Instability of degenerate solitons for nonlinear Schrödinger equations with derivative
aa r X i v : . [ m a t h . A P ] F e b Instability of degenerate solitons for nonlinearSchr¨odinger equations with derivative
Noriyoshi Fukaya and Masayuki Hayashi
Abstract.
We consider the following nonlinear Schr¨odinger equation with de-rivative:(1) iu t = − u xx − i | u | u x − b | u | u, ( t, x ) ∈ R × R , b ∈ R . If b = 0, this equation is a gauge equivalent form of the well-known derivativenonlinear Schr¨odinger (DNLS) equation. The equation (1) for b ≥ b = 0, they arealgebraic solitons. Inspired from the works [29, 8] on instability theory of the L -critical generalized KdV equation, we study the instability of degenerate soli-tons of (1) in a qualitative way, and when b >
0, we obtain a large set of initialdata yielding the instability. The arguments except one step in our proof workfor the case b = 0 in exactly the same way, and in particular the unstable di-rections of algebraic solitons are detected. This is a step towards understandingthe dynamics around algebraic solitons of the DNLS equation. Contents
1. Introduction 12. Structure of the linearized operator 73. Modulation theory 124. Virial identities 195. Proof of instability 21Appendix A. Relation to instability theory on (gKdV) 22Acknowledgments 23References 23 Introduction
We consider the following nonlinear Schr¨odinger equation with derivative: iu t = − u xx − i | u | u x − b | u | u, ( t, x ) ∈ R × R , (1.1)where b ∈ R , and u is the complex-valued unknown function of ( t, x ) ∈ R × R . Itis well-known (see [39]) that (1.1) is locally well-posed in the energy space H ( R )and the following three quantities E ( u ) := 12 k u x k L −
14 ( i | u | u x , u ) L − b k u k L , (Energy) M ( u ) := k u k L , (Mass) P ( u ) := ( iu x , u ) L , (Momentum) Date : February 26, 2021. are conserved by the flow. Here the inner product ( · , · ) L is defined by( v, w ) L = Re Z R v ( x ) w ( x ) dx, and we regard L ( R ) as a real Hilbert space. The equation (1.1) is L -critical(mass-critical) in the sense that (1.1) is invariant under the scaling u λ ( t, x ) = λ / u ( λ t, λx ) , which satisfies k u λ (0) k L = k u (0) k L . By using the energy functional, (1.1) isrewritten as(1.2) iu t ( t ) = E ′ ( u ( t )) . When b = 0 the equation (1.1) is sometimes referred to as the Chen-Lee-Liu equa-tion [5]. This equation is a gauge equivalent form of the well-known derivativenonlinear Schr¨odinger equation iψ t = − ψ xx − i ( | ψ | ψ ) x , ( t, x ) ∈ R × R , (DNLS)which was introduced as a model in plasma physics [32, 33] and shown to be com-pletely integrable [21]. The equation (1.1) can be considered as a generalization of(DNLS) while preserving L -criticality and the Hamiltonian structure (1.2).The equation (1.1) admits a two-parameter family of solitons u ω,c ( t, x ) = e iωt φ ω,c ( x − ct ) , where ( ω, c ) ∈ R satisfies ( − √ ω < c ≤ √ ω if b > − / , − √ ω < c < − κ ∗ √ ω if b ≤ − / , (1.3) κ ∗ = κ ∗ ( b ) := r − γ − γ = r b b ∈ (0 ,
1) when b ≤ − / ,γ = γ ( b ) := 1 + 163 b, and φ ω,c is explicitly written as φ ω,c ( x ) = Φ ω,c ( x ) exp (cid:18) i c x − i Z x −∞ Φ ω,c ( y ) dy (cid:19) , Φ ω,c ( x ) = ω − c ) p c + γ (4 ω − c ) cosh( √ ω − c x ) − c ! / if − √ ω < c < √ ω, (cid:18) c ( cx ) + γ (cid:19) / if c = 2 √ ω. We note that φ ω,c ∈ H ( R ) is the nontrivial solution of the stationary equation(1.4) − φ ′′ + ωφ + ciφ ′ − i | φ | φ ′ − b | φ | φ = 0 , x ∈ R , and that Φ ω,c is the positive even solution of(1.5) − Φ ′′ + (cid:16) ω − c (cid:17) Φ + c | Φ | Φ − γ | Φ | Φ = 0 , x ∈ R . The terminology soliton was originally used in a context of integrable equations, but we alsouse it for nonintegrable equations according to conventions in the literature.
The equation (1.5) has nontrivial H -solutions if and only if ( ω, c ) satisfies (1.3).For ( ω, c ) satisfying (1.3), one can rewrite ( ω, c ) = ( ω, κ √ ω ), where the param-eter κ satisfies − < κ ≤ b > − / , − < κ < − κ ∗ if b ≤ − / . For each parameter κ , the following curve R + ∋ ω ( ω, κ √ ω ) ∈ R gives the scaling of the soliton: φ ω, κ √ ω ( x ) = ω / φ , κ ( √ ωx ) for x ∈ R . When b ≥
0, there exists a unique κ = κ ( b ) ∈ (0 ,
1] such that E ( φ , κ ) = P ( φ , κ ) = 0 , which implies that the soliton u ω, κ √ ω corresponds to the degenerate case. Wenote that 0 < κ ( b ) < b >
0, and κ (0) = 1. Therefore, algebraic solitons of(DNLS) correspond to the degenerate case, while degenerate solitons for b > u ω, κ √ ω in a qualitative way.The degenerate soliton can be also found in a different context, for example, the L -critical NLS(NLS) iu t = − u xx − | u | u, ( t, x ) ∈ R × R , and the L -critical generalized KdV equation(gKdV) u t = − ( u xx + u ) x , ( t, x ) ∈ R × R . The equations (NLS) and (gKdV) have the same conserved quantities: E ( v ) = 12 k v x k L − k v k L , (Energy) M ( v ) = k v k L . (Mass)(NLS) has the standing wave e it Q ( x ) and (gKdV) has the traveling wave Q ( · − t ),where Q ( x ) = / cosh / (2 x ) is the positive even solution of − Q ′′ + Q − Q = 0 , x ∈ R , and Q is an optimizer of the following Gagliardo–Nirenberg inequality (see [44]):16 k f k L ≤ (cid:18) M ( f ) M ( Q ) (cid:19) k f x k L for f ∈ H ( R ) . (1.6)In particular E ( Q ) = 0 holds, which implies that the solitons e it Q ( x ) and Q ( · − t )correspond to the degenerate case. It is also known that these degenerate solitonsare unstable (see [44, 29]).Instability of degenerate solitons is important to understand the global dynamicsof (NLS) and (gKdV). It follows from (1.6) and conservation laws that if the initialdata u ∈ H ( R ) of (NLS) or (gKdV) satisfies M ( u ) < M ( Q ), the corresponding H -solution is global and satisfies12 − (cid:18) M ( u ) M ( Q ) (cid:19) ! k u x ( t ) k L ≤ E ( u ) for all t ∈ R . For (NLS), it is known that finite time blow-up occurs for the initial data satisfying M ( u ) > π and E ( u ) < E ( u ) < , M ( Q ) < M ( u ) < M ( Q ) + α (1.7)and some decay condition, where α > b ≥ it was proved in [46, 16] that if the initial data u ∈ H ( R ) satisfies M ( u ) < M ( φ , κ ) =: M ∗ , then the corresponding H -solution is global and satisfies k u x ( t ) k L ≤ C ( k u k H ) for all t ∈ R , (1.8)where the constant in the right-hand side is composed of the conserved quantities E ( u ), M ( u ), and P ( u ). For (DNLS) this mass condition is nothing but the 4 π -mass condition. In the recent progress of studies on (DNLS), global well-posednesswithout the smallness assumption of the mass was established by taking advantageof completely integrable structure (see [40, 20, 2]). These results give a remarkabledifference with other L -critical equations (NLS) and (gKdV), while the uniformboundedness of the flow as (1.8) is not known for M ( u ) ≥ π . We note that ifwe impose further assumptions with M ( u ) ≥ π , for example, M ( u ) = 4 π and P ( u ) <
0, or highly oscillating data, then the estimate (1.8) still holds (see [10]).It was proved in [16] that the mass threshold M ∗ gives a certain turning pointin variational properties of (1.1). This suggests that global dynamics of (1.1) willchange at the mass of M ∗ . From the variational point of view, M ∗ correspondsto the mass threshold M ( Q ) in (NLS) and (gKdV). Therefore, to investigate thedynamics around the mass of M ∗ is important to understand the global dynamicsof (1.1). To this end, in this paper we study instability properties of degeneratesolitons of (1.1) in a qualitative way.We first give a precise definition of stability and instability of solitons. Definition 1.1.
We say that the soliton u ω,c of (1.1) is stable if for any α > β > u ∈ H ( R ) satisfies k u − φ ω,c k H < β , the solution u ( t ) of (1.1) exists globally in time and satisfiessup t ∈ R inf ( θ,y ) ∈ R k u ( t ) − e iθ φ ω,c ( · − y ) k H < α. Otherwise, we say that the soliton u ω,c is unstable .We now review the known stability results related to our work. When b = 0,Colin and Ohta [6] proved by applying variational approach that if ω > c /
4, thesoliton u ω,c is stable. For the case c = 2 √ ω some kinds of stability propertieswere studied in [22, 23], while the stability or instability in the sense of Definition1.1 remains an open problem. Liu, Simpson and Sulem [27] calculated linearizedoperators of the generalized derivative nonlinear Schr¨odinger equation iu t + u xx + i | u | σ u x = 0 , ( t, x ) ∈ R × R , σ > , (gDNLS)and studied stability of nondegenerate solitons by applying the abstract theoryof Grillakis, Shatah and Strauss [12, 13] (see also [14] for partial results in this The global result for the case b < direction). Although well-posedness in the energy space for (gDNLS) was assumedin [27], the well-posedness problem was later dealt with in [41, 18, 26].When b >
0, Ohta [38] proved by applying variational approach in [43, 11, 6] thatthe soliton u ω,c is stable if − √ ω < c < κ √ ω , and unstable if 2 κ √ ω < c < √ ω .Ning, Ohta and, Wu [35] proved that the algebraic soliton is unstable for small b > P ( φ ω,c ) is positive in the stable region {− √ ω < c < κ √ ω } , negative in the unstable region { κ √ ω < c ≤ √ ω } , andzero on { c = 2 κ √ ω } (see Remarks 2 and 3 of [38]). When b < b <
0, the momentum of allsolitons is positive.It is known that the stability/instability depends on the spectral properties ofthe Hessian matrix of the two-variable function d ( ω, c ) := S ω,c ( φ ω,c ) , where S ω,c is the action defined by S ω,c ( v ) := E ( v ) + ω M ( v ) + c P ( v ) . From a direct computation, we have the identitydet[ d ′′ ( ω, c )] = − P ( φ ω,c ) √ ω − c { c + γ (4 ω − c ) } for ω > c . If P ( φ ω,c ) = 0, then d ′′ ( ω, c ) has a zero eigenvalue, which corresponds to the de-generate case.We note that the abstract theory in [12, 13] is not applicable to degeneratesolitons. In [7, 37, 28] instability of degenerate solitons with one-parameter isstudied in the abstract framework. The first author [9] extended the work of [37] todegenerate solitons with two-parameter. However, these results are not applicableto degenerate solitons of L -critical equations (NLS), (gKdV) and (1.1). Recently,Ning [34] proved the instability of the soliton u ω, κ √ ω of (1.1) for sufficiently small b >
0. The proof was done by combining localized virial identities and modulationanalysis, whose argument was originally developed in [47, 15].Our approach in the present paper is motivated by the works [29, 8] on instabilityof degenerate solitons of (gKdV).We now state our results of this paper. We first organize the spectral propertiesof the linearized operator around the soliton. The linearized operator is explicitlywritten as L ω,c v := S ′′ ω,c ( φ ω,c ) v (1.9) = − v xx + ωv + civ x − i | φ ω,c | v x − i Re( φ ω,c v ) φ ′ ω,c − b | φ ω,c | v − b | φ ω,c | Re( φ ω,c v ) φ ω,c for v ∈ H ( R ). The following claim is used as a basic tool in the proof of our mainresult. Proposition 1.2.
Let b ∈ R and let ( ω, c ) satisfy (1.3) . Then the space H ( R ) isdecomposed as the orthogonal direct sum H ( R ) = N ω,c ⊕ Z ω,c ⊕ P ω,c . Here N ω,c is the negative subspace of L ω,c spanned by the eigenvector χ ω,c corre-sponding to the simple negative eigenvalue λ ω,c , Z ω,c is the kernel of L ω,c spannedby iφ ω,c and φ ′ ω,c , and P ω,c is the positive subspace of L ω,c such that (i) if − √ ω < c < √ ω , then there exists a positive constant k > such that forany p ∈ P ω,c h L ω,c p, p i ≥ k k p k H , (1.10)(ii) if c = 2 √ ω , then for any p ∈ P ω,c \ { }h L ω,c p, p i > . (1.11)We prove Proposition 1.2 by mainly following the argument in [27]. Here wetreat the case c = 2 √ ω , which was not considered in previous works. As in theassertion (ii), the coercivity fails for the case c = 2 √ ω because the essential spectralof L ω,c consists of the interval [0 , ∞ ) for this case.We now state our main result, which concerns the instability of degenerate soli-tons of (1.1). Theorem 1.3.
Let b > and c = 2 κ √ ω and let χ ω,c be as in Proposition 1.2.Then there exist α, β ∈ (0 , such that if ε := u − φ ω,c for the initial data u ∈ H ( R ) satisfies < k ε k H ≤ β | ( ε , φ ω,c ) L | , ε ⊥ { χ ω,c , iφ ω,c , φ ′ ω,c , iφ ′ ω,c } , (1.12) then there exists t = t ( u ) ∈ R such that the solution u ( t ) of (1.1) satisfies inf ( θ,y ) ∈ R k u ( t ) − e iθ φ ω,c ( · − y ) k H ≥ α. In particular, the soliton u ω,c is unstable.Remark . We can construct ε satisfying (1.12) as follows. One can easily showthat the functions χ ω,c , iφ ω,c , φ ′ ω,c , φ ω,c , iφ ′ ω,c are linearly independent. Applyingthe Gram–Schmidt process, we have a function ε ∈ H ( R ) satisfying( ε , φ ω,c ) = 0 , ε ⊥ { χ ω,c , iφ ω,c , φ ′ ω,c , iφ ′ ω,c } . Then ε := δε for small δ > Remark . If we replace the assumption (1.12) by0 < k ε k H ≤ β (cid:12)(cid:12) ( ε , iφ ′ ω,c ) L (cid:12)(cid:12) , ε ⊥ { χ ω,c , iφ ω,c , φ ′ ω,c , φ ω,c } , then the conclusion in Theorem 1.3 still holds. Remark . In [34] some explicit function was used as a negative direction of L ω,c instead of the eigenfunction χ ω,c . The smallness assumption on b > χ ω,c , we construct and control modulation parameters by using thescaling properties of the equation. Moreover, we obtain a large set of initial datayielding the instability while in [34] the only one unstable direction is found. In contexts of (NLS) and (gKdV) one can use the explicit eigenfunction for negative eigenvalueof the linearized operator.
For the proof of Theorem 1.3 we use modulation theory and the virial identity ddt Im Z xu x ( t, x ) u ( t, x ) = 4 E ( u ) , (1.13)but we avoid a direct use of this identity. We consider the decomposition u ( t, x ) = e iθ ( t ) λ ( t ) / ( φ ω,c + ε ) (cid:18) t, x − x ( t ) λ ( t ) (cid:19) , (1.14)where λ ( t ) > θ ( t ) ∈ R , x ( t ) ∈ R , and the function ε ( t, x ) satisfies suitableorthogonal conditions (see Proposition 3.2). If we put the formula (1.14) into(1.13), the left-hand side of (1.13) yields the quantity ddt Im Z ε ( t, x )Λ φ ω,c ( x ) (cid:16) Λ f := f + xf x (cid:17) , (1.15)which plays an essential role in our proof. This quantity has already been effectivelyused on the studies of the blow-up dynamics of (NLS) (see, e.g., [31]), but it seemsto be new in the contexts of (1.1), (DNLS) and (gDNLS). The quantity (1.15)is well-defined in the H -setting, so we do not need any cut-off arguments, whichbecomes a much simpler argument than previous works [47, 15, 34]. Moreover, ourproof gives a close relation to instability theory on (gKdV) (see Appendix A).The arguments except one step (Lemma 3.7) in our proof work for the case b = 0 and c = 2 √ ω , i.e., algebraic solitons of (DNLS), in exactly the same way. Although we could not complete the proof of Theorem 1.3 for the case b = 0, theunstable directions are detected in the same way as the case b > u = φ ω,c + ε with ( ε , φ ω,c ) L >
0, then(1.16) E ( u ) < , M ( φ ω,c ) < M ( u ) < M ( φ ω,c ) + β , where β is a small constant. We note that the condition (1.16) corresponds to theblow-up set of (NLS) and (gKdV), and so Theorem 1.3 gives an important clue toconstruct a singular solution of (1.1).The rest of this paper is organized as follows. In Section 2 we study the spectraof the linearized operator L ω,c and prove Proposition 1.2. In Section 3 we constructthe modulation parameters satisfying suitable orthogonal conditions and controlthese parameters. In Section 4 we organize the virial identities. In Section 5 wecomplete the proof of Theorem 1.3 by using the estimates obtained in previoussections. Structure of the linearized operator
In this section, we study the structure of the linearized operator L ω,c . Through-out this section, we assume that ( ω, c ) satisfies (1.3). For simplicity we often dropthe subscript ( ω, c ) as S = S ω,c , φ = φ ω,c , Φ = Φ ω,c . For the proof of Lemma 3.7, we use the coercivity property of L ω,c which does not hold in thecase c = 2 √ ω . We define the function η ω,c as η ( x ) = η ω,c ( x ) = c x − Z x −∞ Φ ω,c ( y ) dy, (2.1)and define the operator ˜ L ω,c as˜ L = ˜ L ω,c = e − iη ω,c ( x ) L ω,c e iη ω,c ( x ) . For w ∈ H ( R ) we set f = Re w and g = Im w . After a direct calculation, ˜ Lw isexplicitly represented as˜ Lw = − w xx + (cid:18) ω − c (cid:19) w + c w + c Φ Re w − γ Φ w − γ Φ Re w (2.2) + 14 Φ Re w − i w x + i ′ w − i ΦΦ ′ Re w = L f + L g + 14 Φ f + i ( L f + L g ) , where L := − ∂ x + U Φ , U Φ := (cid:18) ω − c (cid:19) + 32 c Φ − γ Φ ,L := 12 Φ ∂ x −
12 ΦΦ ′ ,L := −
12 Φ ∂ x −
32 ΦΦ ′ ,L := − ∂ x + V Φ , V Φ := (cid:18) ω − c (cid:19) + c − γ Φ . Since e iη ( x ) is a unitary operator, the spectral property of ˜ L is the same as that of L . In what follows, we investigate the spectra of the operator ˜ L .We first note that ˜ L can be considered as compact perturbation of the operator − ∂ x + ( ω − c / σ ess ( ˜ L ) = σ ess (cid:16) − ∂ x + (cid:16) ω − c (cid:17)(cid:17) = h ω − c , ∞ (cid:17) and the spectrum of ˜ L in ( −∞ , ω − c /
4) consists of isolated eigenvalues.2.1.
Kernel.
In this subsection we prove the nondegeneracy of the kernel of ˜ L .Our proof depends on the argument in [25]. Lemma 2.1.
The following statement is true. (i) ker L = span { Φ ′ ω,c } , (ii) ker L = span { Φ ω,c } .Proof. Since Φ is a solution of (1.5), we have L Φ = 0. By differentiating theequation (1.5), we also have L Φ ′ = 0. Hence we haveker L ⊃ span { Φ ′ } , ker L ⊃ span { Φ } . It now suffices to show ker L ⊂ span { Φ } because one can show ker L ⊂ span { Φ ′ } by the same argument. Let g ∈ ker L . We consider the Wronskian of Φ and g : W ( x ) := Φ ′ ( x ) g ( x ) − Φ( x ) g ′ ( x ) . From Φ , g ∈ H ( R ), we have W ( x ) → | x | →
0. Since L Φ = L g = 0, weobtain W ′ ( x ) = Φ ′′ g − Φ g ′′ = V Φ Φ g − Φ V Φ g = 0 . Thus, we deduce W ≡
0, which implies that Φ and g are linearly dependent. Thiscompletes the proof. (cid:3) Lemma 2.2.
The kernel ˜ L ω,c is determined by ker ˜ L ω,c = span (cid:26) i Φ ω,c , Φ ′ ω,c − i ω,c (cid:27) , which is equivalent to ker L ω,c = span { iφ ω,c , φ ′ ω,c } .Proof. First we show ker ˜ L ⊃ span (cid:8) i Φ , Φ ′ − i Φ (cid:9) . Since φ is a solution of (1.4),and the equation has symmetries under the phase and spatial translation, we have S ′ ( e iθ φ ( · − y )) = 0 for all ( θ, y ) ∈ R × R . Differentiating this with respect to θ or y at ( θ, y ) = 0, we have(2.3) Liφ = 0 , Lφ ′ = 0 , respectively. Since e − iη ( x ) L = ˜ Le − iη ( x ) and φ = e iη ( x ) Φ, (2.3) is equivalent to˜ Li Φ = 0 , ˜ L (cid:16) Φ ′ + i c − i (cid:17) = 0 . This implies ker ˜ L ⊃ span (cid:8) i Φ , Φ ′ − i Φ (cid:9) .Next we show the inverse inclusion. Let w ∈ ker ˜ L , f = Re w , and g = Im w . Theexpression (2.2) of ˜ L implies that ( f, g ) satisfies the following system of ordinarydifferential equations: L f + L g + 14 Φ f = 0 ,L f + L g = 0 . (2.4)Now we apply the following transformation to g : g = h −
12 Φ Z x −∞ Φ f dy. (2.5)Then we have L g + 14 Φ f = 12 Φ g x −
12 ΦΦ ′ g + 14 Φ f (2.6) = 12 Φ h x −
12 ΦΦ ′ h. Moreover, noting that ∂ x (cid:18)
12 Φ Z x −∞ Φ f dy (cid:19) = 12 Φ ′′ Z x −∞ Φ f dy + 32 ΦΦ ′ f + 12 Φ f x = 12 Φ ′′ Z x −∞ Φ f dy − L f, it follows from L Φ = 0 that L f + L g = L f + L h + ∂ x (cid:18)
12 Φ Z x −∞ Φ f dy (cid:19) − V Φ Φ Z x −∞ Φ f dy (2.7) = L h −
12 ( − Φ ′′ + V Φ Φ) Z x −∞ Φ f dy = L h. Using (2.6) and (2.7) we write the equation (2.4) as L f + 12 Φ (cid:0) Φ h x − Φ ′ h (cid:1) = 0 ,L h = 0 . (2.8)From the second equation in (2.8) and Lemma 2.1 (ii), we have h = α Φ for some α ∈ R . Substituting this into the first equation in (2.8), we get L f = 0. Therefore,Lemma 2.1 (i) implies that f = β Φ ′ for some β ∈ R . Substituting h = α Φ and f = β Φ ′ into (2.5), we have g = α Φ − β Z x −∞ ΦΦ ′ dy = α Φ − β Z x −∞ (Φ ) ′ dy = α Φ − β . Therefore, we obtain that w = f + ig = β Φ ′ + i (cid:18) α Φ − β (cid:19) = αi Φ + β (cid:18) Φ ′ − i (cid:19) ∈ span (cid:26) i Φ , Φ ′ − i (cid:27) . This completes the proof. (cid:3)
Construction of a negative direction.
In this subsection we prove that˜ L ω,c has exactly one negative eigenvalue. Our proof depends on the argument in[27] (see also [14]). The following expression of the quadratic form is useful toconstruct a negative direction. Lemma 2.3.
Let w ∈ H ( R ) , f = Re w , and g = Im w . Then we have h ˜ L ω,c w, w i = h L f, f i + 14 k Φ ω,c f + 2Φ ω,c ∂ x (Φ − ω,c g ) k L . (2.9) Proof.
First, by the expression (2.2), we have h ˜ Lw, w i = h L f, g i + h L g, f i + 14 h Φ f, f i + h L f, g i + h L g, g i . We set ˜ g = Φ − g . It follows from L Φ = 0 that h L g, g i = h ˜ g ( − ∂ x + V Φ )Φ , Φ˜ g i − h ′ ˜ g x + Φ˜ g xx , Φ˜ g i = −h ∂ x (Φ ˜ g x ) , ˜ g i = k Φ˜ g x k L . Next, we calculate the interaction terms as h L g, f i = D
12 Φ ∂ x (Φ˜ g ) −
12 Φ Φ ′ ˜ g, f E = 12 h Φ , f ˜ g x i and h L f, g i = − (cid:28)
12 Φ f x + 32 ΦΦ ′ f, Φ˜ g (cid:29) = − h Φ , ˜ gf x i − h ∂ x (Φ ) , f ˜ g i = 12 h Φ , f ˜ g x i . Therefore we deduce that h ˜ Lw, w i = h L f, g i + 14 h Φ f, f i + h Φ , f ˜ g x i + k Φ˜ g x k L = h L f, f i + 14 k Φ f + 2Φ˜ g x k L . This completes the proof. (cid:3)
Lemma 2.4.
The operator L has exactly one negative eigenvalue.Proof. We note that L is a compact perturbation of the operator − ∂ x +( ω − c / σ ess ( L ) = σ ess (cid:16) − ∂ x + (cid:16) ω − c (cid:17)(cid:17) = h ω − c , ∞ (cid:17) , and the spectrum of L in ( −∞ , ω − c /
4) consists of isolated eigenvalues. We notethat L Φ ′ = 0 and that Φ ′ has exactly one zero point. By Sturm–Liouville theorywe deduce that zero is the second eigenvalue of L , and that L has one negativeeigenvalue. Moreover, one can prove that the negative eigenvalue is simple (see,e.g., [1, Theorem B.59]). This completes the proof. (cid:3) We denote the negative eigenvalue of L in Lemma 2.4 by λ and its normalizedeigenvector by χ , that is,(2.10) L χ = λ χ , k χ k L = 1 . Lemma 2.5.
The operator ˜ L ω,c has exactly one negative eigenvalue.Proof. Let χ := −
12 Φ Z x −∞ Φ χ dy. Then we have Φ ∂ x (Φ − χ ) = −
12 Φ χ . Therefore, it follows from (2.9) and (2.10) that χ ∗ := χ + iχ satisfies h ˜ Lχ ∗ , χ ∗ i = h L χ , χ i = λ < . This means that the operator ˜ L has at least one negative eigenvalue.Now we show that ˜ L has exactly one negative eigenvalue. Assume that ˜ L hastwo negative eigenvalues (including repeats) λ ≤ λ < χ and χ such that˜ Lχ = λ χ , ˜ Lχ = λ χ , k χ k L = k χ k L = 1 , ( χ , χ ) L = 0 . We note that by the formula (2.9) and Lemma 2.4, h ˜ Lp, p i ≥ p ∈ H ( R ) satisfying (Re p, χ ) L = 0. Thus, it follows from h ˜ Lχ , χ i = λ < χ , χ ) L = 0. If we set α = − (Re χ , χ ) L (Re χ , χ ) L , p = χ + αχ , then we have (Re p , χ ) L = 0. Hence we deduce that h ˜ Lp , p i ≥
0. On the otherhand, by a direct calculation we obtain h ˜ Lp , p i = λ + α λ < , which yields a contradiction. This completes the proof. (cid:3) Remark . When b ≥
0, by variational characterization of the solitons (see [6, 10,16]) one can prove that L ω,c has exactly one negative eigenvalue (see the argumentof [25]). Our approach based on the formula (2.9) is more elementary and applicableto the case b < Spectral decomposition.
We now complete the proof of Proposition 1.2.
Proof of Proposition 1.2.
By Lemma 2.2 and Lemma 2.5, we have the followingdecomposition H ( R ) = span { ˜ χ } ⊕ span (cid:26) i Φ , Φ ′ − i (cid:27) ⊕ ˜ P , (2.11)where ˜ χ is the eigenvector of ˜ L corresponding to its negative eigenvalue λ and ˜ P isthe nonnegative subspace of ˜ L . Since ˜ L = e − iη ( x ) Le iη ( x ) , (2.11) is equivalent that H ( R ) = N ⊕ Z ⊕ P , (2.12)where N is spanned by the negative eigenvector χ := e iη ( x ) ˜ χ of L , Z := span { iφ, φ ′ } is its kernel, and P := e iη ( x ) ˜ P is its nonnegative subspace. The rest of the proof isto show the positivity of L on P .(i) We consider the case − √ ω < c < √ ω . Since σ ess ( L ) = h ω − c / , ∞ (cid:17) , thespectra of L except for its negative eigenvalue and zero eigenvalue are positive andbounded away from zero. Therefore, there exists a positive constant δ > h Lp, p i ≥ δ k p k L for all p ∈ P . (2.13)From the explicit formula (1.9), there exists a positive constant C such that h Lv, v i ≥ k v x k L − C k v k L for all v ∈ H ( R ). Combined with (2.13), we have k p k H ≤ h Lp, p i + (1 + 2 C ) k p k L ≤ (cid:18) C δ (cid:19) h Lp, p i for all p ∈ P , which shows the desired inequality (1.10).(ii) We now consider the case c = 2 √ ω . Assume by contradiction that thereexists p ∈ P such that k p k L = 1 and h Lp , p i = 0. Then we obtain the followingrelation: h Lp , p i = min {h Lp, p i : k p k L = 1 , ( χ, p ) L = ( iφ, p ) L = ( φ ′ , p ) L = 0 } . This minimization problem implies that there exist Lagrange multipliers α , α , α , and α such that Lp = α χ + α iφ + α φ ′ + α p . By the orthogonal conditions and h Lp , p i = 0, we have α = α = α = α = 0.Therefore, p ∈ ker L ∩ P = { } , which is a contradiction. Hence (1.11) holds. (cid:3) Modulation theory
In this section we organize modulation theory for three fundamental symmetrieswhich are phase, translation, and scaling.We prepare some notations. For α > φ ω,c by U α = { u ∈ H ( R ) : inf ( θ,z ) ∈ R k e iθ u ( · + z ) − φ ω,c k H < α } . For u ∈ H ( R ), λ >
0, and θ, y ∈ R , we denote the function ε by ε ( λ, θ, x ; u ) = ε ( λ, θ, x ) = λ / e − iθ u ( λ · + x ) − φ ω,c . For λ > f : R → C , we define the rescaling f λ ( y ) = λ / f ( λy ) . Let Λ be the generator of this transformation asΛ f := ∂ λ f λ | λ =1 = f yf y . We note that Λ is skew-symmetric, i.e.,(Λ f, g ) L = − ( f, Λ g ) L . Construction of modulation parameters.
We construct the modulationparameters λ , θ , and x satisfying suitable orthogonal conditions. We first preparethe following lemma. Lemma 3.1.
Assume that ( ω, c ) satisfy (1.3) . Then we have (i) (Λ φ ω,c , iφ ω,c ) L = (Λ φ ω,c , φ ′ ω,c ) L = 0 . If we further assume b ≥ and c = 2 κ ( b ) √ ω , then we have (ii) ( iφ ′ ω,c , Λ φ ω,c ) L = ( iφ ′ ω,c , φ ω,c ) L = 0 , (iii) (Λ φ ω,c , χ ω,c ) L = 0 .Proof. (i) It follows from the explicit formula of η (see (2.1)) that η ′ = c −
14 Φ ,φ ′ = e iη (cid:0) iη ′ Φ + Φ ′ (cid:1) = e iη (cid:18) i c − i + Φ ′ (cid:19) . Since Φ is a real-valued and even function, one computes easily that(Λ φ, iφ ) L = (cid:16) φ + yφ ′ , iφ (cid:17) L = ( yφ ′ , iφ ) L = Re Z y (cid:18) i c − i + Φ ′ (cid:19) · ( − i Φ) = Re Z y (cid:18) c −
14 Φ (cid:19) = 0 , (Λ φ, φ ′ ) L = (cid:16) φ + yφ ′ , φ ′ (cid:17) L = ( yφ ′ , φ ′ ) L = Re Z y (cid:26) (Φ ′ ) + (cid:18) c −
14 Φ (cid:19) (cid:27) = 0 . (ii) Since P ( φ ) = 0 by c = 2 κ √ ω , we have( iφ ′ , Λ φ ) L = ( iφ ′ , φ + yφ ′ ) L = Re Z iy | φ ′ | = 0 . (iii) By twice differentiating the following relation S ( φ λ ) = λ E ( φ ) + ωM ( φ ) + λcP ( φ )at λ = 1, we have h L Λ φ, Λ φ i = 2 E ( φ ) = 0. Suppose that (Λ φ, χ ) L = 0. FromProposition 1.2 and (i) proved just above, we obtain that h L Λ φ, Λ φ i >
0. This is acontradiction and completes the proof. (cid:3)
The next proposition is the foundation of the modulation analysis. Proposition 3.2.
Let b ≥ and c = 2 κ √ ω . Then there exist constants α > , λ > and C -mappings (Λ , Θ , X ) : U α → (1 − λ , λ ) × R such that ( ε (Λ( u ) , Θ( u ) , X ( u )) , χ ω,c ) L = ( ε (Λ( u ) , Θ( u ) , X ( u )) , iφ ω,c ) L = ( ε (Λ( u ) , Θ( u ) , X ( u )) , φ ′ ω,c ) L = 0(3.1) for all u ∈ U α . Moreover, there exists a constant C > such that for any α ∈ (0 , α ) and u ∈ U α k ε (Λ( u ) , Θ( u ) , X ( u )) k H ≤ Cα, | Λ( u ) − | ≤ Cα. (3.2)
Proof.
Let F : (0 , ∞ ) × R × H ( R ) → R be the function defined by F ( λ, θ, x ; u ) = ( ε ( λ, θ, x ; u ) , χ ) L ( ε ( λ, θ, x ; u ) , iφ ) L ( ε ( λ, θ, x ; u ) , φ ′ ) L . We define the open neighborhoods V α of φ and Ω δ ⊂ (0 , ∞ ) × R of (1 , ,
0) by V α = { u ∈ H ( R ) : k u − φ k H < α } , Ω δ = { ( λ, θ, x ) ∈ (0 , ∞ ) × R : | λ − | + | θ | + | x | < δ } . By the orthogonality between ker L and χ , and Lemma 3.1, we have ∂F∂ ( λ, θ, x ) (1 , , φ ) = (Λ φ, χ ) L − ( iφ, χ ) L ( φ ′ , χ ) L (Λ φ, iφ ) L − ( iφ, iφ ) L ( φ ′ , iφ ) L (Λ φ, φ ′ ) L − ( iφ, φ ′ ) L ( φ ′ , φ ′ ) L = (Λ φ, χ ) L −k φ k L
00 0 k φ ′ k L . Since (Λ φ, χ ) L = 0 by Lemma 3.1 (3), we deduce thatdet ∂F∂ ( λ, θ, x ) (1 , , φ ) = 0 . (3.3)Combined with F (1 , , φ ) = 0, the implicit function theorem implies that thereexist constants ¯ α > δ > C -mappings (Λ , Θ , X ) : V ¯ α → Ω ¯ δ such that F (Λ( u ) , Θ( u ) , X ( u ); u ) = 0 for all u ∈ V ¯ α (3.4)and | Λ( u ) − | + | Θ( u ) | + | X ( u ) | . k u − φ k H for all u ∈ V ¯ α . (3.5)By the expression of ε (Λ( u ) , Θ( u ) , X ( u )) and (3.5), one can compute easily that k ε (Λ( u ) , Θ( u ) , X ( u )) k H . k u − φ k H for u ∈ V ¯ α . In particular, for α ∈ (0 , ¯ α ) we have k ε (Λ( u ) , Θ( u ) , X ( u )) k H . α, | Λ( u ) − | . α for u ∈ V α . (3.6)By possibly choosing α smaller, we can extend Λ, Θ, and X to the functions definedon the tubular neighborhood U α (see, e.g., [24] for more details). This completesthe proof. (cid:3) Control of the modulation parameters.
Now we derive the equation for ε and estimate on the modulation parameters.Let u ∈ U α and u ( t ) be the solution of (1.1) with u (0) = u . We denote theexit times from the tubular neighborhood U α by T ± α = inf { t > u ( ± t ) / ∈ U α } . We set I α = ( − T − α , T + α ). Since u ( t ) ∈ U α for t ∈ I α , we can define λ ( t ) = Λ( u ( t )) , θ ( t ) = Θ( u ( t )) , x ( t ) = X ( u ( t )) , (3.7)where the functions (Λ , Θ , X ) are given in Proposition 3.2. We see that λ ( t ), θ ( t ),and x ( t ) are C -functions on I α . For t ∈ I α we denote v ( t ) = v ( t, y ) = λ ( t ) / e − iθ ( t ) u ( t, λ ( t ) y + x ( t ))(3.8)and define the function ε ( t ) by ε ( t ) = ε ( λ ( t ) , θ ( t ) , x ( t ); u ( t )) = v ( t ) − φ ω,c . (3.9)We rescale the time as follows. We set˜ s ( t ) = Z t dτλ ( τ ) , ˜ I α = ˜ s ( I α ) . Obviously t ˜ s ( t ) is strictly increasing, so the inverse function ˜ t := ˜ s − exists.For a function I α ∋ t f ( t ), we define ˜ I α ∋ s ˜ f ( s ) by˜ f ( s ) = f (˜ t ( s )) . We note that ˜ f s ( s ) = f t ( t ) λ ( t ) for s = ˜ s ( t ) . (3.10)For simplicity of notations, in what follows we omit “tilde” over the functions ofthe variable s although it is the same symbol as the function of the variable t . Lemma 3.3.
For s ∈ I α , ε ( s ) satisfies iε s = Lε + ( θ s − ω ) φ ω,c + (cid:16) x s λ − c (cid:17) iφ ′ ω,c + λ s λ i Λ φ ω,c (3.11) + ( θ s − ω ) ε + (cid:16) x s λ − c (cid:17) iε y + λ s λ i Λ ε + R ( ε ) , where R ( ε ) is the sum of second and higher order terms of ε explicitly written as R ( ε ) = − i | ε | φ ′ ω,c − i Re( εφ ω,c ) ε y − b { Re( εφ ω,c ) } φ ω,c − b | φ ω,c | | ε | φ ω,c − b | φ ω,c | Re( εφ ω,c ) ε − i | ε | ε y − b | ε | Re( εφ ω,c ) φ ω,c − b { Re( εφ ω,c ) } ε − b | φ ω,c | | ε | ε − b | ε | φ ω,c − b | ε | Re( εφ ω,c ) ε − b | ε | ε, and there exists C > such that (3.12) Z | R ( ε ) | ≤ C ( k ε k L + k ε k L k ε y k L ) for ε ∈ H ( R ) with k ε k H ≤ .Proof. By direct calculations we see that v ( t ) satisfies the equation iλ v t = − v yy − i | v | v y − b | v | v + λ t λi Λ v + θ t λ v + x t λiv y . By rescaling the time and (3.10), we have iv s = − v yy − i | v | v y − b | v | v + λ s λ i Λ v + θ s v + x s λ iv y . By substituting v ( s ) = φ + ε ( s ), we obtain that iε s = iv s = − v yy − i | v | v y − b | v | v + λ s λ i Λ v + θ s v + x s λ iv y (3.13) = − ( φ + ε ) yy − i | φ + ε | ( φ + ε ) y − b | φ + ε | ( φ + ε )+ λ s λ i Λ( φ + ε ) + θ s ( φ + ε ) + x s λ i ( φ + ε ) y . We now set R ( ε ) = − i | φ + ε | ( φ + ε ) y + i | φ | φ ′ + i | φ | ε y + 2 i Re( εφ ) φ ′ = − i | ε | φ ′ − i Re( εφ ) ε y − i | ε | ε y ,R ( ε ) = − b | φ + ε | ( φ + ε ) + b | φ | φ + b | φ | ε + 4 b | φ | Re( εφ ) φ = − b (cid:16) { Re( εφ ) } φ + | ε | φ + 4 | ε | Re( εφ ) φ + 2 | φ | | ε | φ + 4 { Re( εφ ) } ε + | ε | ε + 4 | φ | Re( εφ ) ε + 4 | ε | Re( εφ ) ε + 2 | φ | | ε | ε (cid:17) , and R ( ε ) = R ( ε ) + R ( ε ). By the Sobolev embedding we have Z ( | R ( ε ) | + | R ( ε ) | ) . k ε k L + k ε k L k ε y k L for ε ∈ H ( R ) with k ε k H ≤ iε s = − ( φ + ε ) yy + R ( ε ) − i | φ | φ ′ − i | φ | ε y − i Re( εφ ) φ ′ + R ( ε ) − b | φ | φ − b | φ | ε − b | φ | φ ε + λ s λ i Λ( φ + ε ) + θ s ( φ + ε ) + x s λ i ( φ + ε ) y = − ε yy − i | φ | ε y − i Re( εφ ) φ ′ − b | φ | ε − b | φ | φ ε − φ ′′ − i | φ | φ ′ − b | φ | φ + λ s λ i Λ( φ + ε ) + θ s ( φ + ε ) + x s λ i ( φ + ε ) y + R ( ε ) . By using the relations − ε yy − i | φ | ε y − i Re( εφ ) φ ′ − b | φ | ε − b | φ | φ ε = Lε − ωε − ciε y , − φ ′′ − i | φ | φ ′ − b | φ | φ = − ωφ − ciφ ′ , we obtain (3.11). (cid:3) We note that from Proposition 3.2,( ε ( s ) , χ ω,c ) L = ( ε ( s ) , iφ ω,c ) L = ( ε ( s ) , φ ′ ω,c ) L = 0 , (3.14) k ε ( s ) k H ≤ Cα, | λ ( s ) − | ≤ Cα (3.15)hold for α ∈ (0 , α ) and s ∈ I α , where C is independent of α and s . Lemma 3.4.
Let b ≥ and c = 2 κ √ ω . For s ∈ I α , the following equalities hold. λ s λ (Λ φ ω,c , χ ω,c ) L = − ( ε, L ω,c iχ ω,c ) L − ( θ s − ω )( ε, iχ ω,c ) L + (cid:16) x s λ − c (cid:17) ( ε, χ ′ ω,c ) L + λ s λ ( ε, Λ χ ω,c ) L − ( R ( ε ) , iχ ω,c ) L , ( θ s − ω ) k φ ω,c k L = − ( ε, L ω,c φ ω,c ) L − ( θ s − ω )( ε, φ ω,c ) L − (cid:16) x s λ − c (cid:17) ( ε, iφ ′ ω,c ) L − λ s λ ( ε, Λ iφ ω,c ) L − ( R ( ε ) , φ ω,c ) L , (cid:16) x s λ − c (cid:17) k φ ′ ω,c k L = − ( ε, L ω,c iφ ′ ω,c ) L − ( θ s − ω )( ε, iφ ′ ω,c ) L + (cid:16) x s λ − c (cid:17) ( ε, φ ′′ ω,c ) L + λ s λ ( ε, Λ φ ′ ω,c ) L − ( R ( ε ) , iφ ′ ω,c ) L . Moreover, there exist
C > and α ∈ (0 , α ) such that for s ∈ I α , the followingestimate holds. (cid:12)(cid:12)(cid:12)(cid:12) λ s λ (cid:12)(cid:12)(cid:12)(cid:12) + | θ s − ω | + (cid:12)(cid:12)(cid:12) x s λ − c (cid:12)(cid:12)(cid:12) ≤ C k ε ( s ) k L . (3.16) Proof.
By differentiating the orthogonal relation ( ε ( s ) , χ ) L = 0 with respect to s ,we have the first relation in the statement as follows:0 = ( ε s , χ ) L = − ( iLε, χ ) L − ( θ s − ω )( iφ, χ ) L + (cid:16) x s λ − c (cid:17) ( φ ′ , χ ) L + λ s λ (Λ φ, χ ) L − ( θ s − ω )( iε, χ ) L + (cid:16) x s λ − c (cid:17) ( ε y , χ ) L + λ s λ (Λ ε, χ ) L − ( iR ( ε ) , χ ) L = ( ε, Liχ ) L + λ s λ (Λ φ, χ ) L + ( θ s − ω )( ε, iχ ) L − (cid:16) x s λ − c (cid:17) ( ε, χ ′ ) L − λ s λ ( ε, Λ χ ) L + ( R ( ε ) , iχ ) L , where we used ( iφ, χ ) L = ( φ ′ , χ ) L = 0 in the last equality.From Lemma 3.1 we recall that the following equalities hold.(Λ φ, iφ ) L = (Λ φ, φ ′ ) L = ( iφ ′ , φ ) L = 0 . By differentiating the relation ( ε ( s ) , iφ ) L = 0 with respect to s , we obtain thesecond relation as0 = ( ε s , iφ ) L = − ( iLε, iφ ) L − ( θ s − ω )( iφ, iφ ) L + (cid:16) x s λ − c (cid:17) ( φ ′ , iφ ) L + λ s λ (Λ φ, iφ ) L − ( θ s − ω )( iε, iφ ) L + (cid:16) x s λ − c (cid:17) ( ε y , iφ ) L + λ s λ (Λ ε, iφ ) L − ( iR ( ε ) , iφ ) L = − ( ε, Lφ ) L − ( θ s − ω ) k φ k L − ( θ s − ω )( ε, φ ) L − (cid:16) x s λ − c (cid:17) ( ε, iφ ′ ) L − λ s λ ( ε, i Λ φ ) L − ( R ( ε ) , φ ) L . Similarly, by differentiating the relation ( ε ( s ) , φ ′ ) L = 0 with respect to s , we obtainthe third relation as0 = ( ε s , φ ′ ) L = − ( iLε, φ ′ ) L − ( θ s − ω )( iφ, φ ′ ) L + (cid:16) x s λ − c (cid:17) ( φ ′ , φ ′ ) L + λ s λ (Λ φ, φ ′ ) L − ( θ s − ω )( iε, φ ′ ) L + (cid:16) x s λ − c (cid:17) ( ε y , φ ′ ) L + λ s λ (Λ ε, φ ′ ) L − ( iR ( ε ) , φ ′ ) L = ( ε, Liφ ′ ) L + (cid:16) x s λ − c (cid:17) k φ ′ k L + ( θ s − ω )( ε, iφ ′ ) L − (cid:16) x s λ − c (cid:17) ( ε, φ ′′ ) L − λ s λ ( ε, Λ φ ′ ) L + ( R ( ε ) , iφ ′ ) L . From three relations above and (3.12), we obtain (cid:12)(cid:12)(cid:12)(cid:12) λ s λ (cid:12)(cid:12)(cid:12)(cid:12) + | θ s − ω | + (cid:12)(cid:12)(cid:12) x s λ − c (cid:12)(cid:12)(cid:12) . k ε k L + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) λ s λ (cid:12)(cid:12)(cid:12)(cid:12) + | θ s − ω | + (cid:12)(cid:12)(cid:12) x s λ − c (cid:12)(cid:12)(cid:12)(cid:19) k ε k L . By (3.15) and taking α small enough, we obtain the estimate (3.16). (cid:3) Error estimates.
In this subsection, we derive the uniform estimate of ε ( s )for s ∈ I α . Assume that ε ∈ H ( R ) satisfies( ε , χ ω,c ) L = ( ε , iφ ω,c ) L = ( ε , φ ′ ω,c ) = 0 . (3.17)We set u = φ ω,c + ε . From (3.4) and (3.17), we have λ (0) = Λ( u ) = 1 , θ (0) = Θ( u ) = 0 , x (0) = X ( u ) = 0 , which implies that ε (0) = ε ( λ (0) , θ (0) , x (0)) = ε (1 , ,
0) = u − φ ω,c = ε . We define E e ( ε ) = E ( φ ω,c + ε ) − E ( φ ) ,M e ( ε ) = M ( φ ω,c + ε ) − M ( φ ω,c ) = 2( φ ω,c , ε ) L + M ( ε ) ,P e ( ε ) = P ( φ ω,c + ε ) − P ( φ ω,c ) = 2( iφ ′ ω,c , ε ) + P ( ε ) ,S e ( ε ) = S ω,c ( φ + ε ) − S ω,c ( φ ) = E e ( ε ) + ω M e ( ε ) + c P e ( ε ) . Lemma 3.5.
For ε ∈ H ( R ) , we have E e ( ε ) = − ω ( φ ω,c , ε ) L − c ( iφ ′ ω,c , ε ) L + O ( k ε k H ) ,M e ( ε ) = 2( φ ω,c , ε ) L + O ( k ε k H ) ,P e ( ε ) = 2( iφ ′ ω,c , ε ) L + O ( k ε k H ) ,S e ( ε ) = 12 h L ω,c ε, ε i + O ( k ε k H ) = O ( k ε k H ) . Proof.
Since S ′ ( φ ) = 0, this is equivalent to E ′ ( φ ) = − ωφ − ciφ ′ . By the Taylor expansion we have E e ( ε ) = E ( φ + ε ) − E ( φ ) = h E ′ ( φ ) , ε i + O ( k ε k H )= − ω ( φ, ε ) L − c ( iφ ′ , ε ) L + O ( k ε k H ) ,S e ( ε ) = S ( φ + ε ) − S ( φ ) = 12 h Lε, ε i + O ( k ε k H ) . The estimates for M e and P e are trivial from the definition. (cid:3) Lemma 3.6.
Let b ≥ and c = 2 κ √ ω . For s ∈ I α , we have M e ( ε ( s )) = M e ( ε ) , P e ( ε ( s )) = λ ( s ) P e ( ε ) , E e ( ε ( s )) = λ ( s ) E e ( ε ) . Proof.
A direct computation shows that M ( φ + ε ( s )) = M ( v ( s )) = M ( u ( s )) = M ( u ) = M ( φ + ε ) . By expanding both sides we deduce that2( φ, ε ( s )) L + M ( ε ( s )) = 2( φ, ε ) L + M ( ε ) , which is the desired equality.Since E ( φ ) = P ( φ ) = 0 from the assumption, we have E e ( ε ( s )) = E ( φ ω,c + ε ( s )) = E ( v ( s )) , P e ( ε ( s )) = P ( φ ω,c + ε ( s )) = P ( v ( s )) . Therefore, we deduce that P e ( ε ( s )) = P ( v ( s )) = λ ( s ) P ( u ( t ( s ))) = λ ( s ) P ( u ) = λ ( s ) P e ( ε ) ,E e ( ε ( s )) = E ( v ( s )) = λ ( s ) E ( u ( t ( s ))) = λ ( s ) E ( u ) = λ ( s ) E e ( ε ) . This completes the proof. (cid:3)
Lemma 3.7.
Let b > and c = 2 κ √ ω . Then there exist C > and α ∈ (0 , α ) such that for any α ∈ (0 , α ) and s ∈ I α , we have k ε ( s ) k H ≤ C (cid:0) α | ω ( φ ω,c , ε ) L + c ( iφ ′ ω,c , ε ) L | (3.18) + α | ω ( φ ω,c , ε ) L + c ( iφ ′ ω,c , ε ) L | + k ε k H (cid:1) . Proof.
Since ω > c / α small enough, S e ( ε ( s )) = 12 h Lε ( s ) , ε ( s ) i + O ( k ε ( s ) k H ) & k ε ( s ) k H . On the other hand, we deduce from Lemmas 3.5 and 3.6 that S e ( ε ( s )) = λ ( s ) E e ( ε ) + ω M e ( ε ) + λ ( s ) c P e ( ε )= S e ( ε ) + ( λ ( s ) − E e ( ε ) + ( λ ( s ) − c P e ( ε )= ( λ ( s ) − (cid:16) E e ( ε ) + c P e ( ε ) (cid:17) + ( λ ( s ) − E e ( ε ) + O ( k ε k H )= ( λ ( s ) − (cid:0) − ω ( φ, ε ) L − c ( iφ ′ , ε ) L (cid:1) − ( λ ( s ) − (cid:0) ω ( φ, ε ) L + c ( iφ ′ , ε ) L (cid:1) + O ( k ε k H ) . Therefore, combined with (3.15), we obtain (3.18). (cid:3) Virial identities
In this section we organize virial identities of (1.1). Let u be the H -solution of(1.1) with u (0) = u ∈ H ( R ), which is defined on a maximal interval ( − T min , T max ). Proposition 4.1 (Virial identity) . For u ∈ H ( R ) such that R x | u | < ∞ , wehave the following relations: ddt Z x | u | = 4 Im Z xu x u + Z x | u | , (4.1) ddt Im Z xu x u = 4 E ( u )(4.2) for t ∈ ( − T min , T max ) . Proof.
See [45, Lemma 2.2] and [3, Proposition 6.5.1]. (cid:3)
The first relation (4.1) is different from the one of (NLS) due to the appearanceof the second term in the right-hand side. On the other hand, the second relation(4.2) is the same as (NLS). We take advantage of the latter relation for the proofof instability.We now assume that u (0) = u ∈ U α . We recall that v ( t ) and ε ( t ) are definedin (3.8) and (3.9), respectively. We rescale the time variable t to s as in Section 3.Following [31], we rewrite the virial relation in terms of ε ( s ). We denote J [ v ] = Im Z yv y v dy = − Re Z iyv y v dy. Then J [ ε ] is represented as follows. Lemma 4.2.
Let b ≥ and c = 2 κ √ ω . Assume that R x | u | < ∞ . For s ∈ I α ,we have J [ ε ( s )] = 2( ε ( s ) , i Λ φ ω,c ) L + J [ u ( s )] + x ( s ) P ( u ) . (4.3) Proof.
From the phase and scaling invariance of J , we have J [ v ( s )] = J [ u ( s, · + x ( s )] = J [ u ( s )] + x ( s ) P ( u ) . (4.4)On the other hand, J [ v ( s )] is rewritten as J [ v ( s )] = J [ ε ( s ) + φ ] = J [ ε ( s )] − ε, i Λ φ ) L + J [ φ ] . (4.5)By Lemma 3.1, J [ φ ] is rewritten as J [ φ ] = ( iφ, yφ ′ ) L = ( iφ, φ + yφ ′ ) L = ( iφ, Λ φ ) L = 0 . (4.6)By combining (4.4), (4.5), and (4.6), we obtain (4.3). (cid:3) The first term in the right-hand side of (4.3)( ε ( s ) , i Λ φ ω,c ) L = Im Z ε ( s )Λ φ ω,c (4.7)plays an essential role in our proof of instability. We note that (4.7) is well-definedwithout the assumption R x | u | < ∞ . From the equation (3.11), we have dds ( ε ( s ) , i Λ φ ) L = − ( iε s ( s ) , Λ φ ) L = − (cid:18) Lε + ( θ s − ω ) φ + i (cid:16) x s λ − c (cid:17) φ ′ + i λ s λ Λ φ + ( θ s − ω ) ε + i (cid:16) x s λ − c (cid:17) ε y + i λ s λ Λ ε + R ( ε ) , Λ φ (cid:19) L for s ∈ I α . We note that ( φ, Λ φ ) L = ( i Λ φ, Λ φ ) L = 0 and ( iφ ′ , Λ φ ) L = 0 byLemma 3.1 (2). Therefore, by (3.12) and (3.16), we deduce that dds ( ε ( s ) , i Λ φ ) L = − ( ε ( s ) , L Λ φ ) L + O ( k ε ( s ) k H )(4.8)for s ∈ I α , where α > L Λ φ = − ωφ − ciφ ′ , we obtain the following claim. Lemma 4.3.
Let b ≥ and c = 2 κ √ ω . There exists C > such that for s ∈ I α , (cid:12)(cid:12)(cid:12)(cid:12) dds ( ε ( s ) , i Λ φ ω,c ) L − ( ε ( s ) , ωφ ω,c + ciφ ′ ω,c ) L (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ε ( s ) k H . (4.9) Proof of instability
We are now in a position to complete the proof of Theorem 1.3. We first notethat by Lemma 3.5, the second term in the left-hand side of (4.9) is rewritten as( ε ( s ) , ωφ + ciφ ′ ) L = ωM e ( ε ( s )) + c P e ( ε ( s )) + O ( k ε ( s ) k H ) . By Lemma 3.6 we have ωM e ( ε ( s )) + c P e ( ε ( s )) = ωM e ( ε ) + c λ ( s ) P e ( ε )= 2 ω ( ε , φ ) L + cλ ( s )( ε , iφ ′ ) L + O ( k ε k H ) . Therefore, we obtain the following expression:( ε ( s ) , ωφ + ciφ ′ ) L = 2 ω ( ε , φ ) L + cλ ( s )( ε , iφ ′ ) L (5.1) + O ( k ε k H ) + O ( k ε ( s ) k H ) . Proof of Theorem 1.3.
Let α, β > ε ∈ H ( R )satisfies 0 < k ε k H ≤ β | ( ε , φ ) L | , ε ⊥ { χ, iφ, φ ′ , iφ ′ } . Let u := φ + ε and let α satisfy0 < α < min { α , α } < α < . In what follows, we only consider the case ( ε , φ ) L > ε , φ ) L < u ( t ) ∈ U α for all t ∈ R . Then it follows that I α = R . FromLemma 3.7, we have sup s ∈ R k ε ( s ) k H . α ( ε , φ ) L + k ε k H . We note that sup s ∈ R | λ ( s ) − | . α by (3.15). Therefore, by Lemma 4.3 and (5.1),we have dds ( ε ( s ) , i Λ φ ) L & ( ε , φ ) L − α ( ε , φ ) L + O ( k ε k H ) & (1 − α − Cβ )( ε , φ ) L , where the constant C is independent of ε , α , β and s . Therefore, by taking α, β > dds ( ε ( s ) , i Λ φ ) L & ( ε , φ ) L > s ∈ R . This inequality yields that( ε ( s ) , i Λ φ ) L → ∞ as s → ∞ . On the other hand, from (3.15) we havesup s ∈ R | ( ε ( s ) , i Λ φ ) L | . k Λ φ k L < ∞ , which is a contradiction. This completes the proof. (cid:3) Appendix A. Relation to instability theory on (gKdV)
By following the argument of [8], we review the instability theory of the soliton Q ( · − t ) for the L -critical generalized KdV equation u t + ( u xx + u ) x = 0 , ( t, x ) ∈ R × R , (gKdV)and see a relation to our proof of Theorem 1.3.We define a tubular neighborhood around Q by U α = { u ∈ H ( R ) : inf y ∈ R k u − Q ( · − y ) k H < α } . The linearized operator L around Q is given by Lv = − v xx + v − Q v for v ∈ H ( R ) . We note that L satisfies the following properties: LQ = − Q , ker L = span { Q ′ } . We consider the initial data u = Q + ε such that ε ∈ H ( R ) satisfies( ε , Q ) L = ( ε , Q ′ ) L = 0 . (A.1)Let u ( t ) be the solution of (gKdV) with u (0) = u . In the same way as in Section3, one can prove that there exist α > C -functions λ ( t ) > x ( t ) ∈ R such that if u ( t ) ∈ U α for all t ≥
0, then ε ( t ) = ε ( t, y ) defined by ε ( t, y ) = λ ( t ) / u ( t, λ ( t ) y + x ( t )) − Q ( y )satisfies ( ε ( t ) , Q ) L = ( ε ( t ) , Q ′ ) L = 0 for all t ≥ . (A.2)We rescale the time t s by dsdt = λ ( t ) . A direct calculation shows that ε ( s )satisfies ε s = ( Lε ) y + λ s λ Λ Q + (cid:16) x s λ − (cid:17) Q y + λ s λ Λ ε + (cid:16) x s λ − (cid:17) ε y − r ( ε ) y , (A.3)where r ( ε ) is the sum of second and higher order terms of ε . By (A.2) and (A.3)one can prove that (cid:12)(cid:12)(cid:12)(cid:12) λ s λ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) x s λ − (cid:12)(cid:12)(cid:12) . k ε ( s ) k L for all s ≥ . (A.4)We now introduce the following functional J ( s ) = Z ε ( s ) Z y −∞ Λ Q, (A.5)which corresponds to (4.7) as a Lyapunov functional. As pointed out in [8], if weconsider the exponentially decaying data as | ε ( x ) | . ce − δ | x | for some δ > , (A.6)it is rather easy to show the L -exponential decay on the right of the soliton. Inparticular, (A.5) is well-defined for all s ≥
0. From (A.3) and (A.4), one can obtaineasily that(A.7) dds J ( s ) = − Z ε ( s ) L Λ Q − λ s λ J ( s ) − (cid:18)Z Q (cid:19) ! + O ( k ε ( s ) k L ) . Here we define a rescaled functional of J by K ( s ) = λ ( s ) / J ( s ) − (cid:18)Z Q (cid:19) ! . It follows from (A.7) that dds K ( s ) = − λ ( s ) / Z ε ( s ) L Λ Q + O ( k ε ( s ) k L ) , which corresponds to (4.8). By using the relation L Λ Q = − Q , we have dds K ( s ) = 2 λ ( s ) / Z ε ( s ) Q + O ( k ε ( s ) k L ) , which corresponds to (4.9). Therefore, if we assume (A.1), (A.6) and0 < k ε k H ≤ b Z ε Q (A.8)for suitably small b >
0, we can complete the proof of instability of the soliton.We conclude that the functionals (4.7) and (A.5) play an essential role in theproof of instability of the degenerate solitons in (1.1) and (gKdV), respectively, andthat the unstable directions are determined by L Λ φ for (1.1) and L Λ Q for (gKdV),respectively. Acknowledgments
N.F. was supported by JSPS KAKENHI Grant Number 20K14349 and M.H. byJSPS KAKENHI Grant Number JP19J01504.
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Fukaya) Department of Mathematics, Tokyo University of Science, Tokyo, 162-8601, Japan Email address : [email protected] (M. Hayashi) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502,Japan Email address ::