Transverse instability for nonlinear Schrödinger equation with a linear potential
aa r X i v : . [ m a t h . A P ] O c t Transverse instability for nonlinear Schr¨odingerequation with a linear potential
YOHEI YAMAZAKI ∗ Department of Mathematics, Kyoto University,Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502, Japan
Abstract
In this paper we consider the transverse instability for a nonlinear Schr¨odingerequation with a linear potential on R × T L , where 2 πL is the period of the torus T L . Rose and Weinstein [18] showed the existence of a stable standing wave fora nonlinear Schr¨odinger equation with a linear potential. We regard the standingwave of nonlinear Schr¨odinger equation on R as a line standing wave of nonlinearSchr¨odinger equation on R × T L . We show the stability of line standing waves for all L >
We consider the nonlinear Schr¨odiner equation with linear potential i∂ t u = − ∆ u + V ( x ) u − | u | p − u, ( t, x, y ) ∈ R × R × T L , (1.1)where p >
1, a potential V : R → R and u = u ( t, x, y ) is an unknown complex-valuedfunction for t ∈ R , x ∈ R and y ∈ T L . Here, T L = R / πL Z and L > V .(V1) There exist C > α > | V ( x ) | ≤ Ce − α | x | .(V2) − ∂ x + V has the lowest eigenvalue − λ ∗ < H ( R × T L ) by using the argumentin [7] and [24]. The equation (1.1) has the following conservation laws: E ( u ) = Z R × T L (cid:18) |∇ u | + 12 V ( x ) | u | − p + 1 | u | p +1 (cid:19) dxdy,Q ( u ) = 12 Z R × T L | u | dxdy, ∗ E-mail addresses: [email protected]
AMS 1991 subject classifications. u ∈ H ( R × T L ).We define a standing wave u ( t ) as a non-trivial solution of (1.1) having the form u ( t ) = e iωt ϕ . Then, e iωt ϕ is a standing wave if and only if ϕ is a non-trivial solution of − ∆ ϕ + ωϕ + V ( x ) ϕ − | ϕ | p − ϕ = 0 , ( x, y ) ∈ R × T L . (1.2)Using the bifurcation theory, Rose and Weinstein [18] showed the existence of the stablestanding wave e iωt ϕ ω for the following nonlinear Schr¨odinger equation i∂ t u = − ∂ x u + V ( x ) u − | u | p − u, ( t, x ) ∈ R × R . (1.3)Then, the standing wave e iωt ϕ ω satisfies the following. Proposition 1.1.
Let ψ ∗ be the eigenfunction of − ∂ x + V ( x ) corresponding to − λ ∗ with ψ ∗ > and k ψ ∗ k L = 1 . Then, there exists ω ∗ > λ ∗ such that for λ ∗ < ω < ω ∗ , e iωt ϕ ω is astable standing wave of (1.3) satisfying ϕ ω = k ψ ∗ k − p +1 p − L p +1 ( R ) ( ω − λ ∗ ) p − ψ ∗ + r ( ω ) , where k r ( ω ) k H ( R ) = O (( ω − λ ∗ ) p − +1 ) . Moreover, L + ω = − ∂ x + ω + V − p | ϕ ω | p − has theexactly one negative eigenvalue − λ ω and does not have the zero eigenvalue. We define the line standing wave e iωt ˜ ϕ ω of (1.1) as˜ ϕ ω ( x, y ) = ϕ ω ( x ) , ( x, y ) ∈ R × T L . In this paper, we consider the transverse instability of the line standing wave e iωt ˜ ϕ ω . Thestability of standing waves is defined as follows. Definition 1.2.
We say the standing wave e iωt ϕ is orbitally stable in H if for any ε > δ > u ∈ H ( R × T L ) with k u − ϕ k H < δ , the solution u ( t ) of (1.1) with the initial data u (0) = u exists globally in time and satisfiessup t ≥ inf θ ∈ R ,y ∈ T L (cid:13)(cid:13) u ( t, · , · − y ) − e iθ ϕ ( · , · − y ) (cid:13)(cid:13) H < ε. Otherwise, we say the standing wave e iωt ϕ is orbitally unstable in H .The transverse instability for KP-I or KP-II equation is treated in [1, 15, 16, 19, 20,21, 22]. In [1], Alexander-Pego-Sachs studied the linear instability for line solitons of KP-Iand KP-II. In [16], Mizumachi-Tzvetkov proved the asymptotic stability for line solitonsof KP-II on R × T . Modulating the local phase and the local amplitude of line solitons,Mizumachi showed the asymptotic stability for line solitons of KP-II on R in [15]. Rousset-Tzvetkov proved the transverse instability for line solitons of KP-I on R in [19] and on R × T L in [20]. In [22], Rousset-Tzvetkov showed the stability of line solitons for KP-I on R × T L with small L >
0. Moreover, Rousset-Tzvetkov proved the existence of the criticalperiod 4 / √ L of the transverse direction. Namely, a line soliton for KP-Ion R × T L is stable for 0 < L < / √ L > / √ R in [19]and on R × T L in [20]. To prove the instability, Rousset-Tzvetkov applied the argumentof Grenier [9]. Rousset-Tzvetkov constructed the high order approximate solution with anunstable eigenmode and showed a precise estimate of the growth of the semi-group gener-ated by the linearized operator. To construct the high order approximate solution, we usethe regularity of the nonlinearity | u | u in the sense of Fr´echet differentiation. In [25], theauthor studied the transverse instability for line standing waves of a system of nonlinearSchr¨odinger equations on R × T L which was treated in [4]. In [25], the existence of thecritical period for a period L was also proved, which was suggested by Rousset-Tzvetkov.Constructing the estimate for high frequency parts of solutions and using the existenceof local solutions, the author showed the transverse instability for line standing waves ofequations with the general power nonlinearity. In [26], the author considers the stabilityfor a line standing wave of (1.1) with V = 0. The application of the argument in [25] yieldsthe existence of the critical period for a line standing wave of (1.1) with V = 0. For (1.1)with V = 0 and the critical period, the linearized operator around the line standing wave isdegenerate. Therefore, we can not directly apply the argument in Grillakis-Shatah-Strauss[10, 11]. Since the linearized operator around the line standing wave with the criticalperiod does not have any unstable eigenvalues, we can not show the instability by theargument based on the occurrence of unstable eigenmode in [5, 10, 20, 25]. Moreover,the third order term of the Lyapunov functional around the line standing wave with thecritical period does not appear. Thus, we can not apply the argument for the degeneratecase of the stability in [14]. The transverse instability comes from the symmetry breakingbifurcation. In [26], applying the bifurcation result for symmetry breaking bifurcation andthe stability result for the degenerate case in [13], the author showed the stability for theline standing wave with critical period for some exponents p ≥ e iωt ˜ ϕ ω and obtain the critical periodbetween the stability and the instability. Theorem 1.3.
There exists ω ∗ , > λ ∗ such that for λ ∗ < ω < ω ∗ , the followings twoassertions hold: (i) If < L < ( λ ω ) − , then the standing wave e iωt ˜ ϕ ω is stable. (ii) If ( λ ω ) − < L , then the standing wave e iωt ˜ ϕ ω is unstable. In the Second theorem, we show the stability for the line standing wave e iωt ˜ ϕ ω withthe critical period L = ( λ ω ) − / . Theorem 1.4.
Let p ≥ and p ∗ = 9 + √ . Then there exists λ ∗ < ω p satisfying the following two properties: If p < p ∗ and λ ∗ < ω < ω p , then the standing wave e iωt ˜ ϕ ω of (1.1) with L = ( λ ω ) − / is stable. (ii) If p ∗ ≤ p and λ ∗ < ω < ω p , then the standing wave e iωt ˜ ϕ ω of (1.1) with L = ( λ ω ) − / is unstable. The proof of Theorem 1.3 follows form the spectrum analysis of the linearized operatorand the estimate of high frequency parts of solution by the argument in [25]. To show thegrowth of the semi-group generated by the linearized operator, we use the assumption ofthe decay for the linear potential V . For the proof of Theorem 1.4, we apply the bifurcationanalysis for the symmetry breaking bifurcation and the argument for the stability in [26].In [26], to prove the stability for the line standing wave with the critical period, we showthe increase of L -norm of the symmetry breaking standing wave with respect to thebifurcation parameter or the decrease of it. To show the increase of L -norm, we need tocalculate an integral of a solution of an ordinary differential equation which comes fromthe linearized equation of one dimensional Schr¨odinger equation around a standing wave.Since it is difficult to obtain the explicit solution of the ordinary differential equation inthe argument in [25], we can not calculate the exact value of the integral and we estimatethe value of the integral. Therefore, it is not known whether the line standing wave isstable or unstable for some nonlinear Schr¨odinger equations with the power nonlinearity | u | p − u which has some exponent p ∈ (2 , − ∂ x + V with respect tothe lowest eigenvalue. Since the line standing wave of the nonlinear Schr¨odinger equationstudied in [26] comes from the standing wave of the one dimensional nonlinear Schr¨odingerequation which has the scale invariant, we need to study the fully nonlinear structure ofthe Lyapunov functional around the line standing wave. In this paper, using the smallnessof the line standing wave of (1.1) and the expansion of the standing wave with respect tothe parameter ω , we weaken the nonlinear structure of the Lyapunov functional aroundthe line standing wave of (1.1). Therefore, we can evaluate a value of the integral andmake a close investigation into the stability for all exponents p ≥ In this section, we investigate properties of the linearized operator of (1.1) around thestanding wave e iωt ˜ ϕ ω .Let H ( X ) = { u : X → C | R X ( |∇ u | + | u | ) dx < ∞} and H ( X, R ) = { u : X → R | R X ( |∇ u | + | u | ) dx < ∞} . Let ψ ω be the eigenfunction of L + ω corresponding to − λ ω with k ψ ω k L ( R ) = 1 and ψ ω >
0. We define the action S ω ( u ) = E ( u ) + ωQ ( u ) . S ω is a conservation law of (1.1) and S ′ ω ( ˜ ϕ ω ) = 0, where S ′ ω is the Fr´echetderivation of S ω . Moreover, we have h S ′′ ω ( ˜ ϕ ω ) u, v i H − ,H = h L + ω (Re u ) , Re v i H − ,H + h L − ω (Im u ) , Im v i H − ,H , where L + ω = − ∆ + ω + V − p | ˜ ϕ ω | p − , L − ω = − ∆ + ω + V − | ˜ ϕ ω | p − . Let
J u = iu = (cid:18) −
11 0 (cid:19) (cid:18) Re u Im u (cid:19) . For u, v ∈ L ( R × T L ), we define h u, v i L = Re Z R × T L u ¯ vdxdy. In the following proposition, we show properties of the spectrum of the linearized operatorfor (1.1) around ˜ ϕ ω . This proposition follows Theorem 1.1 of [21] and Lemma 3.1 of [22](also see Proposition 2.5 of [25].) Proposition 2.1.
Let λ ∗ < ω < ω ∗ . (i) If < L ≤ ( λ ω ) − / , then − J S ′′ ω ( ˜ ϕ ω ) has no positive eigenvalue. (ii) If < L < ( λ ω ) − / , then Ker( S ′′ ω ( ˜ ϕ ω )) = Span { i ˜ ϕ ω } . (iii) If L = ( λ ω ) − / , then Ker( S ′′ ω ( ˜ ϕ ω )) = Span n i ˜ ϕ ω , ψ ω cos yL , ψ ω sin yL o . (iv) If L > ( λ ω ) − / , then − J S ′′ ω ( ˜ ϕ ω ) has a positive eigenvalue and the number of eigen-value of − J S ′′ ω ( ˜ ϕ ω ) with a positive real part is finite.Here, Span { v , . . . , v n } is the real vector space spanned by vectors v , . . . , v n .Proof. We define S ( a ) u = (cid:18) L + ω + a L − ω + a (cid:19) (cid:18) Re u Im u (cid:19) , where u ∈ H ( R ), L + ω = − ∂ x + ω + V − p | ϕ ω | p − and L − ω = − ∂ x + ω + V − | ϕ ω | p − . Then,for u ∈ H ( R × T L ) S ′′ ω ( ˜ ϕ ω ) u ( x, y ) = ∞ X n = −∞ S (cid:16) nL (cid:17) ~u n ( x ) e inyL , where u ( x, y ) = (cid:18) Re u Im u (cid:19) = ∞ X n = −∞ (cid:18) u R,n u I,n (cid:19) e inyL = ∞ X n = −∞ ~u n ( x ) e inyL , u ∈ H ( R × T L ) . − J S ′′ ω ( ˜ ϕ ω ) has an eigenvalue λ if and only if there exists n ∈ Z such that − J S ( n/L ) has the eigenvalue λ .By Proposition 1.1, S ( a ) has no negative eigenvalues for a ≥ ( λ ω ) / . By Theorem3.1 in [17], the number of eigenvalues of − J S ( a ) with the positive real part is less thanor equal to the number of negative eigenvalues of S ( a ). Thus, for a ≥ ( λ ω ) / , J S ( a )has no eigenvalues with the positive real part. (i) follows this. Moreover, the number ofeigenvalues of − J S ′′ ω ( ˜ ϕ ω ) with the positive real part is less than 1 + 2 L/ ( λ ω ) / . Since thekernel of S ( a ) is trivial for a > ( λ ω ) / , the kernel of − J S ( a ) is trivial for a > ( λ ω ) / .Then the kernel of − J S (0) is spanned by i ˜ ϕ ω . Therefore, for a > ( λ ω ) / , the kernel of − J S ′′ ω ( ˜ ϕ ) is spanned by i ˜ ϕ ω and (ii) is verified.The kernel of L + ω + λ ω is spanned by ψ ω and L − ω + λ ω do not has zero eigenvalue. Hence,the kernel of S ′′ ω ( ˜ ϕ ω ) with L = ( λ ω ) − / is spanned by i ˜ ϕ ω , ψ ω cos yL and ψ ω sin yL . This is(iii).Let M ( v, a, λ ) = S ( a )( ψ ω + v ) + J − λ ( ψ ω + v ) , where v ∈ { w ∈ H ( R ) |h v, ψ ω i L ( R ) = 0 } =: ( ψ ω ) ⊥ and a, λ ∈ R . Then, M is a C ∞ function with M (0 , ( λ ω ) / ,
0) = 0 . Since ∂M∂ ( v, a ) (cid:12)(cid:12)(cid:12)(cid:12) ( v,a,λ )=(0 , ( λ ω ) / , ( w, µ ) = 2( λ ω ) / µψ ω + S (( λ ω ) / ) w, by the implicit function theorem, there exist a ( λ ) ∈ R and v ( λ ) ∈ H ( R ) such that a ( λ ) , v ( λ ) are the C ∞ functions, where a (0) = ( λ ω ) / , v (0) = 0 and M ( v ( λ ) , a ( λ ) , λ ) = 0.Then, we have − J S ( a ( λ ))( ψ ω + v ( λ )) = λ ( ψ ω + v ( λ ))for sufficiently small | λ | . Differentiating with respect to λ , we obtain ∂M∂λ ( v ( λ ) , a ( λ ) , λ ) = 2 a ( λ ) a ′ ( λ )( ψ ω + v ( λ )) + S ( a ( λ )) v ′ ( λ ) + J − ( ψ ω + v ( λ ) + λv ( λ )) = 0 . Since v (0) = 0, we have h a (0) a ′ (0) ψ ω , ψ ω i L ( R ) = 0 , and a ′ (0) = 0. Therefore, we have S (( λ ω ) / ) v ′ (0) = − J − ψ ω . Since v ′ (0) ∈ ( ψ ω ) ⊥ and ∂ M∂λ ( v ( λ ) , a ( λ ) , λ ) | λ =0 = 2 a (0) a ′′ (0) ψ ω + S (( λ ω ) / ) v ′′ (0) + J − v ′ (0) = 0 ,a ′′ (0) = − h S (( λ ω ) / ) v ′ (0) , v ′ (0) i H − ( R ) ,H ( R ) λ ω ) / < . From the proof of (i), for a > ( λ ω ) / , − J S ( a ) has no positive eigenvalues. Hence, for suffi-ciently small ε > a ( λ ) on (0 , ε ) has the inverse function λ ( a ) on ( a ( ε ) , ( λ ω ) / )and a ( ε ) < ( λ ω ) / . Namely, − J S ( a ) has the simple positive eigenvalue on ( a ( ε ) , ( λ ω ) / ).Let a = inf { a > | − J S ( b ) has a simple positive eigenvalue for a < b < ( λ ω ) / } , a ∈ ( a , ( λ ω ) / ) the value λ ( a ) be the positive eigenvalue of − J S ( a ). We assume a >
0. By the perturbation theory, there exists { a n } ∞ n =1 ⊂ ( a , ( λ ω ) / ) such that a n → a and lim n →∞ λ ( a n ) = 0or lim n →∞ λ ( a n ) = ∞ . Since there exists
C > |h−
J S ( a ) u, u ) i H − ( R ) ,H ( R ) | ≤ C k u k L ( R ) for a ∈ R , λ ( a )is bounded. Therefore, lim n →∞ λ ( a n ) = 0 . Then, there exists { c n } ∞ n =1 such that k v n k H ( R ) = 1 and − J S ( a n ) v n = λ ( a n ) v n . Here, S ( a ) v n = ( a − a n ) v n − J − λ ( a n ) v n . Since S ( a ) is invertible and ( S ( a )) − is bounded, v n = ( S ( a )) − (( a − a n ) v n − J − λ ( a n ) v n ) → n → ∞ . This is contradiction. Therefore, a = 0.Next we show the coerciveness of L + ω on a function space which follows the proof ofTheorem 3.3 in [10]. Lemma 2.2.
There exist ω ∗ , > λ ∗ and k > such that for λ ∗ < ω < ω ∗ , and u ∈ H ( R , R ) with h ϕ ω , u i L ( R ) = 0 , h L + ω u, u i H − ( R ) ,H ( R ) ≥ k k u k H ( R ) , where h u, v i L ( R ) = Re R R u ¯ vdx .Proof. Let u ∈ H ( R , R ) with h ϕ ω , u i L ( R ) = 0. We decompose u = aψ ∗ + u ⊥ , where a = h u, ψ ∗ i L ( R ) and h ψ ∗ , u ⊥ i L ( R ) = 0. From the spectrum of − ∂ x + V + λ ∗ , there exists k > h ( − ∂ x + V + λ ∗ ) u ⊥ , u ⊥ i H − ( R ) ,H ( R ) ≥ k k u ⊥ k L ( R ) . By k V k L ∞ < ∞ , we have for ε > h ( − ∂ x + V + λ ∗ ) u ⊥ , u ⊥ i H − ( R ) ,H ( R ) ≥ ( k − ε ( k V k L ∞ − λ ∗ )) k u ⊥ k L ( R ) + ε k ∂ x u ⊥ k L ( R ) . Therefore, there exists k ′ > h ( − ∂ x + V + λ ∗ ) u ⊥ , u ⊥ i H − ( R ) ,H ( R ) ≥ k ′ k u ⊥ k H ( R ) . By the assumption h ϕ ω , u i L ( R ) = 0, we have a = − h ϕ ω , u ⊥ i L ( R ) h ϕ ω , ψ ∗ i L ( R ) . h L + ω u, u i H − ( R ) ,H ( R ) = h ( − ∂ x + V + λ ∗ ) u ⊥ , u ⊥ i H − ( R ) ,H ( R ) + λ ω a + h ( ω − λ ∗ − p | ϕ ω | p − ) u ⊥ , u ⊥ i H − ( R ) ,H ( R ) − λ ω h ϕ ω , u ⊥ i L ( R ) h ϕ ω , ψ ∗ i L ( R ) + o ( k u ⊥ k L ( R ) ) ≥ k ′ k u ⊥ k H + λ ω a + h ( ω − λ ∗ − p | ϕ ω | p − ) u ⊥ , u ⊥ i H − ( R ) ,H ( R ) − λ ω k u ⊥ k L ( R ) k ϕ ω k L ( R ) h ϕ ω , ψ ∗ i L ( R ) + o ( k u ⊥ k L ( R ) ) . If | ω − λ ∗ | is sufficiently small, then we obtain the conclusion. In this section, we prove Theorem 1.3. The proof of Theorem 1.3 is similar to the proof ofTheorem 1.5 in [25]. We write the detail of the proof of Theorem 1.3 for readers.
In this subsection, we assume 0 < L < ( λ ω ) − / . The proof of (i) of Theorem 1.3 followsSection 3.1 in [25].The following proposition follows Grillakis-Shatah-Strauss [10] or Colin-Ohta [4](see[3]). Proposition 3.1.
Let e iωt ϕ be a standing wave of (1.1). Assume that there exists aconstant δ > such that h S ′′ ω ( ϕ ) u, u i H − ,H ≥ δ k u k H for all u ∈ H ( R × T L ) satisfying h ϕ, u i L = h J ϕ, u i L = 0 . Then, the standing wave e iωt ϕ is stable. Let u ∈ H ( R × T L ) satisfy h ˜ ϕ ω , u i L = h J ˜ ϕ ω , u i L = 0. Then, We have h S ′′ ω ( ˜ ϕ ω ) u, u i H − ,H = X n ∈ Z h S ( n/L ) u n , u n i H − ,H , where u ( x, y ) = X n ∈ Z u n ( x ) e inyL . Since L − ω and L + ω + λ ω are nonnegative, there exists c > h S ( n/L ) v, v i H − ( R ) ,H ( R ) ≥ c k v k H ( R ) for n ∈ Z \{ } and v ∈ H ( R ). By Proposition 1.1 and Lemma 2.2, there exists c ′ > v ∈ H ( R ) with h ˜ ϕ ω , v i L = 2 πL R R ϕ ω (Re v ) dx = 0 and h J ˜ ϕ ω , v i L =2 πL R R ϕ ω (Im v ) dx = 0 h L + ω (Re v ) , Re v i H − ,H ≥ c ′ k Re v k H , h L − ω (Im v ) , Im v i H − ,H ≥ c ′ k Im v k H . Therefore, (i) of Theorem 1.1 follows from Proposition 3.1.8ransverse instability
The proof of (ii) of Theorem 1.3 follows Section 3.2 in [25].In this subsection, we assume
L > ( λ ω ) / . We define µ ∗ = max { λ > | λ ∈ σ ( − J S ′′ ( ˜ ϕ ω )) } , where σ ( − J S ′′ ( ˜ ϕ ω )) is the spectrum of − J S ′′ ( ˜ ϕ ω ). Then, there exist k ∈ Z and χ ∈ H ( R × T L ) such that k χ k L = 1, χ is eigenfunction of − J S ′′ ω ( ˜ ϕ ω ) corresponding to µ ∗ and χ ( x, y ) = χ ( x ) e ik yL + χ ( x ) e − ik yL , where χ , χ ∈ H ( R ). We define the orthogonal projection P ≤ k as P ≤ k u ( x, y ) = k X n = − k u n ( x ) e inyL , ( x, y ) ∈ R × T L , where u ( x, y ) = ∞ X n = −∞ u n ( x ) e inyL . ( x, y ) ∈ R × T L . A function u ( t ) is a solution of (1.1) if and only if v ( t ) is a solution of the equation ∂ t v = − J ( S ′′ ω ( ˜ ϕ ω ) v + g ( v )) , (3.1)where u ( t ) = e iωt ( ˜ ϕ ω + v ( t )), g ( v ) = (cid:18) | v + ˜ ϕ ω | p − ( v R + ˜ ϕ ω ) − p | ˜ ϕ ω | p − v R − | ˜ ϕ ω | p − ˜ ϕ ω | v + ˜ ϕ ω | p − v I − | ˜ ϕ ω | p − v I (cid:19) , and v R = Re v and v I = Im v . We define u δ ( t ) as the solution of (1.1) with the initialdata ˜ ϕ ω + δχ and v δ ( t ) as the solution of (3.1) with the data δχ . Then, we have that u δ ( t ) = e iωt ( ˜ ϕ ω + v δ ( t )).We show the estimate of nonlinear term in the following lemma which follows Lemma2.4 of [8]. Lemma 3.2.
There exists
C > such that k g ( v ) k L ≤ ( C k v k pH , < p ≤ ,C ( k v k H + k v k pH ) , < p. Proof.
We have || a | p − − | b | p − | ≤ ( | a − b | p − , < p ≤ ,p ( | a | p − + | b | p − ) | a − b | , < p. Since g ( v ( x, y )) = Z ( | θv ( x, y ) + ˜ ϕ ω ( x, y ) | p − − | ˜ ϕ ω ( x, y ) | p − ) v ( x, y ) dθ, k g ( v ) k L ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z ( | θv + ˜ ϕ ω | p − − | ˜ ϕ ω | p − ) vdθ (cid:13)(cid:13)(cid:13)(cid:13) L ≤ Z (cid:13)(cid:13) | θv + ˜ ϕ ω | p − − | ˜ ϕ ω | p − (cid:13)(cid:13) L k v k L dθ ≤ ( C k v k pH , < p ≤ ,C ( k v k H + k v k pH ) , < p. In the following lemma, we estimate the low frequency part of the semi-group.
Lemma 3.3.
For a positive integer k and ε > , there exists C k,ε > such that (cid:13)(cid:13)(cid:13) e − tJS ′′ ω ( ˜ ϕ ω ) P ≤ k v (cid:13)(cid:13)(cid:13) L ≤ C k,ε e ( µ ∗ + ε ) t k v k L , t > , v ∈ L ( R × T L ) . Proof.
By the definition of S ( a ), we have − J S ( a ) = (cid:18) − ∂ x + ω + a + V − | ˜ ϕ ω | p − ∂ x − ω − a − V + p | ˜ ϕ ω | p − (cid:19) . Using the exponential decay rates of V and ˜ ϕ ω and applying the argument for the proofof Proposition [5] and Lemma 6 in [6], we obtain σ ( e − JS ( a ) ) = e σ ( − JS ( a )) . By the definition of µ ∗ , we have that the spectral radius of e − JS ( n/L ) is less than or equalto e µ ∗ for n ∈ Z . Therefore, by Lemma 2 and Lemma 3 in [23] we have (cid:13)(cid:13) e − tJS ( n/L ) v (cid:13)(cid:13) L ( R ) ≤ C n,ε e ( µ ∗ + ε ) t k v k L ( R ) , t > , n ∈ Z , v ∈ L ( R ) . Hence, for t > v ∈ L ( R × T L ), (cid:13)(cid:13)(cid:13) e − tJS ′′ ω ( ˜ ϕ ω ) P ≤ k v (cid:13)(cid:13)(cid:13) L ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X n = − k e − tJS ( n/L ) v n e inyL (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ≤ C k,ε e ( µ ∗ + ε ) t k v k L , where v ( x, y ) = X n ∈ Z v n ( x ) e inyL . In the following lemma, we estimate the high frequency part of v δ ( t ). Lemma 3.4.
There exist a positive integer K and C > such that for δ > and t > k v δ ( t ) k H ≤ C k P ≤ K v δ ( t ) k L + o ( δ ) + o ( k v δ ( t ) k H ) . Proof.
By the Taylor expansion we have that for v ∈ H ( R × T L ) S ω ( ˜ ϕ ω + v ) = S ω ( ˜ ϕ ω ) + h S ′ ω ( ˜ ϕ ω ) , v i H − ,H + 12 h S ′′ ω ( ˜ ϕ ω ) v, v i H − ,H + o ( k v k H ) . Since S ω is conservation law, we have S ω ( ˜ ϕ ω + δχ ) = S ω ( ˜ ϕ ω + v δ ( t )) for t ≥
0. Using S ′ ( ˜ ϕ ω ) = 0 and h S ′′ ω ( ˜ ϕ ω ) χ, χ i H − ,H = h− J S ′′ ω ( ˜ ϕ ω ) χ, J − χ i H − ,H = h µ ∗ χ, J − χ i L = 0 , we have h S ′′ ω ( ˜ ϕ ω ) v δ ( t ) , v δ ( t ) i H − ,H = o ( k v δ ( t ) k H ) + o ( δ ) . We define K as the integer part of 1 + L ( λ ω ) / . Since S ( a ) is positive for a > ( λ ω ) / , weobtain S ′′ ω ( ˜ ϕ ω )( I − P ≤ K ) is positive. By the definition of S ( a ) there exist c, C > h S ( a ) v, v i H − ( R ) ,H ( R ) ≥ c k v k H ( R ) − C k v k L ( R ) for v ∈ H ( R ) and a ∈ R . Thus, k v δ ( t ) k H = k P ≤ K v δ ( t ) k H + k ( I − P ≤ K ) v δ ( t ) k H ≤ C ′ h S ′′ ω ( ˜ ϕ ω )( I − P ≤ K ) v δ ( t ) , ( I − P ≤ K ) v δ ( t ) i H − ,H + C ′ h S ′′ ω ( ˜ ϕ ω ) P ≤ K v δ ( t ) , P ≤ K v δ ( t ) i H − ,H + C ′′ k P ≤ K v δ ( t ) k H ≤ C ′′ k v δ ( t ) k L + o ( δ ) + o ( k v δ ( t ) k H ) . Let ε = min { ( p − µ ∗ / , µ ∗ / } . By Lemma 3.2, Lemma 3.3 and Lemma 3.4, weobtain that k v δ ( t ) k H ≤ Cδe µ ∗ t k χ k L + C Z t (cid:13)(cid:13) e − ( t − s ) JS ω ( ˜ ϕ ω ) P ≤ K g ( v δ ( s )) (cid:13)(cid:13) L ds + o ( δ ) + o ( k v δ ( t ) k H ) ≤ Cδe µ ∗ t + C Z t e (1+ ε ) µ ∗ ( t − s ) ( k v δ ( s ) k H + k v δ ( s ) k pH ) ds + o ( δ ) + o ( k v δ ( t ) k H ) . There exists C > δ > ε > k v δ ( t ) k H ≤ C e µ ∗ t , for t ∈ [0 , T ε ,δ ] , where T ε ,δ = log( ε /δ ) µ ∗ . Then, |h χ, v δ ( T ε ,δ ) i L | = (cid:12)(cid:12)(cid:12)(cid:12) δe µ ∗ T ε ,δ + Z T ε ,δ h χ, − J e − ( T ε ,δ − s ) S ′′ ω ( ˜ ϕ ω ) g ( v δ ( s ) i L ds (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε − C Z T ε ,δ e ( T ε ,δ − s ) µ ∗ ( k v δ ( s ) k H + k v δ ( s ) k pH ) ds ≥ ε − Cε min { p, } . P ≤ ˜ ϕ ω = ˜ ϕ ω , there exists ε > ε > δ > θ ∈ R (cid:13)(cid:13) u δ ( T ε ,δ ) − e iθ ˜ ϕ ω (cid:13)(cid:13) L ≥ (cid:13)(cid:13) ( I − P ≤ )( u δ ( T ε ,δ ) − e iθ ˜ ϕ ω ) (cid:13)(cid:13) L = (cid:13)(cid:13) ( I − P ≤ ) e − iωT ε ,δ u δ ( T ε ,δ ) (cid:13)(cid:13) L = (cid:13)(cid:13) ( I − P ≤ )( e − iωT ε ,δ u δ ( T ε ,δ ) − ˜ ϕ ω ) (cid:13)(cid:13) L . By the definition of χ we have k ( P ≤ k − P ≤ k − ) v k L ≥ |h χ, v i L | , for v ∈ L ( R × T L ) . Therefore, (cid:13)(cid:13) ( I − P ≤ )( e − iωT ε ,δ u δ ( T ε ,δ ) − ˜ ϕ ω ) (cid:13)(cid:13) L ≥ |h χ, v δ ( T ε ,δ ) i L | ≥ ε . This implies that the standing wave e iωt ˜ ϕ ω is unstable. In this section, we assume L = ( λ ω ) − / for 0 < ω − λ ∗ ≪ Lemma 4.1. ϕ ω = k ψ ∗ k − p +1 p − L p +1 ( R ) ( ω − λ ∗ ) p − ψ ∗ + k ψ ∗ k − p +1 p − L p +1 ( R ) ( ω − λ ∗ ) p − +1 ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p + o (( ω − λ ∗ ) p − +1 ) ,∂ ω ϕ ω = k ψ ∗ k − p +1 p − L p +1 ( R ) p − ω − λ ∗ ) p − − ψ ∗ + p k ψ ∗ k − p +1 p − L p +1 ( R ) p − ω − λ ∗ ) p − ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p + o (( ω − λ ∗ ) p − ) . (4.1) where ψ ∗ ,p = k ψ ∗ k − ( p +1) L p +1 ( R ) ψ p ∗ − ψ ∗ , H a = − ∂ x + a + V for a ∈ R and P ⊥ is the orthogonalprojection onto ( ψ ∗ ) ⊥ = { u ∈ L ( R ) |h u, ψ ∗ i L ( R ) = 0 } .Proof. Let ϕ ω, = ( ω − λ ∗ ) − p − ϕ ω = k ψ ∗ k − p +1 p − L p +1 ( R ) ψ ∗ + ˜ r. By the bifurcation argument, h ψ ∗ , ˜ r ( ω ) i L ( R ) = 0. Since ϕ ω is C with respect to ω and( − ∂ x + ω + V ) ϕ ω, − | ϕ ω | p − ϕ ω, = 0 , we have0 = ∂ ω (cid:0) ( − ∂ x + ω + V ) ϕ ω, − | ϕ ω | p − ϕ ω, (cid:1) = k ψ ∗ k − p +1 p − L p +1 ( R ) ψ ∗ + ˜ r + H ω ∂ ω ˜ r − (cid:12)(cid:12)(cid:12)(cid:12) k ψ ∗ k − p +1 p − L p +1 ( R ) ψ ∗ + ˜ r (cid:12)(cid:12)(cid:12)(cid:12) p − (cid:18) k ψ ∗ k − p +1 p − L p +1 ( R ) ψ ∗ + ˜ r (cid:19) − p ( ω − λ ∗ ) (cid:12)(cid:12)(cid:12)(cid:12) k ψ ∗ k − p +1 p − L p +1 ( R ) ψ ∗ + ˜ r (cid:12)(cid:12)(cid:12)(cid:12) p − ∂ ω ˜ r. h ψ ∗ , ˜ r ( ω ) i L ( R ) = h ψ ∗ , k ψ ∗ k − ( p +1) L p +1 ( R ) ψ p ∗ − ψ ∗ i L ( R ) = 0, we obtain ∂ ω ˜ r = k ψ ∗ k − p +1 p − L p +1 ( R ) ( P ⊥ H λ ∗ P ⊥ ) − ( k ψ ∗ k − ( p +1) L p +1 ( R ) ψ p ∗ − ψ ∗ ) + o (1) . In the following lemma, we obtain the derivative of the eigenvalue λ ω . Lemma 4.2.
Let p ≥ . Then, ψ ω = ψ ∗ + p ( ω − λ ∗ )( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p + O (( ω − λ ∗ ) ) ,λ ω =( p − ω − λ ∗ ) + p (2 p − ω − λ ∗ ) k ψ ∗ k − ( p +1) L p +1 ( R ) Z R ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p dx + o (( ω − λ ∗ ) ) . Proof.
There exists δ > { z ∈ C || z + λ ∗ | < δ } ∩ σ ( − ∆ + V ) = {− λ ∗ } . LetΓ = { z ∈ C || z | = δ } be a simple closed curve and projections P ω = 12 πi Z Γ ( L + ω − z ) − dz. Then, for ω > λ ∗ with 0 < ω − λ ∗ ≪ P ω u = h u, ψ ω i L ( R ) ψ ω . Since p ≥ L + ω is C with respect to ω . Therefore, the projection P ω is also C . For ω, ω ′ > λ ∗ , ( h ψ ω ′ , ψ ω i L ( R ) ) − h P ω ′ ψ ω , ψ ω i L ( R ) − o (1) as | ω ′ − ω | →
0. For ω > λ ∗ , ψ ω ′ − ψ ω = P ω ( ψ ω + ψ ω ′ ) − P ω ′ ( ψ ω + ψ ω ′ )1 + h ψ ω ′ , ψ ω i L ( R ) . Thus, ψ ω is C with respect to ω . Let ϕ ω, = ( ω − λ ∗ ) − p − ϕ ω . Since L + ω ψ ω = − λ ω ψ ω , wehave − λ ω = h L + ω ψ ω , ψ ω i H − ( R ) ,H ( R ) . Therefore, − ddω λ ω = 1 − p Z R ( ϕ ω, ) p − ( ψ ω ) dx − p Z R ( p − ω − λ ∗ )( ϕ ω, ) p − ( ∂ ω ϕ ω, )( ψ ω ) dx = 1 − p + O ( ω − λ ∗ ) . (4.2)Since ( − ∂ x + ω + λ ω + V − p | ϕ ω | p − ) ψ ω = 0 , we have0 = (1 + ∂ ω λ ω − p | ϕ ω, | p − − p ( p − ω − λ ∗ ) | ϕ ω, | p − ∂ ω ϕ ω, ) ψ ω + L + ω ∂ ω ψ ω . ∂ ω ψ ω = p ( P ⊥ H λ ∗ P ⊥ ) − ( k ψ ∗ k − ( p +1) L p +1 ( R ) ψ p ∗ − ψ ∗ ) + O ( ω − λ ∗ ) . By (4.2) and lemma 4.1, we obtain d dω λ ω =2 p ( p − Z R ( ϕ ω, ) p − ( ∂ ω ϕ ω, )( ψ ω ) dx + 2 p Z R ( ϕ ω, ) p − ψ ω ∂ ω ψ ω dx + p ( p − Z R ( ω − λ ∗ )( ϕ ω, ) p − ( ∂ ω ϕ ω, ) ( ψ ω ) dx + p ( p − Z R ( ω − λ ∗ )( ϕ ω, ) p − ( ∂ ω ϕ ω, )( ψ ω ) dx + 2 p ( p − Z R ( ω − λ ∗ )( ϕ ω, ) p − ( ∂ ω ϕ ω, ) ψ ω ∂ ω ψ ω dx =2 p (2 p − Z R k ψ ∗ k − ( p +1) L p +1 ( R ) ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ( k ψ ∗ k − ( p +1) L p +1 ( R ) ψ p ∗ − ψ ∗ ) dx + o (1) . The following corollary follows Lemma 4.2.
Corollary 4.3.
There exists ω ∗ , > λ ∗ such that for λ ∗ < ω < ω ∗ , , λ ω > . Moreover, if λ ∗ < ω < ω ∗ , , then the followings are hold. (i) If ω < ω < ω ∗ , , then L + ω has exactly two negative eigenvalue and no kernel. (ii) If λ ∗ < ω < ω , then L + ω has exactly one negative eigenvalue and no kernel. Applying Lyapunov-Schmidt decomposition and Crandall-Rabinowitz Transversality in[12], we show ˜ ϕ ω is a bifurcation point. In this paper, we only write the sketch of theproof of the following proposition(see the proof of Theorem 4 in [12] or Proposition 1 in[26] for the detail of the proof of the following proposition). Proposition 4.4.
Let p ≥ and λ ∗ < ω < ω ∗ , . There exist δ > and φ ω ∈ C ([ − δ, δ ] , H ) such that φ ω ( a ) > , φ ω ( a )( x, y ) = φ ω ( a )( − x, y ) = φ ω ( a )( x, − y ) , ( x, y ) ∈ R × [ − πL, πL ] , − ∆ φ ω ( a ) + ω ω ( a ) φ ω ( a ) + V φ ω ( a ) − | φ ω ( a ) | p − φ ω ( a ) = 0 ,φ ω ( a ) = ˜ ϕ ω + aψ ω cos yL + r ω ( a ) ,ω ω ( a ) = ω + ω ′′ ω (0)2 a + o ( a ) , (4.3) where r ω ( a ) ⊥ ψ ω cos yL , k r ω ( a ) k H = O ( a ) , ω ′′ ω (0) = − p ( p − dλ ω dω | ω = ω (cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13) L h ( ˜ ϕ ω ) p − ( ψ ω cos yL ) , L − ω (( ˜ ϕ ω ) p − ( ψ ω cos yL ) ) i L − p ( p − p − dλ ω dω | ω = ω (cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13) L h ( ψ ω cos yL ) , ( ˜ ϕ ω ) p − ( ψ ω cos yL ) i L , (4.4)14ransverse instability λ ( a ) = dλ ω dω | ω = ω ω ′′ ω (0) a + o ( a ) , (4.5) and k φ ω ( a ) k L = k ˜ ϕ ω k L + R p,ω a + o ( a ) . (4.6) Here, R p,ω = − dλ ω dω | ω = ω (cid:13)(cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13)(cid:13) L + ω ′′ ω (0) d k ˜ ϕ ω k L dω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω , and λ ( a ) is the second eigenvalue of L ( a, ω ) = − ∆ + ω ω ( a ) + V − | φ ω ( a ) | p − .The sketch of the proof. Let F be the function from H sym ( R × T L , R ) → L sym ( R × T L , R )satisfying F ( ϕ, ω ) = − ∆ ϕ + ωϕ + V ϕ − | ϕ | p − ϕ, where L sym ( R × T L , R ) = { u ∈ L ( R × T L , R ) | u ( x, y ) = u ( − x, y ) = u ( x, − y ) , ( x, y ) ∈ R × [ − πL, πL ] } , H sym ( R × T L , R ) = H ( R × T L ) ∩ L sym ( R × T L , R ) and L ( R × T L , R )is the set of real valued L -function on R × T L . Then, Ker( ∂ ϕ F ( ˜ ϕ ω , ω )) is spanned by ψ ω cos yL . Applying the Lyapunov-Schmidt decomposition, we obtain that there exists afunction h ( ω, a ) ∈ H sym ( R × T L , R ) such that P ⊥ F ( ˜ ϕ ω + aψ ω cos yL + h ( ω, a ) , ω ) = 0 , where P ⊥ is the orthogonal projection onto { u ∈ L ( R × T L , R ) |h u, ψ ω cos yL i L = 0 } .Then, the problem F ( ˜ ϕ ω + aψ ω cos yL + h ( ω, a ) , ω ) = 0 is equivalent to the problem F || ( ω, a ) = h F ( ˜ ϕ ω + aψ ω cos yL + h ( ω, a ) , ω ) , ψ ω cos yL i L = 0 . We apply the Crandall-Rabinowitz Transversality and we consider the problem g ( ω, a ) = 0,where g ( ω, a ) = ( F || ( ω,a ) − F || ( ω, a , a = 0 , ∂F || ∂a ( ω, , a = 0 . Here for a = 0, F || ( ω, a ) = 0 if and only if g ( ω, a ) = 0. If p >
2, then F || is a C functionand g is a C function. In the case p = 2, by the positivity of ˜ ϕ ω and the Lebesguedominant converge theorem, we can prove g is C . Then, ∂g∂ω ( ω ,
0) = ∂λ ω ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω (cid:13)(cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13)(cid:13) L , ∂g∂a ( ω ,
0) = 0 . Therefore, by the implicit function theorem there exists ω ω ( a ) such that g ( ω ω ( a ) , a ) = 0.Hence, φ ω ( a ) := ˜ ϕ ω + aψ ω cos yL + h ( ω ω ( a ) , a ) is a solution of F ( φ ω ( a ) , ω ω ( a )) = 0 and ω ′ ω (0) = − ∂g∂a∂g∂ω ( ω ,
0) = 0 . φ ω ( a ), we canobtain ω ′′ ω (0) = lim a → ω ′ ω (0) a = − ∂λ ω ∂ω | ω = ω lim a → a ∂g∂a ( ω ω ( a ) , a ) , and (4.4).Since L ( a, ω ) is C , there exists an eigenfunction χ ∗ ( a ) of L ( a, ω ) corresponding to λ ( a ) such that k χ ∗ ( a ) k L = 1, χ ∗ (0) = (cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13) − L ψ ω cos yL and χ ∗ ( a ) is C with respectto a . In the case p >
2, since F || is C , φ ω ( a ) is C . In the case p = 2, since L ( a, ω ) is C and dφ ω da ( a ) = ψ ω cos yL − ( P ⊥ L ( a, ω ) P ⊥ ) − P ⊥ ( L ( a, ω ) ψ ω cos yL + ω ω ( a )( ˜ ϕ ω + h ( ω ω ( a ) , a ))) ,φ ω ( a ) is C . Since λ ( a ) = h L ( a, ω ) χ ∗ ( a ) , χ ∗ ( a ) i L , we obtain dλ da = ω ′′ ω − p ( p − h ( φ ω ) p − dφ ω da dχ ∗ da , χ ∗ i L − p ( p − * ( p − φ ω ) p − (cid:18) dφ ω da (cid:19) + ( φ ω ) p − d φ ω da ! χ ∗ , χ ∗ + L , and (4.5). Finally, calculating d da k φ ω ( a ) k L | a =0 , we get (4.6). Lemma 4.5.
Let p ≥ . Then, there exists ω p > λ ∗ such that for ω ∈ ( λ ∗ , ω p ) , ω ′′ ω (0) > and R p,ω ( > , ≤ p < √ ,< , √ ≤ p. Remark 4.6.
The first term of R p,ω with respect to ω − λ ∗ yields the critical exponent p ∗ . In Lemma 4.5, we show the following expansion: R p,ω = ( − p + 18 p − π p − / ( ω − λ ∗ ) / + O (( ω − λ ∗ ) / ) . Proof.
First, we prove the positivity of ω ′′ ω (0). Let I = D ( ˜ ϕ ω ) p − (cid:0) ψ ω cos yL (cid:1) , L − ω (cid:16) ( ˜ ϕ ω ) p − (cid:0) ψ ω cos yL (cid:1) (cid:17)E L I = D(cid:0) ψ ω cos yL (cid:1) , ( ˜ ϕ ω ) p − (cid:0) ψ ω cos yL (cid:1) E L . Since ( ϕ ω, ( x )) p − is differentiable with respect to x ∈ R and (cid:12)(cid:12)(cid:12) ω − λ ∗ (cid:18) ( ϕ ω, ( x )) p − − k ψ ∗ k − ( p − p +1) p − L p +1 ( R ) ( ψ ∗ ( x )) p − (cid:19)(cid:12)(cid:12)(cid:12) ≤ C ( ϕ θ ( ω ) , ( x )) p − | ∂ ω ϕ θ ( ω ) , ( x ) | , k ∂ ω ϕ ω, k H ( R ) with respect to ω and certain upper and lower expo-nential decay rates for ϕ ω and ψ ω we have I = 38 ( ω − λ ∗ ) p − p − Z R × T L ˜ ϕ p − ω , ψ ω dxdy = 3 πL k ψ ∗ k p +1) p − L p +1 ( R ) ( ω − λ ∗ ) p − p − + 3(5 p − πL k ψ ∗ k − ( p − p +1) p − L p +1 ( R ) ( ω − λ ∗ ) p − p − +1 Z R ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p dx + o (( ω − λ ∗ ) p − p − +1 L ) , where ϕ ω, = ( ω − λ ∗ ) − / ( p − ϕ ω and λ ∗ < θ ( ω ) < ω . On the other hand, I = 14 h ( ˜ ϕ ω ) p − ψ ω , ( L + ω ) − ( ˜ ϕ ω ) p − ψ ω i L + 18 h ( ˜ ϕ ω ) p − ψ ω , ( L + ω + 4 L ) − (( ˜ ϕ ω ) p − ψ ω ) i L = I ′ + I ′′ . By (cid:13)(cid:13)(cid:13) ( L + ω ) − | ( ψ ω ) ⊥ (cid:13)(cid:13)(cid:13) ≤ C and the similar calculation for I , we obtain I ′ = ( ω − λ ∗ ) p − p − (cid:26)(cid:28) ( ϕ ω , ) p − ( ψ ω ) , ( L + ω ) − (cid:18)Z R ( ϕ ω , ) p − ( ψ ω ) dx (cid:19) ψ ω (cid:29) L + (cid:28) ( ϕ ω , ) p − ( ψ ω ) , ( L + ω ) − (cid:18) ( ϕ ω , ) p − ( ψ ω ) − Z R ( ϕ ω , ) p − ( ψ ω ) dxψ ω (cid:19)(cid:29) L (cid:27) = − πL ( ω − λ ∗ ) p − p − k ψ ∗ k p +1) p − L p +1 ( R ) p − − p + 9 p − πL ( ω − λ ∗ ) p − p − +1 k ψ ∗ k − ( p − p +1) p − L p +1 ( R ) p − Z R ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p dx + o (( ω − λ ∗ ) p − p − +1 L ) , where ( ψ ω ) ⊥ = { u ∈ L ( R ) |h u, ψ ω i L ( R ) = 0 } . By the same calculation of I ′ and the17ransverse instabilityboundedness of (cid:13)(cid:13)(cid:13) ( L + ω + L ) − | ( ψ ω ) ⊥ (cid:13)(cid:13)(cid:13) , I ′′ = ( ω − λ ∗ ) p − p − (cid:28) ( ϕ ω , ) p − ( ψ ω ) , ( L + ω + 4 /L ) − (cid:18)Z R ( ϕ ω , ) p − ( ψ ω ) dx (cid:19) ψ ω (cid:29) L + ( ω − λ ∗ ) p − p − (cid:28) ( ϕ ω , ) p − ( ψ ω ) , ( L + ω + 4 /L ) − (cid:18) ( ϕ ω , ) p − ( ψ ω ) − Z R ( ϕ ω , ) p − ( ψ ω ) dxψ ω (cid:19)(cid:29) L = πL ( ω − λ ∗ ) p − p − k ψ ∗ k p +1) p − L p +1 ( R ) p − p − p + 7) πL ( ω − λ ∗ ) p − p − +1 k ψ ∗ k − ( p − p +1) p − L p +1 ( R ) p − Z R ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p dx + o (( ω − λ ∗ ) p − p − +1 L ) . Since 1 ddω λ ω (cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13) L = 1( p − πL − C ∗ ( ω − λ ∗ )( p − πL + o (( ω − λ ∗ ) L − ) , we obtain ω ′′ ω (0)= p ( p + 3)( ω − λ ∗ ) p − p − k ψ ∗ k p +1) p − L p +1 ( R ) − p (2 p + 3 p + 34 p − ω − λ ∗ ) p − p − +1 k ψ ∗ k − ( p − p +1) p − L p +1 ( R ) p − Z R ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p dx + o (( ω − λ ∗ ) p − p − +1 )where C ∗ = 2 p (2 p − k ψ ∗ k − ( p +1) L p +1 ( R ) R R ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p dxp − . Therefore, if 0 < ω − λ ∗ ≪
1, then ω ′′ ω (0) > R p,ω . Since − dλ ω dω | ω = ω (cid:13)(cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13)(cid:13) L = − p − πL − C ∗ ( p − πL ( ω − λ ∗ ) + o (( ω − λ ∗ ) L )and ddω k ˜ ϕ ω k L | ω = ω = 4 πLp − k ψ ∗ k − p +1) p − L p +1 ( R ) ( ω − λ ∗ ) − p − p − + o (( ω − λ ∗ ) p − L ) , R p,ω = − p − πL − C ∗ ( p − πL ( ω − λ ∗ ) + 2 p ( p + 3) πL p −
1) + o (( ω − λ ∗ ) L ) − p (2 p + 3 p + 34 p − πL p − ( ω − λ ∗ ) k ψ ∗ k − ( p +1) L p +1 ( R ) Z R ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p dx = ( − p + 18 p − πL p −
1) + o (( ω − λ ∗ ) L )+ p ( − p + 57 p − p + 30) πL p − ( ω − λ ∗ ) k ψ ∗ k − ( p +1) L p +1 ( R ) Z R ψ p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p dx. (4.7)Let p ∗ = 9 + √ . Since p ∗ is the root of − p + 18 p − p >
1, the conclusion for p = p ∗ follows(4.7). Finally, we consider the case p = p ∗ . By p ∗ >
4, we have − p ∗ + 57 p ∗ − p ∗ + 30 < . Therefore, R p ∗ ,ω = p ∗ ( − p ∗ + 57 p ∗ − p ∗ + 30) πL ( ω − λ ∗ ) k ψ ∗ k − ( p ∗ +1) L p ∗ +1 ( R ) p ∗ − Z R ψ p ∗ ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p ∗ dx + o (( ω − λ ∗ ) L )= p ∗ ( − p ∗ + 57 p ∗ − p ∗ + 30) πL ( ω − λ ∗ )3( p ∗ − Z R ψ ∗ ,p ∗ ( P ⊥ H λ ∗ P ⊥ ) − ψ ∗ ,p ∗ dx + o (( ω − λ ∗ ) L )The conclusion for p = p ∗ follows this.Using Lemma 4.5 and applying the argument in Section 3 of [26], we obtain Theorem1.4.For the completeness of the proof of Theorem 1.4, we introduce the argument for thestability of standing with the degenerate linearized operator in [13, 26]. Using the followingproposition, we show Theorem 1.4. Proposition 4.7.
Let λ ∗ < ω < ω ∗ , . (i) If R p,ω > , then e iω t ˜ ϕ ω is a stable standing wave of (NLS) on R × T L with L = ( λ ω ) − . (ii) If R p,ω < , then e iω t ˜ ϕ ω is an unstable standing wave of (NLS) on R × T L with L = ( λ ω ) − . y ∈ T L , we define the polar coordinate ~a = ( a , a ) = ( a cos ˜ aL , − a sin ˜ aL ) for ~a ∈ R and φ ω ( ~a )( x, y ) = φ ω ( a )( x, y + ˜ a ) , ω ω ( ~a ) = ω ω ( a ) . In the following lemma, we construct a curve which captures the degeneracy of the lin-earized operator S ′′ ω ( ˜ ϕ ω ). Lemma 4.8.
There exist a neighborhood U of (0 , in R and a C function ρ : U → R such that ρ (0 ,
0) = 0 and for ~a ∈ UQ ( φ ω ( ~a ) + ρ ( ~a ) ∂ ω ˜ ϕ ω ) = Q ( ˜ ϕ ω ) ,ρ ( ~a ) h ˜ ϕ ω , ∂ ω ˜ ϕ ω i L = Q ( ˜ ϕ ω ) − Q ( φ ω ( ~a )) + o ( ρ ( ~a )) . (4.8) Proof.
Since ∂ ρ Q ( φ ω ( ~a ) + ρ∂ ω ˜ ϕ ω ) | ρ =0 ,~a =0 = h ˜ ϕ ω , ∂ ω ˜ ϕ ω i L > , the conclusion follows the implicit function theorem.Let Φ( ~a ) = φ ω ( ~a ) + ρ ( ~a ) ∂ ω ˜ ϕ ω . for ~a ∈ U .In the following lemma, we capture the degeneracy of the action S ω . Lemma 4.9.
For ~a ∈ U , S ω (Φ( ~a )) − S ω ( ˜ ϕ ω ) = ddω λ ω k ψ ω cos( y/L ) k L R p,ω h ˜ ϕ ω , ∂ ω ˜ ϕ ω i L | ~a | + o ( | ~a | ) . Proof.
For ~a ∈ U , S ω (Φ( ~a )) − S ω ( ˜ ϕ ω ) = S ω ω ( ~a ) (Φ( ~a )) − S ω ( ˜ ϕ ω ) + ( ω − ω ω ( ~a )) Q ( ˜ ϕ ω )= S ω ω ( ~a ) ( φ ω ( ~a )) − S ω ( ˜ ϕ ω ) + ( ω − ω ω ( ~a )) Q ( ˜ ϕ ω )+ 12 ( ρ ( ~a )) h S ′′ ω ( ˜ ϕ ω ) ∂ ω ˜ ϕ ω , ∂ ω ˜ ϕ ω i L + o (( ρ ( ~a )) ) . From ω ′′ ω (0) > ω ω ( a ) is increasing on a small interval (0 , δ ). Therefore, thereexists the inverse function a + ( ω ) of ω ω ( a ) form [ ω , ω ω ( δ )) to [0 , δ ). By the differentiabilityof a + for ω > ω , φ ω ( a + ) is differentiable for ω > ω . Thus, for ω, ω with ω = ω S ω ( φ ω ( a + ( ω ))) − S ω ( φ ω ( a + ( ω ))) ω − ω = h S ′′ ω ( φ ω ( a + ( ω ))( φ ω ( a + ( ω )) − φ ω ( a + ( ω ))) , ( φ ω ( a + ( ω )) − φ ω ( a + ( ω ))) i L ω − ω )+ Q ( φ ω ( a + ( ω ))) + o (( φ ω ( a + ( ω )) − φ ω ( a + ( ω ))) ) ω − ω → Q ( φ ω ( a + ( ω ))) as ω → ω . ∂ a φ ω ( a ) | a =0 = ψ ω cos yL , for ω > ω S ω ( φ ω ( a + ( ω ))) − S ω ( ˜ ϕ ω ) ω − ω = h S ′′ ω ( ˜ ϕ ω )( φ ω ( a + ) − ˜ ϕ ω ) , ( φ ω ( a + ) − ˜ ϕ ω ) i L ω ′′ ω (0)( a + ) + o (( a + ) ) + Q ( ˜ ϕ ω ) + o (( φ ω ( a + ) − ˜ ϕ ω ) ) ω ′′ ω (0)( a + ) + o (( a + ) ) → Q ( ˜ ϕ ω ) as ω ↓ ω . Hence, S ω ( φ ω ( a + )) is C and dS ω ( φ ω ( a + )) dω = Q ( φ ω ( a + )) . By the equation (4.6), Q ( φ ω ( a + )) is C on ( ω , ω ω ( δ )) andlim ω ↓ ω Q ( φ ω ( a + )) − Q ( ˜ ϕ ω ) ω − ω = R p,ω ω ′′ ω (0) . Therefore, S ω ( φ ω ( a + )) is C with respect to ω on ( ω , ω ω ( δ )) and S ω ( φ ω ( a + )) − S ω ( ˜ ϕ ω ) + ( ω − ω ) Q ( ˜ ϕ ω ) = R p,ω ω ′′ ω (0) ( ω − ω ) + o (( ω − ω ) )= ω ′′ ω (0) R p,ω
16 ( a + ) + o (( a + ) ) . (4.9)From the equation (4.8), we have the expansion( ρ ( ~a )) h ˜ ϕ ω ∗ , ∂ ω ˜ ϕ ω i L = ( R p,ω ) h ˜ ϕ ω ∗ , ∂ ω ˜ ϕ ω i L | ~a | + o ( | ~a | ) . (4.10)Since S ω ω ( | ~a | ) ( φ ω ( | ~a | )) + ( ω − ω ω ( | ~a | )) Q ( ˜ ϕ ω ) = S ω ω ( ~a ) ( φ ω ( ~a )) + ( ω − ω ω ( ~a )) Q ( ˜ ϕ ω ) , by (4.9) and (4.10) we obtain the conclusion.We introduce the distance and tubular neighborhoods of ˜ ϕ ω as follows. Set for ε > ω ( u ) = inf θ ∈ R (cid:13)(cid:13) u − e iθ ˜ ϕ ω (cid:13)(cid:13) H ,N ε = { u ∈ H ( R × T L ) | dist ω ( u ) < ε } ,N ε = { u ∈ N ε | Q ( u ) = Q ( ˜ ϕ ω ) } . Modulating the symmetry, we eliminate the degeneracy of the linearized operatoraround ˜ ϕ ω . 21ransverse instability Lemma 4.10.
Let ε > sufficiently small. Then, there exist C function θ : N ε → R , α : N ε → R , ~a : N ε → U and w : N ε → H ( R × T L ) such that for u ∈ N ε e iθ ( u ) u = Φ( ~a ( u )) + w ( u ) + α ( u ) φ ω ( ~a ( u )) , where h w ( u ) + α ( u ) φ ω ( ~a ( u )) , ψ ω cos( y/L ) i L = h w ( u ) + α ( u ) φ ω ( ~a ( u )) , ψ ω sin( y/L ) i L = h w ( u ) , φ ω ( ~a ( u )) i L = h w ( u ) , iφ ω ( ~a ( u )) i L = 0 .Proof. Let ψ ω , = ψ ω cos( y/L ) and ψ ω , = ψ ω sin( y/L ). We define G ( u, θ, a , a ) = h e iθ u − Φ( ~a ) , iφ ω ( ~a ) i L h e iθ u − Φ( ~a ) , ψ ω , i L h e iθ u − Φ( ~a ) , ψ ω , i L , where ~a = ( a , a ). Since G ( ˜ ϕ ω , , ,
0) = 0 and ∂G∂ ( θ, a , a ) (cid:12)(cid:12)(cid:12)(cid:12) ( u,θ,a ,a )=( ˜ ϕ ω , , , = k ˜ ϕ ω k L −k ψ ω , k L
00 0 −k ψ ω , k L , by the implicit theorem for sufficiently small ε > C functions θ : N ε → R and ~a : N ε → U such that for u ∈ N ε G ( u, θ ( u ) , ~a ( u )) = 0 . We define α ( u ) = h e iθ ( u ) u − Φ( ~a ( u )) , φ ω ( ~a ( u )) i L k φ ω ( ~a ( u )) k L , and w ( u ) = e iθ ( u ) u − Φ( ~a ( u )) − α ( u ) φ ω ( ~a ( u )) . Then, the conclusion follows the definition of w .In the following lemma, we show the estimate of α ( u ) for u ∈ N ε . Lemma 4.11.
Let ε > sufficiently small. There exists C > such that for u ∈ N ε , | α ( u ) | ≤ C k w ( u ) k L ( ρ ( ~a ( u )) + k w ( u ) k L ) . Proof.
By Lemma 4.10, for u ∈ N ε , Q ( ˜ ϕ ω ) = Q (Φ( ~a ( u )) + w ( u ) + α ( u ) φ ω ( ~a ( u )))= Q ( ˜ ϕ ω ) + α ( u ) k φ ω ( ~a ( u )) k L + ρ ( ~a ( u )) α ( u ) h ∂ ω ˜ ϕ ω , φ ω ( ~a ( u )) i L + ρ ( ~a ( u )) h ∂ ω ˜ ϕ ω , w ( u ) i L + Q ( w ( u )) + ( α ( u )) Q ( φ ω ( ~a ( u ))) . Since ρ ( ~a ( u )) → ε →
0, we obtain the conclusion.Next, we prove the coerciveness of the linearized operator around ˜ ϕ ω .22ransverse instability Lemma 4.12.
There exist k > and ε > such that for a , a , α ∈ ( − ε , ε ) , if w ∈ H ( R × T L ) with h w, φ ω ( ~a ) i L = h w, iφ ω ( ~a ) i L = h w + αφ ω ( ~a ) , ψ ω cos( y/L ) i L = h w + αφ ω ( ~a ) , ψ ω sin( y/L ) i L = 0 , then h S ′′ ω (Φ( ~a )) w, w i H − ,H ≥ k k w k H , where ~a = ( a , a ) .Proof. Let ψ ω , = ψ ω cos( y/L ) and ψ ω , = ψ ω sin( y/L ). For w ∈ H ( R × T L ) with h w, φ ω ( ~a ) i L = h w, iφ ω ( ~a ) i L = h w + αφ ω ( ~a ) , ψ ω , i L = h w + αφ ω ( ~a ) , ψ ω , i L = 0, wedecompose w = b ˜ ϕ ω + b i ˜ ϕ ω + b ψ ω , + b ψ ω , + w ⊥ , where h w ⊥ , ˜ ϕ ω i L = h w ⊥ , i ˜ ϕ ω i L = h w ⊥ , ψ ω , i L = h w ⊥ , ψ ω , i L = 0, b j ∈ R for j ∈ { , , , } . By the non-negativenessof L − ω and L + ω + λ ω , Proposition 2.1 and Lemma 2.2, there exists c > h S ′′ ω ( ˜ ϕ ω ) w ⊥ , w ⊥ i H − ,H ≥ c k w ⊥ k H , where c is independent of w ⊥ . Then, from the or-thogonal conditions for w , we have for j ∈ { , , , } , b j = O (( | ~a | + | α | ) k w ⊥ k L ) as | ~a | + | α | →
0. Therefore, there exist ε , k > a , a , α ∈ ( − ε , ε ), h S ′′ ω (Φ( ~a )) w, w i H − ,H = h S ′′ ω ( ˜ ϕ ω ) w ⊥ , w ⊥ i H − ,H + X j =1 b j + o ( k w ⊥ k L ) ≥ k k w k H . In the following lemma, we investigate the variational structure of S ω around ˜ ϕ ω . Lemma 4.13.
Let ε > sufficiently small. For u ∈ N ε S ω ( u ) − S ω ( ˜ ϕ ω ) = 12 h S ′′ ω ( ˜ ϕ ω ) w ( u ) , w ( u ) i H − ,H + C ∗∗ R p,ω | ~a ( u ) | + o ( k w ( u ) k H ) + o ( | ~a ( u ) | ) , where w ( u ) and ~a ( u ) are defined by Lemma 4.10 and C ∗∗ = ddω λ ω k ψ ω cos( y/L ) k L h ˜ ϕ ω , ∂ ω ˜ ϕ ω i L . Proof.
Let u ∈ N ε . By Lemma 4.10 and Lemma 4.11, we have S ω ( u ) − S ω ( ˜ ϕ ω )= S ω (Φ( ~a ( u )) + w ( u ) + α ( u ) φ ω ( ~a ( u ))) − S ω ( ˜ ϕ ω )= S ω (Φ( ~a ( u ))) − S ω ( ˜ ϕ ω ) + h S ′ ω (Φ( ~a ( u ))) , w ( u ) + α ( u ) φ ω ( ~a ( u )) i H − ,H + 12 h S ′′ ω (Φ( ~a ( u )) w ( u ) , w ( u ) i H − ,H + o ( k w ( u ) k H ) . ρ ( ~a ( u )) = O ( | ~a ( u ) | ) as dist ω ( u ) → h φ ω ( ~a ( u )) , w ( u ) i L = 0 and S ′′ ω ( ˜ ϕ ω ) ∂ ω ˜ ϕ ω = − ˜ ϕ ω , we have h S ′ ω (Φ( ~a ( u ))) , w ( u ) i H − ,H = h S ′ ω ω ( ~a ( u )) (Φ( ~a ( u ))) + ( ω − ω ω ( ~a ))Φ( ~a ( u ))) , w ( u ) i H − ,H = h S ′′ ω ω ( ~a ( u )) ( φ ω ( ~a ( u ))) ρ ( ~a ( u )) ∂ ω ˜ ϕ ω , w ( u ) i H − ,H + o ( | ~a | ) + o ( k w ( u ) k H )= h ( S ′′ ω ω ( ~a ( u )) ( φ ω ( ~a ( u ))) − S ′′ ω ( ˜ ϕ ω )) ρ ( ~a ( u )) ∂ ω ˜ ϕ ω , w ( u ) i H − ,H − ρ ( ~a ( u )) h ˜ ϕ ω − φ ω ( ~a ( u )) , w ( u ) i L + o ( | ~a ( u ) | ) + o ( k w ( u ) k H )= o ( | ~a ( u ) | ) + o ( k w ( u ) k H ) . By Lemma 4.11 and the continuity of S ′ ω (Φ( ~a )) and φ ω ( ~a ) at ~a = 0, we have h S ′ ω (Φ( ~a ( u ))) , α ( u ) φ ω ( ~a ( u )) i H − ,H = o ( | ~a ( u ) | ) + o ( k w ( u ) k H ) . Therefore, from Lemma 4.9, we have the conclusion.
In this subsection, we prove (i) of Proposition 4.7. Let 0 < ε ≪
1. By Lemma 4.13 and R p,ω >
0, for small ε we have that there exists c > u ∈ N ε S ω ( u ) − S ω ( ˜ ϕ ω ) ≥ c ( k w ( u ) k H + | ~a ( u ) | ) , (4.11)where w ( u ) , ~a ( u ) are defined by Lemma 4.10. We suppose that there exist ε >
0, asequence { u n } n of solutions and a sequence { t n } n such that t n > u n (0) → ˜ ϕ ω in H and inf θ ∈ R (cid:13)(cid:13) u n ( t n ) − e iθ ˜ ϕ ω (cid:13)(cid:13) H > ε . Let v n = s Q ( ˜ ϕ ω ) Q ( u n ) u n ( t n ) . Since Q ( v n ) = Q ( ˜ ϕ ω ) and Q ( u n ) → Q ( ˜ ϕ ω ) as n → ∞ , k v n − u n ( t n ) k H → S ω ( v n ) − S ω ( ˜ ϕ ω ) → n → ∞ . By the equation (4.11), ~a ( u n ( t n )) → α ( u n ( t n )) → w ( u n ( t n )) → H as n → ∞ . Therefore,inf θ ∈ R (cid:13)(cid:13) u n ( t n ) − e iθ ˜ ϕ ω (cid:13)(cid:13) H → n → ∞ . This is a contradiction. We complete the proof of (i).
In this subsection, we prove (ii) of Proposition 4.7. Let 0 < ε ≪
1. We define the functions A ( u ) and P ( u ) as A ( u ) = h e iθ ( u ) u, − i [ a ( u ) ∂ a Φ( ~a ( u )) + a ( u ) ∂ a Φ( ~a ( u ))] i L , P ( u ) = h S ′ ω ω ( ~a ( u )) ( u ) , iA ′ ( u ) i H − ,H , for u ∈ N ε , where θ ( u ) and ~a ( u ) are defined by Lemma 4.10.Then A ′ ( u ) = − ie − iθ ( u ) [ a ( u ) ∂ a Φ( ~a ( u )) + a ( u ) ∂ a Φ( ~a ( u ))]+ h ie iθ ( u ) u, − i [ a ( u ) ∂ a Φ( ~a ( u )) + a ( u ) ∂ a Φ( ~a ( u ))] i L θ ′ ( u )+ h e iθ ( u ) u, − i [ ∂ a Φ( ~a ( u )) + a ( u ) ∂ a ∂ a Φ( ~a ( u )) + a ( u ) ∂ a ∂ a Φ( ~a ( u ))] i L a ′ ( u )+ h e iθ ( u ) u, − i [ a ( u ) ∂ a ∂ a Φ( ~a ( u )) + ∂ a Φ( ~a ( u )) + a ( u ) ∂ a ∂ a Φ( ~a ( u ))] i L a ′ ( u ) , h iA ′ ( u ) , Q ′ ( u ) i L = −h A ′ ( u ) , iu i L = dA ( e iθ u ) dθ (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = 0 . Therefore, for any solution u ( t ) of (1.1) dA ( u ( t )) dt = h A ′ ( u ( t )) , − iE ′ ( u ( t )) i H − ,H = h iA ′ ( u ( t )) , E ′ ( u ( t )) + ω ( ~a ( u ( t ))) Q ′ ( u ( t )) i H − ,H = P ( u ( t )) . Next, we investigate the function P . Lemma 4.14.
For ~a ∈ U , P (Φ( ~a )) = −| ~a | ρ ( ~a ) dλ ω dω | ω = ω (cid:13)(cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13)(cid:13) L + o ( ρ ( ~a ) ) . Proof.
Let ~a = ( a , , a , ) ∈ U . Then k Φ( ~a ) k L = k ˜ ϕ ω k L , ~a (Φ( ~a )) = ~a and θ (Φ( ~a )) =0. Therefore, S ′ ω ω ( ~a ) (Φ( ~a )) = S ′′ ω ( ˜ ϕ ω ) ρ ( ~a ) ∂ ω ˜ ϕ ω − ρ ( ~a ) p ( p − ϕ ω ) p − ∂ ω ˜ ϕ ω ψ ω (cid:16) a , cos yL + a , sin yL (cid:17) + o ( ρ ( ~a ) | ~a | ) , (4.12) iA ′ (Φ( ~a ))= a , ∂ a Φ( ~a ) + a , ∂ a Φ( ~a )= ψ ω (cid:16) a , cos yL + a , sin yL (cid:17) + ( a , ∂ a ρ ( ~a ) + a , ∂ a ρ ( ~a )) ∂ ω ˜ ϕ ω + ( S ′′ ω ( ˜ ϕ ω )) − h −| ~a | ω ′′ ω (0) ˜ ϕ ω + p ( p − ϕ ω ) p − ψ ω (cid:16) a , cos yL + a , sin yL (cid:17) i + o ( | ~a | ) (4.13)25ransverse instabilityHence, we have P (Φ( ~a ))= h S ′′ ω ( ˜ ϕ ω ) ρ ( ~a ) ∂ ω ˜ ϕ ω , ψ ω (cid:16) a , cos yL + a , sin yL (cid:17) + ( a , ∂ a ρ ( ~a ) + a , ∂ a ρ ( ~a )) ∂ ω ˜ ϕ ω i L + h ρ ( ~a ) ∂ ω ˜ ϕ ω , −| ~a | ω ′′ ω (0) ˜ ϕ ω + p ( p − ϕ ω ) p − ψ ω (cid:16) a , cos yL + a , sin yL (cid:17) i L + h− ρ ( ~a ) p ( p − ϕ ω ) p − ∂ ω ˜ ϕ ω ψ ω (cid:16) a , cos yL + a , sin yL (cid:17) , ψ ω (cid:16) a , cos yL + a , sin yL (cid:17) i L + o ( ρ ( ~a ) | ~a | )= − ρ ( ~a ) h ˜ ϕ ω , ∂ ω ˜ ϕ ω i L ( a , ∂ a ρ ( ~a ) + a , ∂ a ρ ( ~a ) + | ~a | ω ′′ ω (0)) + o ( ρ ( ~a ) | ~a | ) . (4.14)By (4.10), we have h ˜ ϕ ω , ∂ ω ˜ ϕ ω i L ( a , ∂ a ρ ( ~a ) + a , ∂ a ρ ( ~a ))= − | ~a | R p,ω o ( | ~a | )= − | ~a | (cid:16) − dλ ω dω | ω = ω (cid:13)(cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13)(cid:13) L + ω ′′ ω (0) h ∂ ω ˜ ϕ ω , ˜ ϕ ω i L (cid:17) + o ( | ~a | ) . Hence, the conclusion follows the equation (4.14).
Lemma 4.15.
Let ε > be sufficiently small and u ∈ N ε with S ω ( u ) − S ω ( ˜ ϕ ω ) < .Then P ( u ) = −| ~a | ρ ( ~a ) dλ ω dω | ω = ω (cid:13)(cid:13)(cid:13) ψ ω cos yL (cid:13)(cid:13)(cid:13) L + o ( ρ ( ~a ( u )) ) + o ( k w ( u ) k H ) . Proof.
By the Taylor expansion , we have P ( u ) = h S ′ ω ω ( ~a ( u )) (Φ( ~a ( u )) + w ( u ) + α ( u ) φ ω ( ~a ( u ))) , iA ′ (Φ( ~a ( u )) + w ( u ) + α ( u ) φ ω ( ~a ( u ))) i H − ,H = h S ′ ω ω ( ~a ( u )) (Φ( ~a ( u ))) + S ′′ ω ω ( ~a ( u )) (Φ( ~a ( u )))( w ( u ) + α ( u ) φ ω ( ~a ( u ))) ,iA ′ (Φ( ~a ( u ))) + iA ′′ (Φ( ~a ( u )))( w ( u ) + α ( u ) φ ω ( ~a ( u ))) i H − ,H + o ( ρ ( ~a ( u )) + k w ( u ) k H )By (4.12), (4.13), Lemma 4.11 and Lemma 4.14, P ( u ) = P (Φ( ~a ( u ))) + h S ′ ω ω ( ~a ( u )) (Φ( ~a ( u ))) , iA ′′ (Φ( ~a ( u ))) w ( u ) i L + h S ′′ ω ω ( ~a ( u )) (Φ( ~a ( u ))) w ( u ) , iA ′ (Φ( ~a ( u ))) i H − ,H + o ( ρ ( ~a ( u )) + k w ( u ) k H )By the proof of Lemma 4.10, we obtain that ∂G∂ ( θ, a , a ) θ ′ a ′ a ′ = − ie − iθ ( u ) φ ω ( ~a ( u )) − e − iθ ( u ) ψ ω , − e − iθ ( u ) ψ ω , . Thus θ ′ (Φ( ~a ( u )) , a ′ (Φ( ~a ( u ))) and a ′ (Φ( ~a ( u ))) are linear combinations of iφ ω ( ~a ( u )), ψ ω , and ψ ω , . Since h θ ′ (Φ( ~a ( u ))) , w ( u ) i L = h a ′ (Φ( ~a ( u ))) , w ( u ) i L = h a ′ (Φ( ~a ( u ))) , w ( u ) i L = O ( α ( u ) k w ( u ) k H ), we have iA ′′ (Φ( ~a ( u ))) w ( u ) = O ( α ( u ) k w ( u ) k H ) . w ( u ) and A ′ (Φ( ~a ( u ))) = O ( ~a ( u )) we obtain P ( u ) = P (Φ( ~a ( u ))) + ( ω ω ( ~a ( u )) − ω ) h w ( u ) , iA ′ (Φ( ~a ( u ))) i L + h p ( | Φ( ~a ( u )) | p − − | ˜ ϕ ω | p − ) w ( u ) , iA ′ (Φ( ~a ( u ))) i L + h S ′′ ω ( ˜ ϕ ω ) w ( u ) , ψ ω (cid:16) a ( u ) cos yL + a ( u ) sin yL (cid:17) i H − ,H + h S ′′ ω ( ˜ ϕ ω ) w ( u ) , ( a ( u ) ∂ a ρ ( ~a ( u )) + a ( u ) ∂ a ρ ( ~a ( u ))) ∂ ω ˜ ϕ ω i H − ,H + h w ( u ) , −| ~a ( u ) | ω ′′ ω (0) ˜ ϕ ω + p ( p − ϕ ω ) p − ψ ω (cid:16) a ( u ) cos yL + a ( u ) sin yL (cid:17) i H − ,H + o ( ρ ( ~a ( u )) + k w ( u ) k H )= P (Φ( ~a ( u ))) + o ( ρ ( ~a ( u )) + k w ( u ) k H )Hence, we obtain the conclusion.We assume e iω t ˜ ϕ ω is stable. Let { ~a n } n be a sequence with ~a n → { u n } n be thesequence of solutions with u n (0) = Φ( ~a n ). Since R p,ω < C > S ω (Φ( ~a n )) − S ω ( ˜ ϕ ω ) = CR p,ω | ~a n | + o ( | ~a n | ) , we obtain S ω ( ˜ ϕ ω ) > S ω (Φ( ~a n )) for sufficiently large n >
1. From Lemma 4.13 andLemma 4.15 we have for sufficiently large n > n > < S ω ( ˜ ϕ ω ) − S ω (Φ( ~a n )) ≤ cP ( u n ( t )) . Since ρ ( ~a ( u n ( t ))) is positive and bounded for t ≥ n >
1, there exists δ > t ≥ dA ( u n ( t )) dt = P ( u n ( t )) > δ. This contradicts the boundedness of A on N ε . Hence, e iω t ˜ ϕ ω is unstable. Acknowledgments
The author would like to express his great appreciation to Professor Yoshio Tsutsumi fora lot to helpful advices and encouragements. The author would like to thank ProfessorMashahito Ohta for his helpful indication. 27ransverse instability
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