DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS


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1 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN Abstract. When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated along the edges with a translational parameter and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable because of the irreversible drift along the incoming edge towards the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph. 1. Introduction The classical Lyapunov method establishes stability of minimizers of energy in the time flow of a dynamical system under the condition that the second variation of energy is strictly positive definite. In the context of Hamiltonian PDEs such as the nonlinear Schrödinger (NLS) equation, standing waves are often saddle points of energy but additional conserved quantities such as mass and momentum exist due to symmetries such as phase rotation and space translation. It is now a classical result [15, 32] that the standing waves are stable if they are constrained minimizers of energy when other conserved quantities are fixed and if the second variation of energy is strictly positive definite under the constraints eliminating symmetries. Stability of standing waves in the presence of symmetries is understood in the sense of orbital stability, where the orbit is defined by a set of parameters along the symmetry group. Fixed values of other conserved quantities are realized by Lagrange multipliers which define another set of parameters. Both sets of parameters (along the symmetry group and Lagrange multipliers) satisfy the modulation equations in the time flow of the Hamiltonian PDE [33]. The actual values of parameters along the symmetry group are irrelevant in the definition of an orbit for the standing waves, whereas the actual values of Lagrange multipliers and the remainder terms are controlled if the second variation of energy is strictly positive under the symmetry constraints. The present work is devoted to stability of standing waves in the NLS equation defined on a metric graph, a subject that has seen many recent developments [20]. Existence and variational characterization of standing waves was developed for star graphs [1, 2, 3, 4] and for general Date: February 11,
2 2 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN metric graphs [5, 6, 7, 8, 11]. Bifurcations and stability of standing waves were further explored for tadpole graphs [21], dumbbell graphs [14, 18], doublebridge graphs [22], and periodic ring graphs [12, 13, 23, 24]. A variational characterization of standing waves was developed for graphs with compact nonlinear core [25, 26, 30]. In the context of the simplest star graphs, it was realized in [6] that the infimum of energy under the fixed mass is approached by a sequence of solitary waves escaping to infinity along one edge of the star graph. Consequently, standing waves cannot be energy minimizers, as they represent saddle points of energy under the fixed mass [1]. It was shown in [16] that the second variation of energy at these standing waves is nevertheless positive but degenerate with a zero eigenvalue, hence the standing waves are saddle points beyond the second variation of energy. Orbital instability of these standing waves under the time flow of the NLS equation is developed as a result of the saddle point geometry [16]. Balanced star graphs were introduced in [29] from the condition that the NLS equation on the metric graph reduces to the NLS equation on a real line if the initial conditions satisfy certain symmetry (see also [27, 28, 34] for generalizations). Consequently, solitary waves may propagate across the vertex without any reflection. Standing waves of the NLS equation can be translated along edges of the balanced star graph with a translational parameter and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the shifted states were shown to be spectrally stable if their monotonic tails are located on the outgoing edges [17]. Moreover, these shifted states are constrained minimizers of the energy under fixed mass and the only degeneracies of the second variation of energy are due to phase rotation and the spatial translation along the balanced star graph. Standing waves are orbitally stable in the NLS equation on a real line since the two degeneracies are related to two symmetries of the NLS equation which also conserves mass and momentum. In contrast to this wellknown result, we show in this paper that the shifted states are orbitally unstable in the NLS equation on the balanced star graph. The instability is related to the following observation. If the initial perturbation to the standing wave is symmetric with respect to the exchange of components on the outgoing edges, then the reduction to the NLS equation on a line holds, and the solution has translational symmetry. Perturbations that lack this exchange symmetry also break the translation symmetry and the solution fails to conserve momentum. Moreover, the value of the momentum functional increases monotonically in the time flow of the NLS equation and this monotone increase results in the irreversible drift of the shifted state along the incoming edge towards the outgoing edges of the balanced star graph. When the center of mass for the shifted state reaches the vertex, the shifted state becomes a saddle point of energy under the fixed mass. At this point in time, orbital instability of the shifted state develops as a result of the saddle point geometry similar to the instability studied in [16]. The main novelty of this paper is to show that degeneracy of the positive second variation of energy may lead to orbital instability of constrained minimizers if this degeneracy is not related to the symmetry of the Hamiltonian PDE. The orbital instability appears due to irreversible drift of shifted states from a spectrally stable state towards the spectrally unstable states. We prove rigorously the conjecture posed in the previous work [17] and confirm numerically the instability of the shifted states on the balanced star graphs.
3 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS 3 Note that the drift instability of shifted states on balanced star graphs is different from nonlinear instability of spectrally stable excited states, which are saddle points of energy subject to fixed mass [9, 10]. We emphasize again that the shifted states are degenerate minimizers of energy subject to fixed mass. The paper is structured as follows. Section 2 presents the background material and the main results of this work. Section 3 collects together the linear estimates. Section 4 gives the proof of the irreversible drift along the spectrally stable shifted states. Section 5 gives the proof of nonlinear instability of the limiting shifted state (called the halfsoliton state) due to the saddle point geometry. Section 6 illustrates the analytical results with numerical simulations. Section 7 concludes the paper with a summary. 2. Main results We consider a star graph Γ constructed by attaching N halflines at a common vertex. In the construction of the graph Γ, one edge represents an incoming bond and the remaining N 1 edges represent outgoing bonds. We place the vertex at the origin and parameterize the incoming edge by R and the N 1 outgoing edges by R +. An illustration of the star graph Γ with one incoming and three outgoing edges is shown on Fig Figure 1. A star graph with N = 4 edges. The Hilbert space L 2 (Γ) = L 2 (R ) L 2 (R + ) L 2 (R + ) }{{} (N1) elements is defined componentwise on edges of the star graph Γ. Sobolev spaces HΓ 1 and H2 Γ are also defined componentwise subject to the generalized Kirchhoff boundary conditions: (2.1) H 1 Γ := {Ψ H 1 (Γ) : α 1 ψ 1 (0) = α 2 ψ 2 (0) = = α N ψ N (0)} and (2.2) H 2 Γ := { Ψ H 2 (Γ) H 1 Γ : α 1 1 ψ 1(0) = N j=2 α 1 j ψ j(0) where derivatives are defined as lim x 0 for the incoming edge and lim x 0 + for the (N 1) outgoing edges. The dual space to HΓ 1 H 1 (Γ), which is also defined componentwise. is H 1 Γ },
4 4 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN We consider a balanced star graph defined by the following constraint on the positive coefficients (α 1, α 2,, α N ) in the boundary conditions (2.1) and (2.2): 1 N 1 (2.3) =. α 2 1 α 2 j=2 j This constraint was introduced in [29] as a condition of the reflectionless transmission of a solitary wave across the vertex of the star graph Γ. The time flow on Γ is given by the following nonlinear Schrödinger (NLS) equation: (2.4) i Ψ t = Ψ 2α2 Ψ 2 Ψ, where Ψ = Ψ(t, x), Ψ = (ψ 1, ψ 2,..., ψ N ) is the Laplacian operator defined componentwise with primes denoting derivatives in x, α = (α 1, α 2,..., α N ) R N represents the same coefficients as in (2.1) and (2.2), and the nonlinear term α 2 Ψ 2 Ψ is interpreted as a symbol for (α1 ψ ψ 1, α2 ψ ψ 2,..., αn 2 ψ N 2 ψ N ). Local and global wellposedness of the Cauchy problem for the NLS equation (2.4) is wellknown both for weak solutions in HΓ 1 and for strong solutions in H2 Γ (see Proposition 2.2 in [3] and Lemmas 2.2 and 2.3 in [17]). For any of these solutions, we can define the energy and mass functionals as (2.5) E(Ψ) := Ψ 2 L 2 (Γ) α 1 2 Ψ 4 L 4 (Γ), Q(Ψ) := Ψ 2 L 2 (Γ), respectively. These functionals are constants under the time flow of the NLS equation (2.4) thanks to the fact that : H 2 Γ L2 (Γ) L 2 (Γ) is extended as a selfadjoint operator in the Hilbert space L 2 (Γ) (see Lemma 2.1 in [17]). The NLS equation (2.4) admit standing wave solutions of the form Ψ(t, x) = e it Φ (x), where the realvalued pair (, Φ ) satisfies the stationary NLS equation, (2.6) Φ 2α 2 Φ 2 Φ = Φ, Φ H 2 Γ. The stationary NLS equation is the Euler Lagrange equation for the action functional (2.7) Λ (Ψ) := E(Ψ) + Q(Ψ). It is also wellknown [3] that the set of critical points of Λ in HΓ 1 is equivalent to the set of solutions of the stationary NLS equation in HΓ 2. For > 0, we can set = 1 by employing the following scaling transformation: (2.8) Φ (x) = 1 2 Φ(z), z = 1 2 x. The following lemma states the existence of a family of shifted states in the stationary NLS equation (2.6) with the boundary conditions in (2.1) and (2.2), where the coefficients (α 1, α 2,..., α N ) satisfy the constraint (2.3). Lemma 2.1. For every (α 1, α 2,..., α N ) satisfying the constraint (2.3), there exists a unique oneparameter family of solutions {Φ(x; a)} a R to the stationary NLS equation (2.6) with = 1, where each component of Φ(x; a) is given by (2.9) φ j (x; a) = α 1 j φ(x + a), 1 j n,
5 with φ(x) = sech(x). DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS 5 Proof. The stationary NLS equation (2.6) with = 1 admits a general solution Φ = (φ 1,..., φ N ) H 2 (Γ) of the form: φ j (x) = α 1 j φ(x + a j ), j = 1,..., N, with φ(x) = sech(x). Parameters (a 1,..., a N ) R N are to be defined by the boundary conditions in (2.1) and (2.2). The continuity condition in (2.1) implies that a 1 = = a N, and so, for every j = 1,..., N, there exists σ j { 1, 1} such that a j = σ j a 1. Without loss of generality, we choose σ 1 = 1. The Kirchhoff condition in (2.2) implies that, under the constraint (2.3), N (2.10) φ σ j 1 (a 1 ) = 0. αj 2 j=2 The equation (2.10) holds if either φ (a 1 ) = 0 or N σ j 1 j=2 = 0. The first case has a unique α 2 j solution a 1 = 0. The second case holds for every a 1 R\{0} if and only if for every j = 2,..., N we get σ j = 1, since σ j 1 is either negative or zero. Combining both cases, we have α 2 j a 1 = a 2 = = a N = a, where a R is arbitrary, as is given by (2.9). Remark 2.2. Compared to the parametrization of edges in Γ used in our previous work [17], φ 1 (x) is defined here for x R while all other φ j (x) are defined for x R +. We also replace parameter a R in [17] by a R for convenience. The shifted state (2.9) in Lemma 2.1 satisfies the following symmetry. Definition 2.3. For every fixed α = (α 1, α 2,, α N ) we say that the function Ψ = (ψ 1, ψ 2,, ψ N ) H 1 Γ is αsymmetric if it satisfies for all x R+ : (2.11) α 2 ψ 2 (x) = = α N ψ N (x). The symmetry (2.11) in Definition 2.3 provides the following reduction of the NLS equation (2.4) on the balanced star graph Γ under the constraint (2.3). Lemma 2.4. Assume that Ψ C(R, HΓ 2) C1 (R, L 2 (Γ)) is a strong solution to the NLS equation (2.4) under the constraint (2.3) satisfying the symmetry reduction (2.11) for every t R. The wave function { α1 ψ (2.12) ϕ(t, x) = 1 (t, x), x 0, α 2 ψ 2 (t, x), x 0, is a strong solution ϕ C(R, H 2 (R)) C 1 (R, L 2 (R)) to the following NLS equation on the real line R: (2.13) i ϕ t = ϕ 2 x 2 2 ϕ 2 ϕ. Proof. The NLS equation (2.13) is defined piecewise for x < 0 and x > 0 from the NLS equation (2.4), the symmetry (2.11), and the representation (2.12). Thanks to the boundary conditions in (2.1) and (2.2), the function ϕ(t, x) is continuously differentiable across the vertex point x = 0. Hence if Ψ C(R, HΓ 2) C1 (R, L 2 (Γ)) is a strong solution to (2.4), then ϕ C(R, H 2 (R)) C 1 (R, L 2 (R)) is a strong solution to (2.13).
6 6 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN The NLS equation (2.13) on the real line enjoys the translational symmetry in x. The free parameter a in the family of shifted state (2.9) in Lemma 2.1 is related to the translational symmetry of the NLS equation (2.13) in x. However, the translational symmetry is broken for the NLS equation (2.4) on the star graph Γ due to the vertex at x = 0. As a result, the momentum functional given by (2.14) P (Ψ) := Im Ψ, Ψ L 2 (Γ) = Im ( ) N ψ 1ψ 1 dx + Im ( ) ψ jψ j dx R R + is no longer constant under the time flow of (2.4). It was shown in [17] (see Lemma 6.1) that for every weak solution Ψ C(R, HΓ 1) C1 (R, H 1 (Γ)) to the NLS equation (2.4) the map t P (Ψ) is monotonically increasing, thanks to the following inequality: (2.15) d dt P (Ψ) = 1 2 N j=2 N i=2 i j α 2 1 α 2 j α2 i j=2 α j ψ j(0) α i ψ i(0) 2 0. If the weak solution Ψ satisfies the symmetry (2.11) in Definition 2.3, then P (Ψ) is conserved in t. It was proved in [17] (see Theorem 4.1 and Corollary 4.3) that the oneparameter family of shifted states in Lemma 2.1 is spectrally unstable for a < 0 and spectrally stable for a > 0. The degenerate state at a = 0 called the halfsoliton state is expected to be nonlinearly unstable as was shown in [16] for uniform star graphs with α = 1. In regards to the shifted states with a > 0, it was conjectured in [17] (see Conjectures 7.1 and 7.2) that the shifted state in Lemma 2.1 with a > 0 is nonlinearly unstable under the time flow because the shifted states drifts along the only incoming edge towards the halfsoliton state with a = 0, where it is affected by spectral instability of the shifted states with a < 0. The present work is devoted to the proof of the aforementioned conjectures. Our first main result shows that the monotone increase of the map t P (Ψ) as in (2.15) leads to a drift along the family of shifted states (2.9) in which the parameter a decreases monotonically in t towards a = 0. This drift induces nonlinear instability of the spectrally stable shifted states in Lemma 2.1 with a > 0. The following theorem formulates the result. Theorem 2.5. Fix a 0 > 0. For every ν (0, a 0 ) there exists ɛ 0 > 0 (sufficiently small) such that for every ɛ (0, ɛ 0 ), there exists δ > 0 and T > 0 such that for every initial datum Ψ 0 H 1 Γ with P (Ψ 0 ) > 0 and (2.16) inf θ R Ψ 0 e iθ Φ( ; a 0 ) H 1 (Γ) δ the unique solution Ψ C([0, T ], HΓ 1) C1 ([0, T ], H 1 Γ ) to the NLS equation (2.4) with the initial datum Ψ(0, ) = Ψ 0 satisfies the bound (2.17) inf θ R Ψ(t, ) eiθ Φ( ; a(t)) H 1 (Γ) ɛ, t [0, T ], where a C 1 ([0, T ]) is a strictly decreasing function such that lim t T a(t) = ν. By Theorem 2.5, the shifted state (2.9) with a > 0 drifts towards the halfsoliton state with a = 0. The halfsoliton state is more degenerate than the shifted state with a > 0 because the zero eigenvalue of the Jacobian operator associated with the stationary NLS equation (2.6) is
7 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS 7 simple for a > 0 and has multiplicity N 1 for a = 0. Moreover, while the shifted state Φ( ; a) with a > 0 is a degenerate minimizer of the action functional Λ =1 (Ψ) = E(Ψ) + Q(Ψ), the halfsoliton state Φ Φ( ; a = 0) is a degenerate saddle point of the same action functional [16]. The following theorem shows the nonlinear instability of the halfsoliton state related to the saddle point geometry of the critical point. Theorem 2.6. Denote Φ Φ( ; a = 0). There exists ɛ > 0 such that for every sufficiently small δ > 0 there exists V HΓ 1 with V HΓ 1 δ such that the unique solution Ψ C(R, H1 Γ ) C 1 (R, H 1 Γ ) to the NLS equation (2.4) with the initial datum Ψ(0, ) = Φ + V satisfies (2.18) inf θ R e iθ Ψ(T, ) Φ H 1 (Γ) > ɛ for some T > 0. Consequently, the orbit {Φe iθ } θ R is unstable under the time flow of the NLS equation (2.4). Remark 2.7. The result of Theorem 2.6 is very similar to the instability result in Theorem 2.7 in [16] which was proven for the uniform star graph with α = 1. Finally, the shifted state Φ( ; a) with a < 0 is a saddle point of the action functional Λ =1 (Ψ) = E(Ψ) + Q(Ψ). The saddle point is known to be spectrally unstable [17]. Consequently, it is also nonlinearly unstable under the time flow of the NLS equation (2.4) with fast growing perturbations which break the symmetry (2.11) of the shifted state. Our numerical results collected together in Section 6 illustrate all three stages of the nonlinear instability of the shifted state with a > 0 in the balanced star graph Γ with N = 3. We show the drift instability for the shifted states with a > 0, the weak instability of the halfsoliton state with a = 0, and the fast exponential instability of the shifted states with a < 0. We also illustrate numerically that the monotonic increase of the momentum functional in (2.15) can lead to the drift instability even if the assumption P (Ψ 0 ) > 0 of Theorem 2.5 on the initial datum Ψ 0 is not satisfied. 3. Linear estimates Recall that the scaling transformation (2.8) transforms the normalized shifted states Φ of Lemma 2.1 to the dependent family Φ of the shifted states. We note the following elementary computations: (3.1) D 1 () = Φ ( ; a), Φ ( ; a) L 2 (Γ) = 1 d 2 d Φ 2 L 2 (Γ) = 1 2α and (3.2) D 2 () = Φ ( ; a), ( + a)φ ( ; a) L 2 (Γ) = 1 2 Φ 2 L 2 (Γ) = 1 2. α 2 1 We discuss separately the linearization of the shifted state with a 0 and the halfsoliton state with a = 0.
8 8 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN 3.1. Linearization at the shifted state with a 0. For every standing wave solution Φ ( ; a) we define two selfadjoint linear operators L ± (, a) : HΓ 2 L2 (Γ) L 2 (Γ) by the differential expressions: { L (, a) = + 2α 2 Φ ( ; a) 2, (3.3) L + (, a) = + 6α 2 Φ ( ; a) 2. The operator L (, a) acts on the imaginary part of the perturbation to Φ ( ; a) and the operator L + (, a) acts on the real part of the perturbation; the latter operator is also the Jacobian operator for the stationary NLS equation (2.6). The spectrum of the selfadjoint operators L ± (, a) was studied in [17], from which we recall some basic facts. The continuous spectrum is strictly positive thanks to the fast exponential decay of Φ (x; a) to zero as x and Weyl s Theorem: (3.4) σ c (L ± (, a)) = [, ), where > 0. The discrete spectrum σ p (L ± (, a)) (, ) includes finitely many negative, zero, and positive eigenvalues of finite multiplicities. Figure 2. The spectrum of L + (, a) for = 1. The continuous spectrum is [1, ), while the discrete spectrum is given by the eigenvalues λ = 0, λ = 3, and λ = λ 1 (a) in (3.5). Eigenvalues of σ p (L + (, a)) (, ) are known in the explicit form [17]. For = 1, these eigenvalues are given by: a simple negative eigenvalue λ 0 = 3;
9 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS 9 a zero eigenvalue λ = 0 which is simple when a 0; the additional eigenvalue λ = λ 1 (a) of multiplicity N 2 given by (3.5) λ 1 (a) = 3 [tanh(a) 2 tanh(a) ] sech(a). It is negative ( ) for a < 0, zero for a = 0, and positive for a (0, a ), where a = tanh The eigenvalue merges into the continuous spectrum as a a. The spectrum of L + (, a) for = 1 is illustrated in Fig. 2. Eigenvalues of σ p (L (, a)) (, ) are nonnegative and the zero eigenvalue is simple. If a 0, the zero eigenvalues of L + (, a) and L (, a) are each simple with the eigenvectors given by (3.6) L + (, a)φ ( ; a) = 0, L (, a)φ ( ; a) = 0. The eigenvectors in (3.6) induce the generalized eigenvectors in (3.7) L + (, a) Φ ( ; a) = Φ ( ; a), L (, a)( + a)φ ( ; a) = 2Φ ( ; a). The following lemma gives coercivity of the quadratic forms associated with the operators L + (, a) and L (, a) for a > 0. Lemma 3.1. For every > 0 and a > 0, there exists a positive constant C(, a) such that (3.8) L + (, a)u, U L 2 (Γ) + L (, a)w, W L 2 (Γ) C(, a) U + iw 2 H 1 (Γ) if U and W satisfy the orthogonality conditions W, Φ ( ; a) L 2 (Γ) = 0, (3.9) U, Φ ( ; a) L 2 (Γ) = 0, U, ( + a)φ ( ; a) L 2 (Γ) = 0, Proof. The first orthogonality condition in (3.9) shifts the lowest (zero) eigenvalue of L (, a) to a positive eigenvalue thanks to the condition (3.1) (see Lemma 5.6 in [16]) and yields by Gårding s inequality the following coercivity bound L (, a)w, W L 2 (Γ) C() W 2 H 1 (Γ) independently of a. The second orthogonality condition in (3.9) shifts the lowest (negative) eigenvalue of L (, a) to a positive eigenvalue thanks to the same condition (3.1) (see Lemma 3.8 in [16]) and yields L + (, a)u, U L 2 (Γ) 0 with L + (, a)u, U L 2 (Γ) = 0 if and only if U is proportional to Φ ( ; a). The zero eigenvalue of L + (, a) is preserved by the constraint since Φ ( ; a), Φ ( ; a) L 2 (Γ) = 0. Finally, the third orthogonality condition in (3.9) shifts the zero eigenvalue of L + (, a) to a positive eigenvalue thanks to the condition (3.2). By Gårding s inequality, this yields the coercivity bound L + (, a)u, U L 2 (Γ) C(, a) U 2 H 1 (Γ), where C(, a) depends on a because the gap between the zero eigenvalue and the rest of the positive spectrum in σ p (L + (, a)) exists for a > 0 but vanishes as a 0.
10 10 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN Remark 3.2. For every > 0, the positive constant C(, a) in (3.8) satisfies C(, a) 0 as a 0. This is because the zero eigenvalue in σ p (L + (, a = 0)) has multiplicity (N 1) and the (N 2) eigenvectors of L + (, a = 0) satisfy the last two orthogonality conditions (3.9) as is seen from the proof of Lemma 3.6. Remark 3.3. For a < 0, the result of Lemma 3.1 is invalid because σ p (L + (, a)) includes another negative eigenvalue as is seen from Fig. 2. Remark 3.4. The orthogonality conditions in (3.9) are typically referred to as the symplectic orthogonality conditions, because they express orthogonality of residual terms U and W for real and imaginary parts of the perturbation to Φ ( ; a) to the eigenvectors and generalized eigenvectors of the spectral stability problem expressed by L + (, a) and L (, a) and the symplectic structure of the NLS equation. Note that our approach will not rely on adding one more orthogonality constraint W, Φ ( ; a) L 2 (Γ) = 0. As a result, we will only use three parameters for modulations of the stationary state orbit {e iθ Φ ( ; a)} θ R,a R +, R Linearization at the halfsoliton state. For a = 0, we denote operators L ± () L ± (, a = 0). The kernel of the operator L + () is spanned by an orthogonal basis consisting of N 1 eigenvectors, which we denote by {U (1), U (2),, U (N 1) }. The following lemma specifies properties of these basis eigenvectors. Lemma 3.5. There exists an orthogonal basis {U (1), U (2),, U (N 1) } of the kernel of L + () satisfying the orthogonality condition (3.10) U, Φ L 2 (Γ) = 0. The eigenvectors can be represented in the following way: for j = 1, (3.11) U (1) and for j = 2,..., N 1, := (α 1 1 φ, α 1 2 φ,..., α 1 N φ ), (3.12) U (j) := ( 0,..., 0 }{{}, r j φ, α 1 j+1 φ,..., α 1 N φ ), (j1) elements where φ (x) = 1 2 sech( 1 2 x), x R. r j = ( N α 2 i=j+1 i ) 1 α j, Proof. Let U = (u 1, u 2,..., u N ) HΓ 2 be an eigenvector for the zero eigenvalue of the operator L + (). Each component of the eigenvalue problem L + ()U = 0 satisfies (3.13) u j (x) + u j (x) 6 sech 2 ( x)u j (x) = 0, where x R on the first edge and x R + on the remaining edges. Since H 2 (R ± ) are continuously embedded into C 1 (R ± ), if U H 2 (Γ), then both u j (x) and u j(x) decay to zero as x. Such solutions to the differential equations (3.13) are given uniquely by u j (x) = a j φ (x) up to multiplication by a constant a j. Therefore, the eigenvector U is given by (3.14) U = (a 1 φ, a 2 φ,..., a N φ ).
11 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS 11 The eigenvector U HΓ 2 must satisfy the boundary conditions in (2.2). The continuity conditions hold since φ (0) = 0, whereas the Kirchhoff condition implies a 1 N a j (3.15) =. α 1 α j j=2 Since the scalar equation (3.15) relates N unknowns, the space of solutions for (a 1, a 2,..., a N ) is (N 1)dimensional and the kernel of the operator L + () is (N 1)dimensional. Let {U (1), U (2),..., U (N 1) } be an orthogonal basis of the kernel, which can be constructed from any set of basis vectors by applying the GramSchmidt orthogonalization process. Direct computations show that if U is given by (3.14), then ( N ) a j U, Φ L 2 (Γ) = a 1 φ α j α, φ L 2 (R + ), 1 j=2 which means that the condition (3.15) is equivalent to U, Φ L 2 (Γ) = 0. Therefore, all elements in the orthogonal basis satisfy the orthogonality condition (3.10). It remains to prove that the orthogonal basis can be characterized in the form given in (3.11) (3.12). From the constraint (2.3), we can take a j = α 1 j eigenvector U (1) to be defined by (3.11). The last eigenvector U (N 1) (3.16) U (N 1) := (0,..., 0, r N 1 φ, α 1 N φ ), where r N 1 is defined to satisfy the orthogonality condition U (1) for all j in (3.15) to set the first can be defined by, U (N 1) L 2 (Γ) = 0 and the condition (3.15). In fact, both conditions are equivalent since the first (N 2) entries of U (N 1) are zero and ) ( U (1), U (N 1) L 2 (Γ) = φ 2 rn 1 L 2 (R + ) with φ 2 L 2 (R + ) 0. Hence r N 1 is defined by r N 1 = α N 1. αn 2 α N α 2 N The remaining eigenvectors U (j) in (3.12) are constructed recursively from j = N 2 to j = 2. By direct computations we obtain that the orthogonality condition U (1), U (j) L 2 (Γ) = 0 is equivalent to the constraint (3.15). Moreover, all the eigenvectors are mutually orthogonal thanks to the recursive construction of U (j), We denote the eigenspace for the kernel of L + () by (3.17) X := span{u (1), U (2),, U (N 1) }. For each j = 1, 2,..., N 1, we construct the generalized eigenvector W (j) L ()W (j) = U (j), HΓ 2 by solving which exists thanks to the orthogonality condition (3.10) since Φ spans the kernel of L (). Explicitly, representing U (j) from (3.11) (3.12) by (3.18) U (j) = φ e j
12 12 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN with some xindependent vectors e j R N, we get for the same vectors e j (3.19) W (j) = χ e j, where χ (x) = 1 2 xφ (x), x R. We denote the eigenspace for the generalized kernel of L () by (3.20) X := span{w (1), W (2),, W (N 1) }, Similarly to Lemma 5.4 in [16], the following lemma gives coercivity of the quadratic forms associated with the operators L + () and L (). Lemma 3.6. For every > 0, there exists a positive constant C() such that (3.21) L + ()U, U L 2 (Γ) + L ()W, W L 2 (Γ) C() U + iw 2 H 1 (Γ) if U X and W X satisfying the additional orthogonality conditions { W, Φ (3.22) L 2 (Γ) = 0, U, Φ L 2 (Γ) = 0. Proof. We claim that basis vectors in X and X satisfy the following orthogonality conditions: { U (j) { W (j) { U (j), U (k) L 2 (Γ)} 1 j,k N 1 is a positive diagonal matrix;, W (k) L 2 (Γ)} 1 j,k N 1 is a positive diagonal matrix;, W (k) L 2 (Γ)} 1 j,k N 1 is a positive diagonal matrix. Indeed, orthogonality of {U (1),..., U (N 1) } is established by Lemma 3.5. Therefore, the vectors {e 1,..., e N 1 } in (3.18) are orthogonal in R N 1. Orthogonality of {W (1),..., W (N 1) } follows by the explicit representation (3.19) due to orthogonality of the vectors {e 1,..., e N 1 } in R N 1. The sets {U (1),..., U (N 1) } and {W (1),..., W (N 1) } are mutually orthogonal by the same reason. Finally, we have for every j = 1,..., N ( N ) ( N ) 1 1 (3.23) U (j), W (j) L 2 (Γ) = α2 j 4 i=j α 2 i α 2 i=j+1 i φ 2 L 2 (R + )(> 0). The rest of the proof is similar to the proof of Lemma 3.1 with the only difference that the third orthogonality condition (3.9) is replaced by the (N 1) orthogonality conditions in U X. The constraint U X provide the shift of the zero eigenvalue of L + () of algebraic multiplicity (N 1) to positive eigenvalues thanks to the condition that { U (j), W (k) L 2 (Γ)} 1 j,k N 1 is a positive diagonal matrix. 4. Drift of the shifted states with a > 0 The proof of Theorem 2.5 is divided into several steps. First, we decompose a unique global solution Ψ to the NLS equation (2.4) into the modulated stationary state {e iθ Φ ( ; a)} θ R,a R, R + and the symplectically orthogonal remainder terms. Second, we estimate the rate of change of the modulation parameter a(t) in time t and show that a (t) < 0 for t > 0. Third, we use energy estimates to control the time evolution of the modulation parameter (t) and the remainder terms. Although the decomposition works for any a(t), we only consider a(t) > 0.
13 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS Step 1: Symplectically orthogonal decomposition. Any point in HΓ 1 close to an orbit {e iθ Φ( ; a 0 )} θ R for some a 0 R can be represented by a superposition of a point on the family {e iθ Φ ( ; a)} θ R,a R, R + and a symplectically orthogonal remainder term. Here and in what follows, we denote Φ Φ =1. The following lemma provides details of this symplectically orthogonal decomposition. Lemma 4.1. Fix a 0 R. There exists some δ 0 > 0 such that for every Ψ H 1 Γ satisfying (4.1) δ := inf θ R Ψ eiθ Φ( ; a 0 ) H 1 (Γ) δ 0, there exists a unique choice for realvalued (θ,, a) R R + R and realvalued (U, W ) HΓ 1 H1 Γ in the decomposition (4.2) Ψ(x) = e iθ [Φ (x; a) + U(x) + iw (x)], subject to the orthogonality conditions W, Φ ( ; a) L 2 (Γ) = 0, (4.3) U, Φ ( ; a) L 2 (Γ) = 0, U, ( + a)φ ( ; a) L 2 (Γ) = 0, where, a, and (U, W ) satisfy the estimate (4.4) 1 + a a 0 + U + iw H 1 (Γ) Cδ, for some positive constant C > 0. Moreover, the map from Ψ H 1 Γ to (θ,, a) R R+ R and (U, W ) H 1 Γ H1 Γ is C. Proof. Define the following vector function G(θ,, a; Ψ) : R R + R HΓ 1 R3 given by G(θ,, a; Ψ) := Im(Ψ eiθ Φ ( ; a)), Φ ( ; a) L 2 (Γ) Re(Ψ e iθ Φ ( ; a)), Φ ( ; a) L 2 (Γ), Re(Ψ e iθ Φ ( ; a)), ( + a)φ ( ; a) L 2 (Γ) the zeros of which represent the orthogonality constraints in (4.3). The function G(θ,, a; Ψ) is C with respect to its arguments. Let θ 0 be the argument of inf θ R Ψ e iθ Φ( ; a 0 ) H 1 (Γ) for a given Ψ HΓ 1. The vector function G is a C map from R R + R to R 3 since the map R + R (, a) Φ ( ; a) L 2 (Γ) is C in both variables. Moreover, if Ψ HΓ 1 satisfies (4.1), then (4.5) G(θ 0, 1, a 0 ; Ψ) R 3 Cδ for a δindependent constant C > 0. Also we have D (θ,,a) G(θ 0, 1, a 0 ; Ψ) = D + B, where D = diag(d 1, d 1, d 2 ) with entries d 1 D 1 ( = 1) and d 2 D 2 ( = 1) given by (3.1) and (3.2), whereas B is a matrix satisfying the estimate B M3 3 Cδ for a δindependent constant C > 0. Since d 1, d 2 0, the matrix D is invertible and there exists δ 0 > 0 such that the Jacobian D (θ,,a) G(θ 0, 1, a 0 ; Ψ) is invertible for every δ (0, δ 0 ) with the bound (4.6) [D (θ,,a) G(θ 0, 1, a 0 ; Ψ)] 1 M3 3 C for a δindependent constant C > 0. By the local inverse mapping theorem, for the given Ψ HΓ 1 satisfying (4.1), the equation G(θ,, a; Ψ) = 0 has a unique solution (θ,, a) R3 in
14 14 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN a neighborhood of the point (θ 0, 1, a 0 ). Since G(θ,, a; Ψ) is C with respect to its arguments, the solution (θ,, a) R R + R is C with respect to Ψ HΓ 1. The Taylor expansion of G(θ,, a; Ψ) = 0 around (θ 0, 1, a 0 ), 0 = G(θ 0, 1, a 0 ; Ψ)+D (θ,,a) G(θ 0, 1, a 0 ; Ψ)(θ θ 0, 1, a a 0 ) T +O( θ θ a a 0 2 ), together with the bounds (4.5) and (4.6) implies the bound (4.4) for 1 and a a 0. From the decomposition (4.2) and with use of the triangle inequality for (θ,, a) near (θ 0, 1, a 0 ), it follows that (U, W ) are uniquely defined in HΓ 1 and satisfy the bound in (4.4). In addition, (U, W ) HΓ 1 are C with respect to Ψ HΓ 1. If δ > 0 in the initial bound (2.16) is sufficiently small, we can represent the initial datum Ψ 0 HΓ 1 to the Cauchy problem associated with the NLS equation (2.4) in the form: (4.7) Ψ 0 (x) = Φ(x; a 0 ) + U 0 (x) + iw 0 (x), U 0 + iw 0 H 1 (Γ) δ, subject to the orthogonality conditions W 0, Φ =1 ( ; a 0 ) L 2 (Γ) = 0, (4.8) U 0, Φ( ; a 0 ) L 2 (Γ) = 0, U 0, ( + a 0 )Φ( ; a 0 ) L 2 (Γ) = 0. By Lemma 4.1, the orthogonal decomposition (4.7) with (4.8) implies that θ(0) = 0 and (0) = 1 initially. Although this is not the most general case for the initial datum satisfying (2.16), this simplification is used to illustrate the proof of Theorem 2.5. A generalization for initial datum Ψ 0 HΓ 1 with θ(0) 0 and (0) 1 is straightforward. By the global wellposedness theory [3, 17], the NLS equation (2.4) with the initial datum Ψ 0 HΓ 1 generates a unique solution Ψ C(R, H1 Γ ) C1 (R, H 1 Γ ). By continuous dependence of the solution on the initial datum and by Lemma 4.1, for every ɛ (0, δ 0 ) with δ 0 in the bound (4.1) there exists t 0 > 0 such that the unique solution Ψ satisfies (4.9) inf θ R e iθ Ψ(t, ) Φ H 1 (Γ) ɛ, t [0, t 0 ] and can be uniquely decomposed in the form: (4.10) Ψ(t, x) = e iθ(t) [ Φ (t) (x; a(t)) + U(t, x) + iw (t, x) ], subject to the orthogonality conditions W (t, ), Φ =(t) ( ; a(t)) L 2 (Γ) = 0, (4.11) U(t, ), Φ (t) ( ; a(t)) L 2 (Γ) = 0, U(t, ), ( + a(t))φ ( ; a(t)) L 2 (Γ) = 0. By the smoothness of the map in Lemma 4.1 and by the wellposedness of the time flow of the NLS equation (2.4), we have U, W C([0, t 0 ], HΓ 1) C1 ([0, t 0 ], H 1 Γ ) and (θ,, a) C 1 ([0, t 0 ], R R + R). In order to prove Theorem 2.5, we control (t), U(t, ), and W (t, ) from energy estimates and a(t) from modulation equations, whereas θ(t) plays no role in the bound (2.17). Note that the modulation of a(t) captures the irreversible drift of the shifted states along the incoming edge towards the vertex of the balanced star graph. We would not see this drift without using the parameter a(t) and we would not be able to control (t), U(t, ), and W (t, ) from energy
15 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS 15 estimates without the third constraint in (4.11) because of the zero eigenvalue of L + (, a), see Lemma Step 2: Monotonicity of a(t). We use the orthogonal decomposition (4.10) with (4.11) in order to obtain the evolution system for the remainder terms (U, W ) and for the modulation parameters (θ,, a). By analyzing the modulation equation for a(t), we relate the rate of change of a(t) and the value of the momentum functional P (Ψ) given by (2.14). Lemma 4.2. Assume that the unique solution Ψ C([0, t 0 ], HΓ 1) C1 ([0, t 0 ], H 1 Γ ) represented by (4.10) and (4.11) satisfies (4.12) (t) 1 + U(t, ) + iw (t, ) H 1 (Γ) ɛ, t [0, t 0 ] with ɛ (0, δ 0 ) and δ 0 defined in (4.1). The time evolution of the translation parameter a(t) is given by (4.13) ȧ(t) = α P (Ψ) [ 1 + O( U + iw H 1 (Γ)) ] + O( U + iw 2 H 1 (Γ) ), where P (Ψ) is given by (2.14). Proof. By substituting (4.10) into the NLS equation (2.4) and by using the rotational and translation symmetries, we obtain the time evolution system for the remainder terms: ( ) ( ) ( ) ( ) d U 0 L = (, a) U + ( dt W L + (, a) 0 W θ W ) (Φ + U) ( ) ( ) ( ) Φ Φ ȧ RU (4.14) +, 0 0 R W where Φ Φ (x; a), the prime denotes derivative in x, the dot denotes derivative in t, the linearized operators are given by (3.3), and the residual terms are given by { RU = 2α 2 (2Φ U + U 2 + W 2 ) W, (4.15) R W = 2α 2 [Φ (3U 2 + W 2 ) + (U 2 + W 2 )U]. By using the orthogonality conditions (4.11), we obtain the modulation equations for parameters (θ,, a) from the system (4.14): θ 0 Φ ( ; a), R U L 2 (Γ) (4.16) A = 0 + Φ, R W L 2 ȧ 2 Φ (Γ), ( ; a), W L 2 (Γ) ( + a)φ ( ; a), R W L 2 (Γ) where the matrix A is given by Φ ( ; a), W L 2 (Γ) Φ ( ; a), Φ ( ; a) U L 2 (Γ) Φ ( ; a), U L 2 (Γ) A = Φ ( ; a), Φ ( ; a) + U L 2 (Γ) Φ 2 ( ; a), W L 2 (Γ) Φ ( ; a), W L 2 (Γ). ( + a)φ ( ; a), W L 2 (Γ) ( + a) Φ ( ; a), U L 2 (Γ) ( + a)φ ( ; a), Φ ( ; a) U L 2 (Γ) If (U, W ) = (0, 0), the matrix A is invertible since 0 D 1 () 0 A 0 = D 1 () D 2 ()
16 16 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN has nonzero elements thanks to (3.1) and (3.2). Therefore, under the assumption (4.12) with small ɛ > 0, we have (4.17) A 1 M3 3 C for an ɛindependent constant C > 0. translation parameter a(t) is given by (4.18) ȧ = 2 Φ ( ; a), W L 2 (Γ) D 1 () This bound implies that the timeevolution of the [ 1 + O( U + iw H 1 (Γ)) ] + O( U + iw 2 H 1 (Γ) ). On the other hand, the momentum functional P (Ψ) in (2.14) can be computed at the solution Ψ in the orthogonal decomposition (4.10) as follows (4.19) P (Ψ) = Φ ( ; a), W L 2 (Γ) Φ ( ; a), W L 2 (Γ) + O( U + iw 2 H 1 (Γ) ) = 2 Φ ( ; a), W L 2 (Γ) + O( U + iw 2 H 1 (Γ) ), where the integration by parts does not result in any contribution from the vertex at x = 0 thanks to the boundary conditions in (2.2) and the constraint (2.3). Combining (4.18) and (4.19) with the exact computation (3.2) yields expansion (4.13). Corollary 4.3. In addition to (4.12), assume that Ψ 0 in (4.7) is chosen such that P (Ψ 0 ) > 0. There exists ɛ 0 sufficiently small such that for every ɛ (0, ɛ 0 ) there exists δ > 0 such that the map t a(t) is strictly decreasing for t [0, t 0 ]. Proof. The map t P (Ψ) is monotonically increasing as it can be seen from the expression (2.15). Therefore, if the initial datum Ψ 0 in (4.7) satisfies P (Ψ 0 ) > 0, then (4.20) P (Ψ) P (Ψ 0 ) > 0 for all t [0, t 0 ]. It follows from (4.7) and (4.19) that there are δindependent constants C, C + > 0 such that (4.21) C δ P (Ψ 0 ) C + δ. Then, it follows from (4.12), (4.13), (4.20) and (4.21) that there exist δ and ɛindependent constants C 1, C 2 > 0 such that ȧ C 1 δ C 2 ɛ 2. If δ satisfies δ Cɛ 2 for a given small ɛ > 0 with an ɛindependent constant C > C1 1 C 2 then ȧ (C 1 C C 2 )ɛ 2 > 0 so that the map t a(t) is strictly decreasing for t [0, t 0 ] Step 3: Energy estimates. The coercivity bound (3.8) in Lemma 3.1 allows us to control the time evolution of (t), U(t, ), and W (t, ), as long as a(t) is bounded away from zero. The following result provides this control from energy estimates. Lemma 4.4. Let Ψ be the unique global solution to the NLS equation (2.4) given by (4.10) (4.11) for t [0, t 0 ] with some t 0 > 0 such that the initial data Ψ(0, ) = Ψ 0 satisfies (4.7) (4.8). Assume that a(t) ā for t [0, t 0 ]. For every ā > 0, there exists a δindependent positive constant K(ā) such that (4.22) (t) U(t, ) + iw (t, ) 2 H 1 (Γ) K(ā)δ2, t [0, t 0 ].
17 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS 17 Proof. Recall that the shifted state Φ ( ; a) is a critical point of the action functional Λ (Ψ) = E(Ψ) + Q(Ψ) in (2.7). By using the decomposition (4.10) and the rotational invariance of the NLS equation (2.4), we define the following energy function: (4.23) (t) := E(Φ (t) +U(t, )+iw (t, )) E(Φ)+(t) [ Q(Φ (t) + U(t, ) + iw (t, )) Q(Φ) ]. Expanding into Taylor series, we obtain (4.24) = D() + L + (, a)u, U L 2 (Γ) + L (, a)w, W L 2 (Γ) + N (U, W ), where N (U, W ) = O( U + iw 3 H 1 (Γ)) and D() is defined by D() := E(Φ ) E(Φ) + [Q(Φ ) Q(Φ)]. Since D () = Q(Φ ) Q(Φ) thanks to the variational problem for the standing wave Φ, we have D(1) = D (1) = 0, and (4.25) D() = ( 1) 2 Φ, Φ =1 L 2 (Ω) + O( 1 3 ). It follows from the initial decomposition (4.7) (4.8) that (4.26) (0) = E(Φ + U 0 + iw 0 ) E(Φ) + Q(Φ + U 0 + iw 0 ) Q(Φ) satisfies the bound (4.27) (0) C 0 δ 2 for a δindependent constant C 0 > 0. On the other hand, the energy and mass conservation in (2.5) imply that (4.28) (t) = (0) + ((t) 1) [Q(Φ + U 0 + iw 0 ) Q(Φ)], where the remainder term also satisfies (4.29) Q(Φ + U 0 + iw 0 ) Q(Φ) C 0 δ 2 for a δindependent constant C 0 > 0. The representation (4.28) together with the expression (4.24) allows us to control (t) near (0) = 1 and the remainder terms (U, W ) in HΓ 1 as follows: (4.30) (0) = ( 1) 2 Φ, Φ =1 L 2 (Ω) ( 1) [Q(Φ + U 0 + iw 0 ) Q(Φ)] + L + (, a)u, U L 2 (Γ) + L (, a)w, W L 2 (Γ) + O( U + iw 3 H 1 (Γ) ). By using the expansion (4.30), the coercivity bound (3.8), and the bounds (4.27) and (4.29), we obtain C 0 δ 2 (0) 1 ( 1) 2 C 2α δ 2 1 +C(, a) U+iW 2 H 1 (Γ) +O( U+iW 3 H 1 (Γ) ), 2 from which the bound (4.22) follows. Remark 4.5. By Remark 3.2, for every > 0, we have C(, a) 0 as a 0. Therefore, we have K(ā) as ā 0.
18 18 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN 4.4. Proof of Theorem 2.5. The initial datum satisfies the initial decomposition (4.7) (4.8) with small δ and initial conditions θ(0) = 0, (0) = 1, and a(0) = a 0 with a 0 > 0. Thanks to the continuous dependence of the solution of the NLS equation (2.4) on initial datum, the solution is represented by the orthogonal decomposition (4.10) (4.11) on a short time interval [0, t 0 ] for some t 0 > 0. Hence, it satisfies the apriori bound (4.9). The modulation parameters θ(t), (t), and a(t) are defined for t [0, t 0 ] and a(t) ā for some ā > 0 for t [0, t 0 ]. By energy estimates in Lemma 4.4, the parameter (t) and the remainder terms (U, W ) H 1 Γ satisfy the bound (4.22) with a δindependent positive constant K(ā). For given small ɛ > 0 and ν > 0 in Theorem 2.5, let us define (4.31) K ν := max ā [ν,a 0 ] K(ā), 2 δ := K ν ɛ. Then, the bound (4.22) provides the bound (4.12) of Lemma 4.2 for all t [0, t 0 ]. Assume that the initial datum also satisfies P (Ψ 0 ) > 0. By Corollary 4.3, the map t a is strictly decreasing for t [0, t 0 ] if δ satisfies δ Cɛ 2 for a δ and ɛindependent constant C > 0. The definition of δ in (4.31) is compatible with the latter bound if ɛ (0, ɛ 0 ) with ɛ 0 := 1 C K ν. If in addition ɛ 0 δ 0, where δ 0 is defined by (4.1) in Lemma 4.1, then the solution Ψ(t, ) H 1 Γ for t [0, t 0 ] satisfies the conditions of Lemma 4.1 so that the orthogonal decomposition (4.10) with (4.11) is continued beyond the short time interval [0, t 0 ] to the maximal time interval [0, T ] as long as a(t) ν for t [0, T ]. Thanks to the monotonicity argument in Lemma 4.2 and Corollary 4.3, for every ɛ (0, ɛ 0 ), there exists a finite T > 0 such that lim t T a(t) = ν. Note that T = O(ɛ 2 ) as ɛ 0. Theorem 2.5 is proved. Remark 4.6. It follows that K ν as ν 0 by Remark 4.5 so that ɛ 0 may be sufficiently small for a fixed small ν > Instability of the halfsoliton state The proof of Theorem 2.6 is divided into several steps. First, we decompose a unique global solution Ψ to the NLS equation (2.4) into the modulated stationary state {e iθ Φ } θ R, R + and the symplectically orthogonal remainder terms, where Φ Φ ( ; a = 0). Next, we provide a secondary decomposition of the remainder terms as a superposition of projections to the bases in X and X in (3.17) and (3.20) and the symplectically orthogonally remainder terms. Energy estimates are used to control the time evolution of (t) and the remainder terms. Finally, we consider the perturbed Hamiltonian system for projections to the bases in X and X and prove that the truncated system is unstable near the zero equilibrium point. This instability drives the instability of the solution Ψ under the time flow of the NLS equation (2.4) away from the halfsoliton state {e iθ Φ } θ R, R Step 1: Primary decomposition. For every sufficiently small δ > 0, we consider the initial datum Ψ 0 HΓ 1 to the Cauchy problem associated with the NLS equation (2.4) in the form: (5.1) Ψ 0 = Φ + U 0 + iw 0, U 0 + iw 0 H 1 (Γ) δ, 1
19 DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS 19 subject to the orthogonality conditions { W0, (5.2) Φ =1 L 2 (Γ) = 0, U 0, Φ L 2 (Γ) = 0, where Φ Φ( ; a = 0). By the global wellposedness theory [3, 17], this initial datum generates a unique global solution Ψ C(R, HΓ 1) C1 (R, H 1 Γ ) to the NLS equation (2.4). We use the decomposition of the unique global solution Ψ C(R, HΓ 1) C1 (R, H 1 Γ ) into the modulated stationary state {e iθ Φ } with close to 0 = 1 and the symplectically orthogonal remainder terms. This decomposition is similar to Lemma 4.1 with a 0 = 0 with the only change that a = 0 is set in the decomposition (4.2) and the remainder terms satisfy the first two of the three orthogonality conditions in (4.3). By continuity of the global solution and by Lemma 4.1, for every ɛ (0, δ 0 ) with δ 0 in the bound (4.1) there exists t 0 > 0 such that the unique solution Ψ satisfies (5.3) inf θ R e iθ Ψ(t, ) Φ H 1 (Γ) ɛ, t [0, t 0 ] and can be uniquely represented as (5.4) Ψ(t, x) = e iθ(t) [ Φ (t) (x) + U(t, x) + iw (t, x) ], subject to the orthogonality conditions { W (t, ), Φ (5.5) =(t) L 2 (Γ) = 0, U(t, ), Φ (t) L 2 (Γ) = 0. Since Ψ C(R, HΓ 1) C1 (R, H 1 Γ ) and the map R Φ HΓ 1 is smooth, we obtain (θ, ) C 1 ([0, t 0 ], R R + ), hence U, W C([0, t 0 ], HΓ 1) C1 ([0, t 0 ], H 1 Γ ). The choice (5.1) with (5.2) implies that (0) = 1 and θ(0) = 0. Although this is again special, it is nevertheless sufficient for the proof of the instability result. In order to prove Theorem 2.6 we fix ɛ (0, δ 0 ) and we intend to show that there exists such T > 0 that the bound (5.3) is satisfied for all t [0, T ), but fails to satisfy for t T. We substitute the decomposition (5.4) into the NLS equation (2.4) to get the time evolution system for the remainder terms (U, W ). The following lemma reports the estimates on the evolution of the modulation parameters θ(t) and (t). Lemma 5.1. Assume that the unique solution Ψ C([0, t 0 ], HΓ 1) C1 ([0, t 0 ], H 1 Γ ) represented by (5.4) and (5.5) satisfies (5.6) (t) 1 + U(t, ) + iw (t, ) H 1 (Γ) ɛ, t [0, t 0 ] with ɛ (0, δ 0 ) and δ 0 defined in (4.1). There exists an ɛindependent constant A > 0 such that for every t [0, t 0 ], { (5.7) θ(t) ( ) (t) A U(t, ) 2 H 1 (Γ) + W (t, ) 2 H 1 (Γ), (t) A U(t, ) H 1 (Γ) W (t, ) H 1 (Γ). Proof. The time evolution system for the remainder terms is the same as the system (4.14) but with a(t) = 0 and ȧ(t) = 0 for t [0, t 0 ]. Using the orthogonality conditions (5.5) yields the same system of modulation equations as in (4.16) but constrained by the first two equations and a 2 2 coefficient matrix denoted by A. The estimates (5.7) are obtained from invertibility
20 20 ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN of A satisfying A 1 M2 2 C for an ɛindependent constant C > 0 and the explicit form R U and R W in (4.15) Step 2: Secondary decomposition. Recall the eigenspaces X and X in (3.17) and (3.20). We decompose the remainder terms (U, W ) in (5.4) as follows: (5.8) U(t, x) = N 1 j=1 N 1 c j (t)u (j) (t) (x) + U (t, x), W (t, x) = j=1 b j (t)w (j) (t) (x) + W (t, x), subject to the orthogonality conditions for U (t, ) X(t) and W (t, ) X (t). The coefficients c = (c 1, c 2,..., c N 1 ) R N 1, b = (b 1, b 2,..., b N 1 ) R N 1 and the remainder terms (U, W ) in (5.8) are uniquely determined for each (U, W ) due to the mutual orthogonality of the basis vectors used in Lemma 3.6. We also have c, b C 1 ([0, t 0 ], R N 1 ) and U, W C([0, t 0 ], HΓ 1) C1 ([0, t 0 ], H 1 Γ ) since C1 ([0, t 0 ], R) and U, W C([0, t 0 ], HΓ 1) C 1 ([0, t 0 ], H 1 Γ ). Remark 5.2. Thanks to the explicit form (3.11), the parameter c 1 (t) in the decomposition (5.4) and (5.8) plays the same role as the parameter a(t) in the decomposition (4.10), whereas a(0) = a 0 in the initial decomposition (4.7) is set to a 0 = 0. By substituting (5.8) into the evolution equation for the remainder terms and using orthogonality conditions for the new remainder terms U (t, ) X(t) and W (t, ) X (t), we obtain the timeevolution system for the coefficients in the form: (5.9) with R (j) c = R (j) b = [ [ { W (j), U L 2 (Γ) U (j), W L 2 (Γ) W (j) W (j) N 1 i=1 N 1 i=1, U (j), U (j) ( dcj L 2 (Γ) dt db j L 2 (Γ) dt b j ) = R (j) b, c i W (j), U (i) L 2 (Γ) b i U (j), W (i) L 2 (Γ) ] ] = R (j) c, + ( θ ) W (j) ( θ ) U (j) where the terms R U and R W are given by (4.15) and the orthogonality conditions, W L 2 (Γ) W (j), R U L 2 (Γ),, U L 2 (Γ) + U (j), R W L 2 (Γ), U (j), Φ L 2 (Γ) = W (j), Φ L 2 (Γ) = 0, 1 j N 1 have been used. The timeevolution system (5.9) will be truncated and studied in Step 3, whereas (t) and the remainder terms (U, W ) are controlled as small perturbations by using the energy estimates. The following lemma gives the estimates on these terms. Lemma 5.3. Assume that there exists a positive constant A such that for every ɛ > 0 and some t 0 > 0, the remainder terms of the solution Ψ decomposed as (5.4) and (5.8) satisfy (5.10) (t) 1 + c(t) + b(t) + U (t, ) + iw (t, ) H 1 (Γ) Aɛ, t [0, t 0 ]. There exist an ɛindependent constant C > 0 such that (5.11) (t) U (t, ) + iw (t, ) 2 H 1 (Γ) C ( δ 2 + b 2 + c 3), t [0, t 0 ], where δ is given in (5.1) for the initial datum Ψ 0.
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