geng in the contextof the KdV equation [2]. The finite-dimensional Hamiltonian systems were obtained for theDNLS equation (1.1) in [3, 36, 44]. Quasi-periodic (algebro-geometric) solutions to the DNLSequation have been analyzed using the complex finite-dimensional Hamiltonian systems in[8] (see also [17, 40, 43] for other studies of quasi-periodic solutions in the DNLS equation).
ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 3
In the context of the periodic standing waves, the algebraic method gives the location ofparticular eigenvalues of the KN spectral problem which correspond to bounded periodiceigenfunctions (see [4, 5, 6] for analysis of other integrable equations). These particulareigenvalues play an important role in the study of modulational stability of the standingwaves [26, 27]. Namely, if the Floquet spectrum is obtained in the KN spectral problemanalytically or numerically, then the stability spectrum in the time-evolution of the linearizedDNLS equation is obtained by a simple transformation (see [7, 12] for an application of thistechnique to the NLS equation). Some examples of using this technique for the DNLSequation and other equations can be found in [14].Here we study the periodic standing waves of the DNLS equation comprehensively by usingthe algebraic method with two eigenvalues. In fact, we relate the periodic standing waveswith eight eigenvalues of the KN spectral problem. These eigenvalues could form either fourpairs of purely imaginary eigenvalues or two quadruplets of four eigenvalues in four quadrantsof the complex plane or the mixture of both (two pairs of purely imaginary eigenvalues andone quadruplet of complex eigenvalues). Performing numerical computations of the Floquetspectrum for the spectral bands connecting the eight eigenvalues, we show that all periodicwaves associated with one or two quadruplets of complex eigenvalues are spectrally unstable,whereas the periodic waves associated with four pairs of purely imaginary eigenvalues arespectrally stable. For numerical computations, we use the Hill’s method developed in [11]and rigorously justified in [25].
The main result of this work is the precise characterization of the spectrally stable periodicstanding waves of the DNLS equation. We show that the location of the eight eigenvaluesof the algebraic method on the imaginary axis gives the criterion for the spectral stability ofthe standing waves. In particular, we show that the spectrally stable periodic waves of theform (1.4) correspond to 4 ω < ν , ω > , ν > , (1.7)whereas the periodic waves are spectrally (and modulationally) unstable for every otherparameter choices. Moreover, we show that a narrow region in the parameter space for thestable periodic waves satisfying (1.7) is surrounded by a wider region, where the periodicwaves are unstable.The paper is organized as follows. Properties of eigenvalues of the KN spectral problem arereviewed in Section 2. Section 3 describes the algebraic method with two eigenvalues whichresults in the stationary DNLS equation. Eight roots of the algebraic method characterizeparameters of the stationary DNLS equation and coincide with the eigenvalues of the KNspectral problem. The connection between the eight eigenvalues, the Floquet spectrum, andthe stability spectrum is described in Section 4. Section 5 describes all possible periodicstanding waves and relates them to the location of eight roots of the algebraic method.Numerical results on the Floquet and stability spectra are given in Section 6 for the particularfamily of the periodic standing waves. The paper is concluded with Section 7 where wedescribe open directions of this study. JINBING CHEN, DMITRY E. PELINOVSKY, AND JEREMY UPSAL Properties of eigenvalues for the DNLS equation
The following definition of eigenvalues is used in what follows.
Definition 1.
Assume that u ( x ) = R ( x ) e i Θ( x ) with L -periodic R and Θ (cid:48) . Then λ is calledan eigenvalue of the KN spectral problem (1.2) w.r.t. periodic (anti-periodic) boundary con-ditions if a nonzero eigenvector ϕ = ( p, q ) T is given by p ( x ) = P ( x ) e i Θ( x ) / and q ( x ) = Q ( x ) e − i Θ( x ) / with L -periodic ( L -anti-periodic) P and Q . Remark 1. If P ( x ) and Q ( x ) are L -periodic, then P ( x ) and Q ( x ) could be either L -periodicor L -anti-periodic. The following two propositions describe two symmetries of eigenvalues in the spectralproblem (1.2) related to the DNLS equation (1.1).
Proposition 1.
Assume λ ∈ C \ R is an eigenvalue of the spectral problem (1.2) with theeigenvector ϕ = ( p, q ) T . Then, ¯ λ is also an eigenvalue with the eigenvector ϕ = (¯ q, − ¯ p ) T . If λ ∈ R \{ } is an eigenvalue, then it is at least double with two eigenvectors ϕ = ( p, q ) T and ϕ = (¯ q, − ¯ p ) T .Proof. If ϕ = ( p, q ) T satisfies (1.2), then p x = − iλ p + λuq, q x = − λ ¯ up + iλ q. (2.1)Taking the complex-conjugate equation, we verify that ϕ = (¯ q, − ¯ p ) T satisfies the sameequation (1.2) but with λ replaced by ¯ λ . Since the periodicity properties for ϕ = ( p, q ) T and ϕ = (¯ q, − ¯ p ) T are the same as in Definition 1, if λ is an eigenvalue, then ¯ λ is an eigenvalue.If λ ∈ C \ R , then ¯ λ (cid:54) = λ . If λ ∈ R , then ¯ λ = λ but ϕ = (¯ q, − ¯ p ) T is linearly independentfrom ϕ = ( p, q ) T , because if there is a nonzero constant c ∈ C such that ¯ q = cp and − ¯ p = cq ,then | c | = −
1, a contradiction. Hence, λ is at least a double eigenvalue. (cid:3) Proposition 2.
Assume λ ∈ i R \{ } is a simple eigenvalue of the spectral problem (1.2)with the eigenvector ϕ = ( p, q ) T . Then there is c ∈ C with | c | = 1 such that p = c ¯ q .Proof. If ϕ = ( p, q ) T satisfies (1.2) with λ = iβ , β ∈ R , then p x = iβ p + iβuq, q x = − iβ ¯ up − iβ q. (2.2)Taking the complex conjugate equation, we verify that ϕ = (¯ q, ¯ p ) T satisfies the same equation(1.2) with the same λ = iβ . Since λ = iβ is a simple eigenvalue, then ϕ = (¯ q, ¯ p ) T is linearlydependent on ϕ = ( p, q ) T , so that there is a nonzero constant c ∈ C such that ¯ q = cp and¯ p = cq . The two relations yield the constraint | c | = 1. (cid:3) Remark 2.
The symmetry of eigenvalues and eigenvectors in Propositions 1 and 2 holdsfor the second Lax equation (1.3).
ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 5 Algebraic method with two eigenvalues
In order to develop the algebraic method, it is natural to extend the DNLS equation (1.1)as a reduction v = ¯ u of the following coupled system: (cid:26) iu t + u xx + i ( u v ) x = 0 , − iv t + v xx − i ( uv ) x = 0 . (3.1)The coupled DNLS system (3.1) appears as a compatibility condition of the Lax pair oflinear equations on ϕ ∈ C given by ϕ x = U ϕ, U = (cid:18) − iλ λu − λv iλ (cid:19) , (3.2)and ϕ t = V ϕ, V = (cid:18) − iλ + iλ uv λ u + λ ( iu x − u v ) − λ v + λ ( iv x + uv ) 2 iλ − iλ uv (cid:19) . (3.3)We only consider the KN spectral problem (3.2) and ignore the time-dependent equation(3.3) for now. As a result, we replace partial derivatives in x with ordinary derivatives. Remark 3.
The time evolution of constraints in the algebraic method for the periodic stand-ing waves is trivial (see, e.g., [4, 5] for other integrable equations). The time evolution ofthe eigenvector ϕ = ( p, q ) T satisfying (3.3) is defined in (4.4) below. Assume that ϕ = ( p , q ) T is a solution to the spectral problem (3.2) for λ = λ and ϕ = ( p , q ) T is a solution to the spectral problem (3.2) for λ = λ such that both solutionsare linearly independent. We set the following constraint between the potentials ( u, v ) andthe squared eigenfunctions: (cid:26) u = λ p + λ p ,v = λ q + λ q . (3.4)With the constraints (3.4), the spectral problem (3.2) for λ = λ and λ = λ can be writtenas the complex Hamiltonian system dp dx = − iλ p + λ ( λ p + λ p ) q = − ∂H∂q , dq dx = iλ q − λ ( λ q + λ q ) p = ∂H∂p ,dp dx = − iλ p + λ ( λ p + λ p ) q = − ∂H∂q , dq dx = iλ q − λ ( λ q + λ q ) p = ∂H∂p , generated by the complex-valued Hamiltonian H = iλ p q + iλ p q −
12 ( λ p + λ p )( λ q + λ q ) . (3.5)The complex Hamiltonian system admits another complex conserved quantity F given by F = i ( p q + p q ) , (3.6)where the normalization factor i is used for convenience. JINBING CHEN, DMITRY E. PELINOVSKY, AND JEREMY UPSAL If v = ¯ u , we need to restrict the eigenvalues λ and λ in order to ensure that the conservedquantities H and F are real-valued. This is done in agreement with the symmetries inPropositions 1 and 2. • Let λ ∈ C \ i R and set λ = ¯ λ with p = ¯ q , q = − ¯ p , (3.7)The constraint (3.4) is compatible with the complex-conjugate symmetry (cid:26) u = λ p + ¯ λ ¯ q , ¯ u = λ q + ¯ λ ¯ p , (3.8)whereas H and F in (3.5) and (3.6) become real-valued: H = i ( λ p q − ¯ λ ¯ p ¯ q ) − (cid:12)(cid:12) λ p + ¯ λ ¯ q (cid:12)(cid:12) (3.9)and F = i ( p q − ¯ p ¯ q ) . (3.10) • Let λ , λ ∈ i R such that λ (cid:54) = ± λ and set λ = iβ , q = − i ¯ p and λ = iβ , q = − i ¯ p . (3.11)The constraint (3.4) is compatible with the complex-conjugate symmetry (cid:26) u = iβ p + iβ p , ¯ u = − iβ ¯ p − iβ ¯ p , (3.12)whereas H and F in (3.5) and (3.6) become real-valued: H = − β | p | − β | p | − (cid:12)(cid:12) β p + β p (cid:12)(cid:12) (3.13)and F = | p | + | p | . (3.14)Let us now derive and integrate the differential equations on ( u, v ) from compatibility ofthe constraint (3.4) with the Hamiltonian system generated by the Hamiltonian (3.5). Fromnow on, we use the complex-conjugate reduction v = ¯ u in all subsequent computations,hence, we only use the constraints (3.8) and (3.12).By taking one derivative of the constraint (3.4) and using (3.5), we obtain the first-orderequation on u , dudx + i | u | u + 2 iHu + 2 i ( λ p + λ p ) = 0 . (3.15)The first-order equation (3.15) is not closed on u . However, by taking another derivative of(3.15), using (3.5), (3.6), and (3.15), we obtain the closed second-order differential equation d udx + i ddx ( | u | u ) + 2 ic dudx − bu = 0 , (3.16) ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 7 where we have introduced two real-valued parameters b = λ λ (1 + F ) , c = λ + λ + H. (3.17) Remark 4.
The second-order equation (3.16) arises in the standing wave reduction of theDNLS equation (1.1) for the solutions of the form u ( x, t ) = ˜ u ( x + 2 ct ) e ibt , (3.18) where ˜ u satisfies (3.16) with tilde notations dropped. It follows from [8] that the complex Hamiltonian system generated by the Hamiltonian(3.5) is equivalent to the Lax equation ddx
Ψ = [ U, Ψ] , (3.19)where U is obtained from (3.2) with u and v given by (3.4) and Ψ is given byΨ := (cid:18) Ψ Ψ Ψ − Ψ (cid:19) , (3.20)where Ψ = − i − λ p q λ − λ − λ p q λ − λ , (3.21)Ψ = λ (cid:18) λ p λ − λ + λ p λ − λ (cid:19) , (3.22)Ψ = − λ (cid:18) λ q λ − λ + λ q λ − λ (cid:19) . (3.23)It follows from (3.5), (3.6), and (3.21)–(3.23) thatdet Ψ = − Ψ − Ψ Ψ = 1 − Hλ − λ λ F ( F + 2)( λ − λ )( λ − λ ) . (3.24) Remark 5.
It follows from (3.24) that if λ (cid:54) = ± λ , then det Ψ only contains simple polesat ( ± λ , ± λ ) with the residue terms being independent of x . By using (3.4) – (3.6), (3.15), and (3.17), the entries of the Lax matrix Ψ can be rewrittenin terms of ( u, ¯ u ) byΨ = − i ( λ − λ )( λ − λ ) (cid:20) λ − λ (cid:18) c + 12 | u | (cid:19) + b (cid:21) , (3.25)Ψ = λ ( λ − λ )( λ − λ ) (cid:20) λ u + i dudx − u | u | − cu (cid:21) , (3.26)Ψ = − λ ( λ − λ )( λ − λ ) (cid:20) λ ¯ u − i d ¯ udx −
12 ¯ u | u | − c ¯ u (cid:21) . (3.27) JINBING CHEN, DMITRY E. PELINOVSKY, AND JEREMY UPSAL
The (1 , − Ψ − Ψ Ψ = P ( λ )( λ − λ ) ( λ − λ ) , (3.28)where P ( λ ) is the eight-degree polynomial given by P ( λ ) = (cid:20) λ − λ (cid:18) c + 12 | u | (cid:19) + b (cid:21) + λ (cid:20) λ u + i dudx − u | u | − cu (cid:21) (cid:20) λ ¯ u − i d ¯ udx −
12 ¯ u | u | − c ¯ u (cid:21) . (3.29)Since det Ψ is x independent, then P ( λ ) is x independent. The coefficients of P ( λ ) are x independent if and only if solutions to the second-order equation (3.16) also satisfy thefollowing two first-order invariants2 i (cid:18) ¯ u dudx − u d ¯ udx (cid:19) − | u | − c | u | = 4 a, (3.30)2 (cid:12)(cid:12)(cid:12)(cid:12) dudx (cid:12)(cid:12)(cid:12)(cid:12) − | u | − c | u | − a + 2 b ) | u | = 8 d, (3.31)where a and d are two real-valued parameters in addition to real-valued parameters b and c of the second-order equation (3.16). Substituting (3.30) and (3.31) into (3.29) yields P ( λ ) = λ − cλ + ( a + 2 b + c ) λ + ( d − c ( a + 2 b )) λ + b . (3.32) Remark 6.
Since det Ψ in (3.24) has only simple poles at ( ± λ , ± λ ) if λ (cid:54) = ± λ , the twoeigenvalues λ and λ of the algebraic method must be chosen from the roots of P ( λ ) . Similarly to the expressions (3.17), we shall relate parameters a and d to λ , λ , H , and F . By equating (3.24) and (3.28) and substituting (3.32) for P ( λ ), we derive coefficients foreven powers of λ . The coefficient of λ is satisfied identically. The coefficients of λ and λ recover relations (3.17) for parameters c and b respectively, and the coefficients of λ and λ yield the following relations for parameters a and d respectively, a = λ λ F − H , d = λ λ F H ( F + 2) − H ( λ + λ + H ) . (3.33)The polynomial P ( λ ) in (3.32) has generally four pairs of distinct roots, two of which mustbe chosen as the eigenvalues λ and λ of the algebraic method satisfying either the reduction λ = ¯ λ if Re( λ ) (cid:54) = 0 or the reduction λ , λ ∈ i R if λ (cid:54) = ± λ . We label the four pairsof distinct roots of P ( λ ) as {± λ , ± λ , ± λ , ± λ } , where the complementary eigenvalues λ and λ are not used in the algebraic method. The polynomial P ( λ ) can be factorized by itsroots as P ( λ ) = ( λ − λ )( λ − λ )( λ − λ )( λ − λ ) . (3.34) ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 9
It follows by expanding (3.34) in even powers of λ and comparing it with (3.32) that λ + λ + λ + λ = 2 c, ( λ + λ )( λ + λ ) + λ λ + λ λ = a + 2 b + c ,λ λ ( λ + λ ) + λ λ ( λ + λ ) = ac + 2 bc − d,λ λ λ λ = b . (3.35)It follows from (3.17) and the first equation of (3.35) that H = 12 ( λ + λ − λ − λ ) . (3.36)It follows from the last equation of (3.35) that two cases are possible for b : either b = λ λ λ λ or b = − λ λ λ λ . The second choice, however, follows from the first one byreplacing λ (cid:55)→ − λ , so we will only consider the case b = λ λ λ λ . It follows from (3.17)with b = λ λ λ λ that F = λ λ λ λ − . (3.37)Substituting (3.36) and (3.37) into (3.17) and (3.33) allows us to express parameters a , b , c ,and d in terms of the eigenvalues { λ , λ , λ , λ } : a = − [( λ + λ ) − ( λ + λ ) ][( λ − λ ) − ( λ − λ ) ] ,b = λ λ λ λ ,c = ( λ + λ + λ + λ ) ,d = − ( λ + λ − λ − λ )( λ − λ + λ − λ )( λ − λ − λ + λ ) . (3.38)We have checked that these expressions recover all four equations of system (3.35).To summarize the outcome of the algebraic method, the standing waves of the DNLSequation (1.1) of the form (3.18) satisfy the second-order equation (3.16) and the first-orderreductions (3.30) and (3.31) with four parameters a , b , c , and d . These parameters generallydetermine four distinct pairs of roots of the polynomial P ( λ ) in (3.32) and (3.34). Theconnection formulas (3.35) are inverted in the form (3.38). Picking any two distinct rootsof the polynomial P ( λ ) as two eigenvalues λ and λ of the algebraic method allows us torelate the standing wave of the form (3.18) to squared eigenfunctions of the KN spectralproblem by either (3.8) if λ = ¯ λ with Re( λ ) (cid:54) = 0 or (3.12) if λ , λ ∈ i R with λ (cid:54) = ± λ . Ifthe standing wave is L -periodic, so are the squared eigenfunctions due to relations (3.4) and(3.15). Then, the eigenvectors ϕ = ( p , q ) T and ϕ = ( p , q ) T for the eigenvalues λ and λ are either L -periodic or L -anti-periodic.4. Modulational instability of periodic waves
Spectral stability of the standing waves of the form (3.18) in the time evolution of theDNLS equation (1.1) can be studied by adding a perturbation w of the form u ( x, t ) = e ibt [˜ u ( x + 2 ct ) + w ( x + 2 ct, t )] . (4.1) Substituting (4.1) into (1.1) and dropping the quadratic terms in w yields the linearizedsystem of equations (cid:26) iw t − bw + 2 icw x + w xx + i [2 | u | w x + u ¯ w x + 2( u ¯ u x + u x ¯ u ) w + 2 uu x ¯ w ] = 0 , − i ¯ w t − b ¯ w − ic ¯ w x + ¯ w xx − i [2 | u | ¯ w x + ¯ u w x + 2( u ¯ u x + u x ¯ u ) ¯ w + 2¯ u ¯ u x w ] = 0 , (4.2)where the tilde notation for u has been dropped as before. Variables can be separated in thelinearized system (4.2) by w ( x, t ) = w ( x ) e t Λ , ¯ w ( x, t ) = w ( x ) e t Λ , (4.3)where w (cid:54) = ¯ w if Λ / ∈ R . Our goal is to find the admissible values of Λ for which w and w are bounded functions of x on R . By the Floquet-Bloch theory, the admissible valuesof Λ form continuous spectral bands on the complex Λ-plane [16]. We give the followingdefinition of spectral and modulational instability of the standing wave. Definition 2.
If there exists Λ with Re(Λ) > for which ( w , w ) ∈ L ∞ ( R ) in (4.2) and(4.3), then the standing wave of the form (3.18) is called spectrally unstable. It is calledmodulationally unstable if the unstable spectral band in the Λ -plane intersects the origin. There exists an explicit relation between the admissible values of Λ and solutions of theLax equations (1.2) and (1.3). By substituting the standing waves of the form (3.18) intothe Lax equations (1.2) and (1.3) and separating the variables in the form ϕ ( x, t ) = e ibtσ ˜ ϕ ( x + 2 ct, t ) , (4.4)where σ = diag(1 , − ϕ x = U ϕ, ϕ t + 2 ibσ ϕ + 2 cϕ x = V ϕ, (4.5)where U = (cid:18) − iλ λu − λ ¯ u iλ (cid:19) , V = (cid:18) − iλ + iλ | u | λ u + λ ( iu x − | u | u ) − λ ¯ u + λ ( i ¯ u x + | u | ¯ u ) 2 iλ − iλ | u | (cid:19) , (4.6)and the tilde notations for ϕ and u have been dropped again. We note that potentials U and V in (4.6) are t -independent since the transformed solution u (former ˜ u ) is a function of x only. The following proposition summarizes the result obtained in [9]. It is verified hereby explicit computations. Proposition 3.
Let ϕ = ( ϕ , ϕ ) T be the eigenvector of the Lax system (4.5) for the eigen-value λ ∈ C . Then the perturbation w satisfying the linearized DNLS equation (4.2) isexpressed by w = ∂ x ϕ , ¯ w = ∂ x ϕ . (4.7) Consequently, if ϕ ( t, x ) = χ ( x ) e t Ω , then w = ∂ x χ , w = ∂ x χ , and the eigenvalue Λ = 2Ω in (4.3) is related to λ by Λ = ± i (cid:112) P ( λ ) , (4.8) where P ( λ ) is given by (3.32). ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 11
Proof.
By the linear superposition principle, it suffices to show that w = − iλϕ + uϕ ϕ , ¯ w = iλϕ − ¯ uϕ ϕ , (4.9)satisfies the linearized DNLS equation (4.2) if ϕ = ( ϕ , ϕ ) T satisfies the Lax equations (4.5).The two terms in (4.9) are inspected separately as follows: i∂ t ( ϕ ) − bϕ + 2 ic∂ x ( ϕ ) + ∂ x ( ϕ )+ i [2 | u | ∂ x ( ϕ ) − u ∂ x ( ϕ ) + 2( u ¯ u x + u x ¯ u ) ϕ − uu x ϕ ]= 4 u ϕ λ + 4 i | u | uϕ ϕ λ + 2 i ( u ¯ u x + u x ¯ u ) ϕ − iuu x ϕ , and i∂ t ( uϕ ϕ ) − buϕ ϕ + 2 ic∂ x ( uϕ ϕ ) + ∂ x ( uϕ ϕ )+ i [2 | u | ∂ x ( uϕ ϕ ) − u ∂ x (¯ uϕ ϕ ) + 2( u ¯ u x + u x ¯ u ) uϕ ϕ − uu x ¯ uϕ ϕ ]= 4 iu ϕ λ − | u | uϕ ϕ λ − u ¯ u x + u x ¯ u ) ϕ λ + 2 uu x ϕ λ. Summing the first equality multiplied by ( − iλ ) and the second equality yields zero whichverifies the relations (4.9).In order to show (4.8), we recall that the potentials U and V in (4.6) are t -independent.Therefore, we can separate the variables in the form ϕ ( t, x ) = χ ( x ) e t Ω and obtain Ω fromthe characteristic equation det(Ω + 2 ibσ + 2 cU − V ) = 0 . (4.10)By expanding the determinant and using first-order invariants (3.30) and (3.31), we verifythat the characteristic equation (4.10) is equivalent toΩ + 4 P ( λ ) = 0 , (4.11)which yields (4.8) after extracting the square root. (cid:3) Remark 7.
Roots of the polynomial P ( λ ) are mapped to the origin of the Λ plane and belongsto the stability spectrum. The following two propositions state explicitly the stability results on the DNLS equationwhich follow from Theorem 9 in [14] (see also their Section 6.1).
Proposition 4.
Assume that P ( λ ) is given by (3.34) with the roots ( ± λ , ± λ , ± λ , ± λ ) ∈ C \ R and λ ∈ R . Then, Λ ∈ i R .Proof. It follows from (3.34) that if ( ± λ , ± λ , ± λ , ± λ ) ∈ C \ R , then P ( λ ) > λ ∈ R . This implies that Λ ∈ i R in (4.8). (cid:3) Proposition 5.
Assume that P ( λ ) is given by (3.34) with the roots ( ± λ , ± λ , ± λ , ± λ ) and λ ∈ i R . If the roots form complex quadruplets or double real eigenvalues, then Λ ∈ i R in (4.8). If two pairs of roots are purely imaginary, e.g. λ , = iβ , with < β < β , then Λ ∈ i R if Im( λ ) ∈ ( −∞ , − β ] ∪ [ − β , β ] ∪ [ β , ∞ ) . (4.12) If four pairs of roots are purely imaginary, e.g. λ , , , = iβ , , , with < β < β < β < β ,then Λ ∈ i R if Im( λ ) ∈ ( −∞ , − β ] ∪ [ − β , − β ] ∪ [ − β , β ] ∪ [ β , β ] ∪ [ β , ∞ ) . (4.13) Proof.
For λ ∈ i R , we can rewrite (3.34) in the form: P ( z ) = ( z + λ )( z + λ )( z + λ )( z + λ ) , (4.14)where z = − λ and the notation for P ( z ) is overwritten. If the roots form complex quadru-plets or double real eigenvalues, then P ( z ) > z > ∈ i R follows from(4.8). If two or four pairs of purely imaginary eigenvalues occur, then P ( z ) > z ∈ (0 , β ) ∪ ( β , ∞ ) or z ∈ (0 , β ) ∪ ( β , β ) ∪ ( β , ∞ ), respectively. This gives the stabilityconditions (4.12) and (4.13) respectively. (cid:3) Remark 8.
It was also proven in Theorem 9 in [14] that if Λ ∈ i R for a given λ ∈ R ∪ i R ,then λ ∈ R ∪ i R belongs to the Floquet spectrum of the KN spectral problem (1.2). ByPropositions 4 and 5, this implies that R ∪ i R \ S belongs to the Floquet spectrum of the KNspectral problem (1.2), where S ⊂ i R includes either two or four spectral gaps in (4.12) and(4.13), respectively. To summarize the separation of variables for the time-dependent problem, if we computethe admissible values of λ in the Floquet spectrum of the Lax system (4.5) for the standingwaves of the form (3.18), then we can obtain the admissible values of Λ in the spectral bandsof the stability problem using the transformation (4.8). By Propositions 4 and 5, spectralinstability of the standing waves may only arise if there are admissible values of λ in openquadrants of the complex plane or on the imaginary axis in the complement of the intervalsin (4.12) and (4.13).5. Classification of periodic standing waves
Here we characterize the periodic standing waves from analysis of solutions to the second-order equation (3.16) closed with the first-order invariants (3.30) and (3.31).We use the polar form u ( x ) = R ( x ) e i Θ( x ) with real-valued R ( x ) and Θ( x ) for the periodicstanding waves. The polar form is non singular if R ( x ) > x . Substituting the polarform into the first-order invariants (3.30) and (3.31) leads to4 R d Θ dx + 3 R + 4 cR = − a ⇒ d Θ dx = − aR − R − c, (5.1)and 2 (cid:18) dRdx (cid:19) + 2 R (cid:18) d Θ dx (cid:19) − R − cR − a + 2 b ) R = 8 d. (5.2)Inserting (5.1) into (5.2) yields the first-order quadrature: (cid:18) dRdx (cid:19) + a R + 116 R + c R + R (cid:16) c − b − a (cid:17) + 2 ac − d = 0 . (5.3) ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 13
Two cases are distinguished here: a (cid:54) = 0 and a = 0. In the remainder of this section, wewill consider the general case a (cid:54) = 0 and obtain the periodic solutions in the explicit form.In the following section, we will set a = 0 and investigate the periodic solutions and theirmodulational instability in more details.If a (cid:54) = 0, the singularity R = 0 of the quadrature (5.3) is unfolded with the transformation ρ = R which yields (cid:18) dρdx (cid:19) + Q ( ρ ) = 0 , (5.4)where Q ( ρ ) is the quartic polynomial given by Q ( ρ ) = ρ + 4 cρ + 2(2 c − a − b ) ρ + 4( ac − d ) ρ + a . (5.5)The polynomial Q ( ρ ) can be factorized by its roots ( u , u , u , u ) as Q ( ρ ) = ( ρ − u )( ρ − u )( ρ − u )( ρ − u ) . (5.6)Equating coefficients of the same powers in (5.5) and (5.6) yields u + u + u + u = − c,u u + u u + u u + u u + u u + u u = 2(2 c − a − b ) ,u u u + u u u + u u u + u u u = 4(2 d − ac ) ,u u u u = a . (5.7)Recall that the parameters a , b , c , and d are related to the roots ( ± λ , ± λ , ± λ , ± λ ) ofthe polynomial P ( λ ) by using (3.35) and (3.38). The following proposition shows that theroots of P ( λ ) are related to the roots of Q ( ρ ) by using simple and explicit expressions. Proposition 6.
Let ( ± λ , ± λ , ± λ , ± λ ) be the roots of P ( λ ) in (3.34) and ( u , u , u , u ) be roots of Q ( ρ ) in (5.6). Then u = − ( λ − λ + λ − λ ) ,u = − ( λ − λ − λ + λ ) ,u = − ( λ + λ − λ − λ ) ,u = − ( λ + λ + λ + λ ) . (5.8) Proof.
If the roots ( u , u , u , u ) are related to the roots {± λ , ± λ , ± λ , ± λ } by (5.8), thenthe last equation of system (5.7) yields the first equation of system (3.38) after extractingthe negative square root. The first equation of (5.7) yields the third equation of (3.38), u + u + u + u = − λ + λ + λ + λ ) = − c. Similarly, the second equation of (5.7) is compatible with system (3.38),( u + u )( u + u ) + u u + u u = ( λ − λ ) + ( λ − λ ) + 2( λ + λ )( λ + λ ) − λ λ λ λ + 12 ( λ + 6 λ λ + λ ) + 12 ( λ + 6 λ λ + λ ) − ( λ + λ )( λ + λ ) − λ λ λ λ = ( λ + λ + λ + λ ) − λ λ λ λ + 12 [( λ + λ ) − ( λ + λ ) ][( λ − λ ) − ( λ − λ ) ]= 4 c − a − b. Compatibility of the third equation of (5.7) is checked with Wolfram’s Mathematica. (cid:3)
Because the coefficients of Q are real-valued, we have three cases to consider: (i) fourroots of Q are real, (ii) two roots of Q are real and one pair of roots is complex-conjugate,and (iii) two pairs of roots of Q are complex-conjugate. Each case is considered separately.5.1. Four roots of Q are real. For simplicity, we order the four real roots of Q as u ≤ u ≤ u ≤ u . (5.9)Periodic solutions to the quadrature (5.4) with (5.6) and (5.9) can be expressed explicitly(see, e.g., [6]) by ρ ( x ) = u + ( u − u )( u − u )( u − u ) + ( u − u )sn ( µx ; k ) , (5.10)where positive parameters µ and k are uniquely expressed by µ = 12 (cid:112) ( u − u )( u − u ) , k = (cid:112) ( u − u )( u − u ) (cid:112) ( u − u )( u − u ) . (5.11)The periodic solution ρ in (5.10) is located in the interval [ u , u ] and has period L =2 K ( k ) µ − . The solution is meaningful for ρ = R ≥ u ≥
0. The four pairsof eigenvalues {± λ , ± λ , ± λ , ± λ } generate real roots of Q if and only if they satisfy thefollowing three configurations: (i) they form two complex quadruplets; (ii) they form fourpairs of purely imaginary eigenvalues; or (iii) they form four pairs of real eigenvalues. Eachcase is considered separately.Assume that the four pairs of eigenvalues {± λ , ± λ , ± λ , ± λ } form two complex quadru-plets with λ = ¯ λ = α + iβ , λ = ¯ λ = α + iβ . (5.12)Then the roots ordered as (5.9) satisfy the more precise ordering u ≤ u ≤ ≤ u ≤ u . (5.13) ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 15 If α , α , β , β are all positive, so that λ and λ are located in the first quadrant, we deducethe explicit expressions (cid:26) α = √ ( √− u + √− u ) ,α = √ ( √− u − √− u ) , (cid:26) β = √ ( √ u + √ u ) ,β = √ ( √ u − √ u ) , (5.14)so that α ≤ α and β ≤ β .Assume that the four pairs of eigenvalues {± λ , ± λ , ± λ , ± λ } are purely imaginary with λ = iβ , λ = iβ , λ = iβ , λ = iβ . (5.15)Then the roots ordered as (5.9) satisfy the more precise ordering0 ≤ u ≤ u ≤ u ≤ u . (5.16)If β , β , β , β are all positive, we deduce the explicit expressions β = √ ( √ u + √ u + √ u + √ u ) ,β = √ ( √ u + √ u − √ u − √ u ) ,β = √ ( √ u − √ u + √ u − √ u ) ,β = √ ( √ u − √ u − √ u + √ u ) , (5.17)so that β ≤ β ≤ β ≤ β .Finally, if we assume that all pairs of eigenvalues {± λ , ± λ , ± λ , ± λ } are real, the rootssatisfy u ≤ u ≤ u ≤ u ≤ . (5.18)The solutions are not meaningful in this case as they do not give ρ = R ≥ Case: two roots of Q are real and one pair of roots is complex-conjugate. Let u , be real roots ordered as u ≤ u and u , = γ ± iη be complex-conjugate roots with u ≤ u , u = γ + iη, u = γ − iη. (5.19)Periodic solutions to the quadrature (5.4) with (5.6) and (5.19) can be expressed explicitly(see, e.g., [6]) by ρ ( x ) = u + ( u − u )(1 − cn( µx ; k ))1 + δ + ( δ − µx ; k ) , (5.20)where positive parameters δ , µ , and k are uniquely expressed by δ = (cid:112) ( u − γ ) + η (cid:112) ( u − γ ) + η , µ = (cid:112) [( u − γ ) + η ] [( u − γ ) + η ] , (5.21)and 2 k = 1 − ( u − γ )( u − γ ) + η (cid:112) [( u − γ ) + η ] [( u − γ ) + η ] . (5.22)The periodic solution ρ in (5.20) is located in the interval [ u , u ] and has period L =4 K ( k ) µ − . The solution is meaningful for ρ = R ≥ u ≥
0. The four pairs of eigenvalues {± λ , ± λ , ± λ , ± λ } generate the two roots and one pair of complex-conjugate roots of Q if and only if they satisfy the following two configurations: (i) theyform one complex quadruplet and two pairs of purely imaginary roots; or (ii) they form onecomplex quadruplet and two pairs of real eigenvalues. Each case is considered separately.Assume that the four pairs of eigenvalues {± λ , ± λ , ± λ , ± λ } form one quadruplet {± λ , ± ¯ λ } of complex eigenvalues and two pairs {± λ , ± λ } of purely imaginary eigen-values. Then, we have 0 ≤ u ≤ u and u = ¯ u . By writing λ = ¯ λ = α + iβ , λ = iβ , λ = iβ . (5.23)with positive α , β , β , β and β > β , we deduce the explicit expressions (cid:40) α = (cid:113)(cid:112) γ + η − γ,β = √ ( √ u + √ u ) , β = η (cid:113) √ γ + η − γ + √ ( √ u − √ u ) ,β = η (cid:113) √ γ + η − γ − √ ( √ u − √ u ) , (5.24)so that β ≤ β . Since u ≥
0, the periodic solution (5.20) is meaningful for ρ = R ≥ {± λ , ± ¯ λ } of complexeigenvalues and two pairs {± λ , ± λ } of real eigenvalues. Then u = ¯ u and u ≤ u ≤ ρ = R ≥ Case: two pairs of roots of Q are complex-conjugate. In the case of no realroots of Q , we have Q ( ρ ) > ρ ∈ R . There exist no periodic wave solutions to thequadrature (5.4) with Q ( ρ ) > ρ . Hence, this case doesnot result in periodic wave solutions.6. Periodic standing waves in the case of a = 0The family of periodic standing waves u ( x ) = R ( x ) e i Θ( x ) can be made explicit in the case a = 0. This case for the NLS equation is referred to as the waves of trivial phase in [12] (seealso [7]). For the DNLS equation, the phase is still non trivial due to the dependence of Θfrom R in (5.1). The case of a = 0 was the only case of periodic standing wave solutions ofthe DNLS equation considered in [20].It follows from (5.3) with a = 0 that the amplitude function R satisfies the quadrature (cid:18) dRdx (cid:19) + F ( R ) = 4 d, (6.1)where F ( R ) = 116 R + c R + ( c − b ) R . (6.2)Introducing again ρ := R and abusing notations for F , we can rewrite (6.2) in the form F ( ρ ) = 12 ρ + 2 cρ + 2( c − b ) ρ. (6.3) ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 17
One root of the cubic polynomial F ( ρ ) is at zero and the other two roots are given by ρ ± = − c ± √ b. (6.4)The graph of F versus ρ is shown in Fig. 1 for c < b (left) and for c > b , c <
0, and b > ρ − < < ρ + and 0 < ρ − < ρ + , respectively.The other cases of c > b and c > c > b , c <
0, and b < ρ − < ρ + < ρ ± , respectively, so that F ( ρ ) > ρ > -4 -2 0 2 4-10010203040 F 4 d + - -2 0 2 4 6 8 10-40-20020406080 4 d + - Figure 1.
The graph of F versus ρ in (6.3) for c < b (left) and for c > b , c <
0, and b > -3 -2 -1 0 1 2 3 R -3-2-10123 R' -4 -2 0 2 4 R -6-4-20246 R' Figure 2.
Phase portrait in the phase plane (
R, R (cid:48) ) for c < b (left) and for c > b , c <
0, and b > If c < b , sign-definite periodic solutions exist for d ∈ ( d − , d − := min ρ ∈ [0 , ∞ ) F ( ρ )(see the left panel of Fig. 1). As d → d − , the family of periodic solutions degenerates tothe constant-amplitude solution. As d →
0, the family of periodic solutions approaches thesolitary wave satisfying R ( x ) → | x | → ∞ , which corresponds to the exact solution (1.4)with (1.5). Sign-indefinite periodic solutions exist for d ∈ (0 , d + ) and d ∈ ( d + , ∞ ), where d + := max ρ ∈ ( −∞ , F ( ρ ). It should be noted that R ( x ) in the quadrature (6.1) is allowed to benegative but both positive and negative values of R correspond to positive values of ρ = R .Also note that the local maximum point d + occurring for ρ ∈ ( −∞ ,
0) affects the analyticalconstruction of the periodic solutions but does not change the qualitative behavior of R . Aphase portrait for the quadrature (6.1) with c < b is shown on the phase plane ( R, R (cid:48) ) inFig. 2 (left).If c > b and either c > c < b <
0, then F ( ρ ) > ρ >
0. Sign-indefiniteperiodic solutions exist for every d > c < b with d ∈ (0 , ∞ ), therefore, we will not considerexamples of such periodic solutions for these parameter ranges.If c > b , c <
0, and b >
0, sign-definite periodic solutions exist for d ∈ ( d − , d + ), andsign-indefinite periodic solutions exist for d ∈ (0 , d + ) and d ∈ ( d + , ∞ ) (see the right panelof Fig. 1). When d → d − , the family of sign-definite periodic solutions degenerates to theconstant-amplitude solution. As d → d + , the first and third families approach the solitarywave satisfying R ( x ) → R as | x | → ∞ with R > F ( R ) = 4 d with F ( R )given by (6.2). The second family approaches the kink solution satisfying R ( x ) → ± R as x → ±∞ . Phase portrait for the quadrature (6.1) with c > b , c <
0, and b >
R, R (cid:48) ) in Fig. 2 (right).If a = 0, one root of Q in (5.5) is zero. The other three roots are given by the intersectionof the graph of F ( ρ ) given by (6.3) with the constant level 4 d . In the remainder of thissection we study the two cases (i) c < b and (ii) c > b , c <
0, and b >
0. In each case wegive exact analytical expressions for the periodic wave solutions and create representativefigures of the Floquet spectrum in the KN spectral problem (1.2) using the numerical Hill’smethod [11, 25]. This allows us to study the modulational stability or instability of theperiodic standing waves using the correspondence (4.8).6.1.
Case: c < b . If d ∈ ( d − ,
0) (see the left panel of Fig. 1), then the roots of Q are allreal and ordered as u < u < u < u . The exact analytical expression for the periodic wave solutions is given by (5.10) for ρ in[ u , u ]. The period of the periodic wave is L = 2 K ( k ) µ − . The roots of P ( λ ) in (3.34) formtwo quadruplets of complex-conjugate eigenvalues in (5.12) with α = α in (5.14).As d → d − , we have u → u and β → d →
0, we have u → u = 0 and β → β in (5.14), hence two quadruplets ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 19 − . − . . . . − . − . − . − . . . . . . (a) u = 0 . , u = 0 . u = 0, u = − . − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . − . − . − . − . . . . . . (b) u = 1 . , u = 0 . u = 0, u = − . − . − . − . − . . . . . . − . − . − . − . . . . . . (c) u = 1 . , u = 0 . u = 0, u = − . − . − . − . . . . . − . − . − . . . . . (d) u = 3 . , u = 0 . u = 0, u = − . Figure 3.
Numerical computations of the Floquet (left) and stability (right)spectra on the complex λ and Λ planes (Re. vs. Im.) for the four representativecases found for c < b and d ∈ ( d − , P ( λ ). coalesce in the complex plane to a single quadruplet. This corresponds to the solitary wave(1.4)–(1.5).For d ∈ ( d − , d ∈ (0 , d + ), the roots of Q ( ρ ) are all real and ordered by u < u < u < u . The exact analytical expression for the periodic wave solutions is given by (5.10) for ρ in[0 , u ]. However, the case when ρ ( x ) may vanish corresponds to the case of the sign-indefinite R ( x ). If u = 0 is used in the expression (5.10), the expression can be written as ρ ( x ) = u cn ( µx ; k )1 + | u | − u sn ( µx ; k ) . (6.5)Extracting the square root analytically yields the exact expression for the periodic wavesolutions, R ( x ) = √ u cn( µx ; k ) (cid:112) | u | − u sn ( µx ; k ) . (6.6)The period of the periodic wave is now L = 4 K ( k ) µ − (which is double compared to the caseof sign-definite solutions). The roots of P ( λ ) in (3.34) form two quadruplets of complex-conjugate eigenvalues in (5.12) with β = β in (5.14).For d ∈ (0 , d + ), we find four typical configurations for the Floquet spectrum shown inFigure 4. Each case is similar to one in Fig. 3.As d → d + , we have u → u and α → d ∈ ( d + , ∞ ) but now corresponds to the case of two real roots of Q ( ρ ) with0 = u < u and a pair of complex-conjugate roots u , = γ ± iη . The exact analyticalexpression for the periodic wave is given by (5.20) for ρ in [0 , u ]. Again, the case when ρ ( x ) may vanish corresponds to the case of the sign-indefinite R ( x ). If u = 0 is used in theexpression (5.20), the expression can be written as ρ ( x ) = u δ µx ; k )1 + δ + ( δ − µx ; k ) . (6.7) ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 21 − . − . − . − . . . . . . − . − . − . − . . . . . . (a) u = 0 . u = 0 , u = − . u = − − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . − . − . − . − . . . . . . (b) u = 1 . u = 0 , u = − . u = − . − . − . − . . . . . − . − . − . . . . . (c) u = 1, u = 0 , u = − . u = − − . − . − . − . . . . . . − . − . − . − . . . . . . (d) u = 1 . u = 0 , u = − . u = − . Figure 4.
The same as Figure 3 but for c < b and d ∈ (0 , d + ). Extracting the square root analytically yields the exact expression for the periodic wavesolutions, R ( x ) = √ u δ cn( µx ; k ) (cid:113) δ cn ( µx ; k ) + sn ( µx ; k )dn ( µx ; k ) . (6.8)The period of the periodic wave is L = 8 K ( k ) µ − (double compared to the case of sign-definite solutions). The roots of P ( λ ) in (3.34) form one quadruplet of complex-conjugateeigenvalues ( ± λ , ± ¯ λ ) and two pairs of purely imaginary eigenvalues ( ± iβ , ± iβ ) as in(5.23). − . − . − . − . . . . . . − . − . − . − . . . . . . (a) u = 1 . u = 0, u = − . − . i , u = − . . i . − . − . − . − . . . . . . − . − . − . − . . . . . . (b) u = 3 . u = 0, u = − . . i , u = − . − . i .(c) u = 8, u = 0, u = − . . i , u = − . − . i . Figure 5.
The same as Figure 3 but for c < b and d ∈ ( d + , ∞ ). ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 23
For d ∈ ( d + , ∞ ), we find three typical configurations for the Floquet spectrum shown inFigure 5. All three have two spectral gaps on the imaginary axis. The three configurationsdiffer by: (a) has two spectral bands intersecting the real axis, (b) has two spectral bandsintersecting the imaginary axis in the inner spectral band on the imaginary axis, and (c)has two spectral bands intersecting the imaginary axis in the spectral gaps. The stabilityspectrum in (a) and (b) cases represent a single figure 8. The stability spectrum in (c) isa different (new) shape, which was not seen for the periodic standing waves in the NLSequation. The gap on the imaginary axis satisfies the stability condition (4.12). However,each periodic wave has a complex band connected to the origin and hence is modulationallyunstable according to Definition 2.6.2. Case: c > b , c < , and b > . If d ∈ ( d − ,
0) (see the right panel of Fig. 1), thenthe roots of Q are all real and ordered as u < u < u < u . The exact analytical expression for the periodic wave solutions is given by (5.10) for ρ in[ u , u ] with the period L = 2 K ( k ) µ − . The roots of P ( λ ) in (3.34) form two quadruplets ofcomplex-conjugate eigenvalues in (5.12) with α = α in (5.14). This case leads to similarfigures of the Floquet and stability spectra as those in Figure 3. All periodic standing wavesare modulationally unstable.When d → d − , we have u → u and one quadruplet coalesces on the real axis. Thiscorresponds to the constant-amplitude wave. When d →
0, we have u → u = 0 and bothquadruplets coalesce on the imaginary axis. At this point, the sign-definite periodic solutionis continued for d ∈ (0 , d + ) but another sign-indefinite periodic solution arises.If d ∈ (0 , d + ), then the roots of Q are ordered as u = 0 < u < u < u . As is described above, two periodic solutions coexist: one sign-definite solution is givenby (5.10) for ρ ∈ [ u , u ] and the other sign-indefinite solution is given by (5.10) after theexchange u with u and u with u for ρ ∈ [0 , u ]. Extracting the square root analyticallyyields the sign-indefinite solution in the exact form, R ( x ) = √ u cn( µx ; k ) (cid:112) − u − u sn ( µx ; k ) . (6.9)The roots of P ( λ ) in (3.34) for both periodic solutions form four pairs of purely imaginaryroots {± iβ , ± iβ , ± iβ , ± iβ } in (5.15) with β < β < β < β .For d ∈ (0 , d + ) and for either sign-definite or for sign-indefinite solutions, we find only onetypical configuration for the Floquet spectrum shown on Figure 6. The Floquet spectrumconsists of the real axis and the imaginary axis with four spectral gaps. Location of thespectral gaps satisfy the stability condition (4.13). By Propositions 4 and 5, the periodicstanding waves are spectrally and modulationally stable. Indeed, the stability spectrum ison the imaginary axis. − . − . . . . − . − . − . − . . . . . . Figure 6.
The same as Figure 3 but for c > b , c < b > d ∈ (0 , d + ).The parameters are: u = 3, u = 2, u = 1, u = 0When d → d + , we have u → u and β → β so that two spectral bands on the purelyimaginary axis coalesce into two points, after which the spectral bands re-emerge in thecomplex plane transversely to the imaginary axis. These complex spectral bands intersectthe imaginary axis inside the spectral gaps between the eigenvalues ± iβ and ± iβ .If d ∈ ( d + , ∞ ), the roots of Q can be re-enumerated and ordered as u = 0 < u with u = ¯ u = γ + iη being complex-conjugate. The exact solution is given by (6.8). The rootsof P ( λ ) in (3.34) corresponds to one quadruplet of complex eigenvalues ( ± λ , ± ¯ λ ) and twopairs of purely imaginary eigenvalues {± iβ , ± iβ } in (5.23). This case leads to similarfigures of the Floquet and stability spectra as those in Figure 5. All periodic standing wavesare modulationally unstable.To summarize the outcomes of the Floquet and stability spectra for the periodic standingwaves in the case a = 0, the only difference between the cases c < b and c > b , c < b > d ∈ (0 , d + ). For c < b , there is only one sign-indefiniteperiodic wave for each d ∈ (0 , d + ) and it is modulationally unstable according to Figure 4.For c > b , c < b >
0, there are two periodic waves (one is sign-definite and the other oneis sign-indefinite) for each d ∈ (0 , d + ); both are spectrally stable according to Figure 6. For d ∈ ( d − ,
0) and d ∈ ( d + , ∞ ), the periodic standing waves between the two cases are similarand the spectral pictures are given on Figures 3 and 5 respectively.7. Conclusion
In this work we have developed the algebraic method in order to classify all periodicstanding waves of the DNLS equation in terms of the location of eight complex eigenvaluesof the KN spectral problem. With the assistance of the numerical Hill’s method, we havecomputed the location of the Floquet spectrum in the KN spectral problem. This allowedus to conclude that the periodic standing waves with all eight eigenvalues on the imaginary
ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 25 axis were spectrally and modulationally stable, whereas all other periodic standing waveswere modulationally unstable.We showed these results for the periodic standing waves in the particular case a = 0.However, since the eight roots of the polynomial P ( λ ) in (3.32) have similar location forthe periodic standing waves in the general case a (cid:54) = 0, we expect that the same stabilityconclusions hold for a (cid:54) = 0.A number of other new directions in the context of the DNLS equation are opened followingthis work. Even if the periodic standing waves are modulationally unstable, they can beorbitally stable with respect to periodic perturbations of the same or multiple period, as wasexplored for the NLS equation in [13]. Nonlinear stability analysis for the DNLS equationis an open problem, whereas some results in this direction for the perturbations of the sameperiod were found in [20].Another interesting problem is to locate the Floquet spectrum in the complex λ planeanalytically. For example, we do not have the analytical proof that the Floquet spectrumcannot be located in between the purely imaginary eigenvalues, so that the spectral gapsappear in the stability conditions (4.12) and (4.13). There is no proof that there are noother spectral bands of the Floquet spectrum in the open quadrants of the complex planein addition to those connecting the eight eigenvalues of the algebraic method. It is alsointeresting that in the case d ∈ ( d + , ∞ ), we find that the complex spectral band in theFloquet spectrum can only intersect the imaginary axis inside the spectral gap or in theinterior spectral band but not in the exterior spectral bands. It would be interesting tosee why this is the case analytically. Analysis of the KN spectral problem for the periodicstanding waves in the DNLS equation is open for further studies. Acknowledgements.
This work was supported in part by the National Natural ScienceFoundation of China (No. 11971103).
References [1] H.A. Biagioni and F. Linares, “Ill-posedness for the derivative Schr¨odinger and generalized Benjamin-Ono equations”, Trans. Amer. Math. Soc. (2001), 3649–3659.[2] C.W. Cao and X.G. Geng, “Classical integrable systems generated through nonlinearization of eigenvalueproblems”,
Nonlinear physics (Shanghai, 1989) , pp. 68–78 (Research Reports in Physics, Springer,Berlin, 1990).[3] C.W. Cao and X. Yang, “A (2+1)-dimensional derivative Toda equation in the context of the Kaup–Newell spectral problem”, J. Phys. A: Math. Theor. (2008), 025203 (19 pages).[4] J. Chen and D.E. Pelinovsky, “Rogue periodic waves in the modified Korteweg-de Vries equation”,Nonlinearity (2018), 1955–1980.[5] J. Chen and D.E. Pelinovsky, “Rogue periodic waves in the focusing nonlinear Schr¨odinger equation”,Proc. R. Soc. Lond. A (2018), 20170814 (18 pages).[6] J. Chen and D.E. Pelinovsky, “Periodic travelling waves of the modified KdV equation and rogue waveson the periodic background”, J. Nonlin. Sci. (2019), 2797–2843. [7] J. Chen, D.E. Pelinovsky, and R.E. White, “Periodic standing waves in the focusing nonlinearSchr¨odinger equation: Rogue waves and modulation instability”, Physica D (2020), 132378 (13pages).[8] J. Chen and R. Zhang, “The complex Hamiltonian systems and quasi-periodic solutions in the derivativenonlinear Schr¨odinger equations”, Stud. Appl. Math. (2020), 153–178.[9] X.J. Chen and J. Yang, “Direct perturbation theory for solitons of the derivative nonlinear Schr¨odingerequation and the modified nonlinear Schr¨odinger equation”, Phys. Rev. E (2002) 066608.[10] M. Colin and M. Ohta, “Stability of solitary waves for derivative nonlinear Schr¨odinger equation”, Ann.I.H. Poincar´e-AN (2006), 753–764.[11] B. Deconinck and J.N. Kutz, “Computing spectra of linear operators using the Floquet-Fourier-Hillmethod”, J. Comput. Phys. (2006), 296–321.[12] B. Deconinck and B.L. Segal, “The stability spectrum for elliptic solutions to the focusing NLS equa-tion”, Physica D (2017), 1–19.[13] B. Deconinck and J. Upsal, “The orbital stability of elliptic solutions of the focusing nonlinearSchr¨odinger equation”, SIAM J. Math. Anal. (2020), 1–41.[14] B. Deconinck and J. Upsal, “Real Lax spectrum implies spectral stability”, Stud. Appl. Math. (2020),in press.[15] K.W. Chow and T.W. Ng, “Periodic solutions of a derivative nonlinear Schr¨odinger equation: Ellipticintegrals of the third kind”, J. Comp. Appl. Math. (2011), 3825–3830.[16] R. Gardner, “On the structure of the spectra of periodic traveling waves”, J. Math. Pures Appl. (1993), 415–439.[17] X.G. Geng, Z. Li, B. Xue, and L. Guan, “Explicit quasi-periodic solutions of the Kaup-Newell hierarchy”,J Math Anal Appl. (2015), 1097–1112.[18] B.L. Guo and Y. Wu, “Orbital stability of solitary waves for the nonlinear derivative Schr¨odingerequation”, J. Diff. Eqs. (1995), 35–55.[19] N. Fukaya, M. Hayashi, and T. Inui, “A sufficient condition for global existence of solutions to ageneralized derivative nonlinear Schr¨odinger equation”, Analysis & PDEs (2017), 1149–1167.[20] S. Hakkaev, M. Stanislavova, and A. Stefanov, “All non-vanishing bell-shaped solutions for the cubicderivative NLS are stable”, preprint (2020).[21] M. Hayashi, “Long-period limit of exact periodic traveling wave solutions for the derivative nonlinearSchr¨odinger equation”, Annales de l´Institut Henri Poincar´e C, Analyse non lin´eaire (2019), 1331–1360.[22] N. Hayashi and T. Ozawa, “On the derivative nonlinear Schr¨odinger equation”, Physica D (1992),14–36.[23] N. Hayashi and T. Ozawa, “Finite energy solution of nonlinear Schr¨odinger equations of derivativetype”, SIAM J. Math. Anal. (1994), 1488–1503.[24] R. Jenkins, J. Liu, P. A. Perry, and C. Sulem, “Global well-posedness for the derivative nonlinearSchr¨odinger equation”, Comm. Part. Diff. Eqs. (2018), 1151–1195.[25] M.A. Johnson and K. Zumbrun, “Convergence of Hill’s method for nonselfadjoint operators”, SIAM J.Numer. Anal. (2012), 64–78.[26] A.M. Kamchatnov, “On improving the effectiveness of periodic solutions of the NLS and DNLS equa-tions”, J Phys A: Math. Gen. (1990), 2945–2960.[27] A.M. Kamchatnov, “New approach to periodic solutions of integrable equations and nonlinear theoryof modulational instability”, Phys. Rep. (1997), 199–270.[28] D.J. Kaup and A.C. Newell, “An exact solution for a derivative nonlinear Schr¨odinger equation”, J.Math. Phys. (1978), 798–801. ODULATIONAL INSTABILITY OF PERIODIC STANDING WAVES 27 [29] S. Kwon and Y. Wu, “Orbital stability of solitary waves for derivative nonlinear Schr¨odinger equation”,Journal d´Analyse Math´ematique (2018), 473–486.[30] P.D. Lax, “Integrals of nonlinear equation of evolution and solitary waves”, Comm. Pure Appl. Math. (1968), 467–490.[31] J. Liu, P.A. Perry, and C. Sulem, “Global existence for the derivative nonlinear Schr¨odinger equationby the method of inverse scattering”, Comm. Part. Diff. Eqs. (2016), 1692–1760.[32] C. Miao, Y. Wu, and G. Xu, “Global well-posedness for Schr¨odinger equation with derivative in H / ( R )”, J. Diff. Eqs., (2011), 2164–2195.[33] C. Miao, X. Tang, and G. Xu, “Stability of the traveling waves for the derivative Schrdinger equationin the energy space”, Calc. Var. PDEs (2017), 45 (20 pages).[34] W. Mio, T. Ogino, K. Minami, and S. Takeda, “Modified nonlinear Schrdinger equation for Alfv´enwaves propagating along the magnetic field in cold plasmas”, J. Phys. Soc. Japan (1976), 265–271.[35] E. Mjolhus, “On the modulational instability of hydromagnetic waves parallel to the magnetic field”, J.Plasma Phys. (1976), 321–334.[36] W.X. Ma and R. Zhou, “On the relationship between classical Gaudin models and BC-type Gaudinmodels”, J. Phys. A: Math. Gen. (2001), 3867–880.[37] D. E. Pelinovsky, A. Saalmann, and Y. Shimabukuro, “The derivative NLS equation: global existencewith solitons”, Dynamics PDEs. (2017), 271–294.[38] D. E. Pelinovsky and Y. Shimabukuro, “Existence of global solutions to the derivative NLS equationwith the inverse scattering transform method”, Int. Math. Res. Notices (2018), 5663–5728.[39] H. Takaoka, “Well-posedness for the one dimensional Schr¨odinger equation with the derivative nonlin-earity”, Adv. Diff. Eq. (1999), 561–680.[40] O.C. Wright, “Maximal amplitudes of hyperelliptic solutions of the derivative nonlinear Schr¨odingerequation”, Stud Appl Math. (2020), 1–30.[41] Y. Wu, “Global well-posedness for the nonlinear Schr¨odinger equation with derivative in energy space”,Anal. PDE (2013), 1989–2002.[42] Y. Wu, “Global well-posedness on the derivative nonlinear Schr¨odinger equation”, Anal. PDE, (2015),1101–1112.[43] P. Zhao and E.G. Fan, “Finite gap integration of the derivative nonlinear Schr¨odinger equation: aRiemann–Hilbert method”, Physica D (2020), 132213 (31 pages).[44] R.G. Zhou, “An integrable decomposition of the derivative nonlinear Schr¨odinger equation”, Chin. Phys.Lett. (2007), 589–591.(J. Chen) School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, P.R.China (D.E. Pelinovsky)
Department of Mathematics, McMaster University, Hamilton, Ontario,Canada, L8S 4K1 (J. Upsal)(J. Upsal)