A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schrodinger Equations

A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schrodinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions. 1. MAIN RESULTS The local behavior near some distinguished solution, such as a steady state, of an evolution equation, can be determined through a decomposition into invariant manifolds, that is, stable, unstable and center manifolds. These (locally invariant) manifolds are characterized by decay estimates. While the flows on the stable and unstable manifolds are determined by exponential decay in forward and backward time respectively, that on the Indiana University Mathematics Journal c ©, Vol. 49, No. 1 (2000) 221 222 F. GESZTESY, C. K. R. T. JONES, Y. LATUSHKIN & M. STANISLAVOVA center manifold is ambiguous. Nevertheless, a determination of the flow on the center manifold can lead to a complete characterization of the local flow and thus this decomposition, when possible, leads to a reduction of this problem to one of identifying the flow on the center manifold. This strategy has a long history for studying the local behavior near a critical point of an ordinary differential equation, or a fixed point of a map, and it has gained momentum in the last few decades in the context of nonlinear wave solutions of evolutionary partial differential equations. Extending the ideas to partial differential equations has, however, introduced a number of new issues. In infinite dimensions, the relation between the linearization and the full nonlinear equations is more delicate. This issue, however, turns out to be not so difficult for the invariant manifold decomposition and has largely been resolved, see, for instance, [2], [3]. A more subtle issue arises at the linear level. All of the known proofs for the existence of invariant manifolds are based upon the use of the group (or semigroup) generated by the linearization. The hypotheses of the relevant theorems are then formulated in terms of estimates on the appropriate projections of these groups onto stable, unstable and center subspaces. These amount to spectral estimates that come directly from a determination of the spectrum of the group. However, in any actual problem, the information available will, at best, be of the spectrum of the infinitesimal generator, that is, the linearized equation and not its solution operator. Relating the spectrum of the infinitesimal generator to that of the group is a spectral mapping problem that is often non-trivial. In this paper, we resolve this issue for nonlinear Schrodinger equations. We formulate the results for the case of space-dependent nonlinearities in arbitrary dimensions. This class of equations is motivated by the one space dimension case that appears in the study of optical waveguides, see [18], and has attracted the attention of many authors. In particular, there is extensive literature on the existence and instability of standing waves, see, for instance, [10, 11, 12, 15, 16] and the references therein. In many instances the questions of the existence of standing waves and the structure of the spectrum of the linearization of the nonlinear equation around the standing wave are well-understood, see [18]. In the case considered in this paper, the interesting examples are known to have the spectrum of their linearization A enjoying a disjoint decomposition: the essential spectrum is positioned on the imaginary axis, and there are several isolated eigenvalues off the imaginary axis, see [10, 15, 16]. However, as mentioned above, this spectral information about the linearization Spectral Mapping Theorems and Invariant Manifolds 223 A is not sufficient to guarantee the existence of invariant manifolds. The general theory gives the existence of these manifolds for a semilinear equation with linear part A only when the spectrum of the operator etA, t > 0, rather than that ofA, admits a decomposition into disjoint components. It is not a priori clear that the spectrum of the operator etA is obtained from the spectrum of A by exponentiation. Indeed, in the present case, the operator A does not generate a semigroup for which this property is known (such as for analytic semigroups). Thus, we prove such a spectral mapping theorem (cf. Theorem 1) in this paper. This spectral mapping theorem is derived from a known abstract result in the theory of strongly continuous semigroups of linear operators (see Theorem 3). To apply this abstract result one needs to prove that the norm of the resolvent of A is bounded along vertical lines in the complex plane. The corresponding proof is based on Lemmas 5 and 6. The main technical step in the proof of these lemmas concerns a result about the high-energy decay of the norm of a BirmanSchwinger-type operator (cf. Proposition 8), a well-known device borrowed from quantum mechanics. We consider the following Schrodinger equation with space-dependent nonlinearity, iut = ∆u+ f(x, |u|2)u+ βu, (1) u = u(x, t) ∈ C, x ∈ Rn, t ≥ 0, β ∈ R, where ∆ = ∑nj=1 ∂/∂xj denotes the Laplacian, n ∈ N, and f is real-valued. Rewriting (1) in terms of its real and imaginary parts, u = v + iw, one obtains vt = ∆w + f(x,v2 +w2)w + βw, (2) wt = −∆v − f(x,v2 +w2)v − βv. A standing wave of frequency β for (1) is a time-independent real-valued solution û = û(x) of (2). Suppose the standing wave û is given a priori. Consider the linearization of (2) around û (recalling ŵ = 0), pt = ∆q + f(x, û2)q + βq, qt = −∆p − f(x, û2)p − 2∂2f(x, û2)û2p − βp, 224 F. GESZTESY, C. K. R. T. JONES, Y. LATUSHKIN & M. STANISLAVOVA where ∂2f(x,y) = fy(x,y). Thus, the linearized stability of the standing wave is determined by the operator A= [ 0 −LR LI 0 ] , (3) where LR = −∆ − β +Q1, LI = −∆ − β +Q2, and the potentials Q1 and Q2 are explicitly given by the formulas Q1(x) = −f(x, û2(x)), Q2(x) = −f(x, û2(x))− 2∂2f(x, û2(x))û2(x). We impose the following conditions on f , β, and the standing wave û (see [10, 15, 16]): (H1) f : Rn+1 → R is C3 and all derivatives are bounded on a set of the form Rn ×U , where U is a neighborhood of 0 ∈ R; (H2) f(x,0)→ 0 exponentially as |x| → ∞; (H3) β < 0; (H4) |û(x)| → 0 exponentially as |x| → ∞. As a result, the potentials Q1 and Q2 exponentially decay at infinity. The operator A is considered on L2(Rn) ⊕ L2(Rn); the domain D (−∆) is chosen to be the standard Sobolev space H2(Rn), and the domain of A is then H2(Rn)⊕H2(Rn). Note that [ 0 ∆ −∆ 0 ] generates a strongly continuous group on L2(Rn)⊕L2(Rn). Thus, its bounded perturbation A generates a strongly continuous group {etA}t∈R as well. It was proved in [10, Thm. 3.1] that σess(A) = {iξ | ξ ∈ R, |ξ| ≥ −β}. In addition, it was proved in [10, 15, 16] that, under the above hypotheses, σ(A)\σess(A) consists of finitely many eigenvalues, symmetric with respect to both coordinate axes. We prove the following result that relates the spectrum of the semigroup {etA}t≥0 and the spectrum of its generator. Theorem 1 (Spectral Mapping Theorem). For each n ∈ N, one has σ(etA) = etσ(A) for all t > 0. Spectral Mapping Theorems and Invariant Manifolds 225 Recall that the spectral inclusion etσ(A) ⊂ σ(etA) always holds. Also, since {etA}t∈R is a group, 0 6∈ σ(etA). See [8, 9, 22, 26, 28] for a discussion of the spectral mapping theorems for strongly continuous semigroups and examples where the spectral mapping property as in Theorem 1 fails. A spectral mapping theorem was proved in [19] for a nonlinear Schrodinger equation with a specific potential and in the case n = 1. Their proof also uses Theorem 3 that is the key to our result. To the best of our knowledge the work of Kapitula and Sandstede [19] was the first to use the GearhartGreiner-Herbst-Pruss Theorem 1 in this context. Theorem 3 was also used by Miller and Weinstein in [21] to prove asymptotic stability of solitary waves for the regularized long-wave equation (see also related work [23] by Pego and Weinstein and the bibliography in these papers). Since the spectral mapping theorem always holds for the point spectrum, Theorem 1 implies, in particular, that σess(etA) ⊆ T = {|z| = 1}. It follows that there will be only finitely many eigenvalues of etA off the unit circle and therefore general results on the existence of invariant manifolds for semilinear equations can be invoked (see, e.g., [3] and compare also with [4] and the literature cited in [20, p. 4]). The (local) stable manifold is defined as the set of initial data whose solutions stay in the prescribed neighborhood and tend to û exponentially as t → +∞. The unstable manifold is defined analogously but in backward time. The center manifold is complementary to these two and contains solutions with neutral decay behavior (although they can decay, they will not do so exponentially). In particular, the center manifold contains all solutions that stay in the neighborhood in both forward and backward time. For details see, for instance, [3]. Concerning the equations under consideration here, we have the following main theorem. Theorem 2. Assuming (H1)-(H4), in a neighborhood of the standing wave solution û of (1) there are locally invariant stable, unstable, and center manifolds. Moreover, the stable and unstable manifolds are of (equal) finite dimension, and the center manifold is infinite-dimensional. All of the examples of standing waves considered in [17] and [18] satisfy the hypotheses given here and thus enjoy a local decomposition of the flow by invariant manifolds. Some of these waveguide modes are stable, while others are unstable. In the unstable cases, the above results show that the instabilities are controlled by finite-dimensional (mostly, just onedimensional) unstable manifolds. A natural question is whether the waveguide modes are stable relative to the flow on the center manifold. Such a 226 F. GESZTESY, C. K. R. T. JONES, Y. LATUSHKIN & M. STANISLAVOVA result was shown for the case of nonlinear Klein-Gordon equations in [4] using an energy argument. Whether such an argument will work for nonlinear Schrodinger equations is open. It is more than of academic interest, as stability on the center manifold has the consequence that the center manifold is unique, see [3], and armed with such a result, a complete description of the local flow can be legitimately claimed. Cases of standing waves in higher dimensions are given in [16]. Some of these are unstable and the above considerations again apply. We also wish to stress that the spectral mapping theorem developed here is not restricted to a single equation. Indeed, the results formulated here are easily adaptable to systems of nonlinear Schrodinger equations. This is particularly important as such systems arise in, among other problems, second harmonic generation in waveguides and wave-division multiplexing in optical fibers. The case of systems is considered in Section 4. In the next section, we give the basic set-up that will be used and formulate the necessary lemmas for proving Theorem 1. The proofs are given in Section 3. 2. BASIC LEMMAS To prove Theorem 1, we will use the following abstract result known as the Gearhart-Greiner-Herbst-Pruss theorem, see, e.g., [22, p. 95]. Theorem 3. Let A be a generator of a strongly continuous semigroup on a complex Hilbert space. Then for each t > 0, the following spectral mapping theorem is valid: σ(etA) \ {0} = { eλt | either μk := λ+ 2πik t ∈ σ(A) for some k ∈ Z, or the sequence {‖(μk −A)−1‖}k∈Z is unbounded } . According to the results in [10, Thm.3.1] and [15, 16], the essential spectrum of the generator A in (3) is given by σess(A) = {iξ | ξ ∈ R, |ξ| ≥ −β}. Moreover, σ(A) \ σess(A) consists of finitely many eigenvalues. In particular, {z ∈ C : |z| = 1} ⊂ etσess(A). Also, for sufficiently large |τ| and each a ∈ R \ {0} we have a + iτ ∉ σ(A). We claim that Theorem 1 is implied by the following assertion: For each a ∈ R \ {0}, the function τ , ‖(a + iτ −A)−1‖ is bounded as |τ| → ∞. Indeed, let us suppose that for some λ ∈ C there exists etλ ∈ σ(etA) \ etσ(A). Let a = Reλ. Since {z ∈ C : |z| = 1} ⊂ etσ(A), we conclude that a ≠ 0. If μk := λ+2πik/t, then Spectral Mapping Theorems and Invariant Manifolds 227 etμk = etλ 6∈ etσ(A) and hence μk 6∈ σ(A) for all k ∈ Z. Since etλ ∈ σ(etA), Theorem 3 implies that the sequence {‖(μk −A)−1‖}k∈Z is unbounded as k→∞ or k→ −∞. But μk = a+ iτ for τ = Im (λ)+2πk/t, and we arrive at a contradiction with the above assertion. Therefore, in what follows we will fix a ∈ R \ {0}, let ξ = a + iτ for sufficiently large |τ| such that ξ 6∈ σ(A), and will show that the function τ , ‖(ξ −A)−1‖ is bounded as |τ| → ∞. We denote D = −∆ − β and recall that β < 0 by (H3). Moreover, we have σ(D) = σ(−∆)− β = [−β,∞). We note that D2 with domain H4(Rn) is a self-adjoint operator. Thus, for ξ = a+ iτ with τ ≠ 0 one has −ξ2 ∉ σ(D2). Moreover, we write ξ −A = ( ξ LR −LI ξ ) = ( ξ D −D ξ ) + ( 0 Q1 −Q2 0 ) (4) = ( ξ D −D ξ )[ I + ( ξ D −D ξ )−1( 0 Q1 −Q2 0 )] , where, by a direct computation with operator-valued matrices, (4a) ( ξ D −D ξ )−1 = ξ[ξ2 +D2]−1 −[ξ2 +D2]−1D [ξ2 +D2]−1D ξ[ξ2 +D2]−1  . Lemma 4. For ξ = a+ iτ, a ∈ R \ {0}, τ ∈ R, the norm of the operator (4a) remains bounded as |τ| → ∞. The elementary proof of this lemma is given in the next section. Next, we denote T(ξ) = ( ξ D −D ξ )−1 ( 0 Q1 −Q2 0 ) (5) =  [ξ2 +D2]−1DQ2 ξ[ξ2 +D2]−1Q1 −ξ[ξ2 +D2]−1Q2 [ξ2 +D2]−1DQ1  . The main step in the proof of Theorem 1 is contained in the next two lemmas. They imply that the norm of the operator (I +T(ξ))−1 is bounded as |τ| → ∞ (in the case n = 1 we give two proofs of this fact). Assume P and Q are real-valued continuous potentials exponentially decaying at infinity and let ξ = a+ iτ. 228 F. GESZTESY, C. K. R. T. JONES, Y. LATUSHKIN & M. STANISLAVOVA Lemma 5. If n = 1, then (a) ‖Q[ξ2 +D2]−1D‖ → 0, and (b) ‖ξ[ξ2 +D2]−1Q‖ → 0 as |τ| → ∞. Lemma 6. If n ≥ 1, then ‖P[ξ2 +D2]−1DQ‖ → 0, and ‖Pξ[ξ2 +D2]−1Q‖ → 0 (6) as |τ| → ∞. The proof of Lemmas 5 and 6 are given in the next section. We proceed finishing the proof of Theorem 1. Proof of Theorem 1. In the case n = 1, by passing to the adjoint operator of ξ[ξ2+D2]−1Q1 and [ξ2+D2]−1DQ2, Lemma 5 implies that the norm of each of the four block-operators in the right-hand side of (5) is strictly less than 1 for |τ| sufficiently large. Thus, ‖T(ξ)‖ < 1. By (4), one infers ‖(a+ iτ −A)−1‖ = ∥∥∥∥(I + T(ξ))−1 ( ξ D −D ξ )−1 ∥∥∥∥ ≤ 1 1− ‖T(ξ)‖ ∥∥∥∥ ( ξ D −D ξ )−1 ∥∥∥∥. Using Lemma 4, we have that ‖(a+iτ−A)−1‖ remains bounded as |τ| → ∞, and hence Theorem 3 implies the result. In the case n ≥ 1 we will use Lemma 6. For j = 1, 2 denote |Qj|(x) = |Qj(x)| and Q j (x) = |Qj|sgn (Q(x)), (7) so that Qj = Q j |Qj| for the potentials Qj, j = 1, 2, in (4). Also, for T(ξ) defined in (5) we write T(ξ) = A(ξ)B, where A(ξ) = ( ξ D −D ξ )−1 0 Q 1 −Q 2 0  , (8) B = |Q2|1/2 0 0 |Q1|  . Spectral Mapping Theorems and Invariant Manifolds 229 Recall the following elementary fact: If A and B are bounded operators, then I +AB is invertible provided I + BA is invertible; moreover, (I +AB)−1 = I −A(I + BA)−1B. By a direct calculation using (4a), one obtains BA(ξ) =  |Q2|[ξ +D2]−1DQ 2 |Q2|ξ[ξ +D2]−1Q 1 −|Q1|ξ[ξ +D2]−1Q 2 |Q1|[ξ +D2]DQ 1  . By Lemma 6, applied to each of the four blocks in the right-hand side of this identity, we obtain ‖BA(ξ)‖ → 0 as |τ| → ∞. Thus, for some τ0 > 0, one infers ‖BA(ξ)‖ < 1 and sup|τ|≥τ0 ‖(I + BA(ξ))−1‖ < ∞. Since supτ≥τ0 ‖A(ξ)‖ <∞ by Lemma 4, we conclude that sup |τ|≥τ0 ‖(I + T(ξ))−1‖ = sup |τ|≥τ0 ‖(I +A(ξ)B)−1‖ <∞. ❐ 3. PROOFS OF LEMMAS 4-6 In this section we give the proofs of Lemmas 4-6. The proof of Lemma 5 is based on a direct estimate of the trace norms. We will give two proofs of Lemma 6. The first proof is applicable to all n ≥ 1 and uses estimates for the norm of the resolvent on weighted spaces of L2-functions. The second proof works for the cases n = 1, 2, 3, and is based on explicit estimates for the integral kernel of the resolvent of the Laplacian. The main tool in the proof of Lemma 5 is the following well-known result. Denote by Jq(L(R)) the set of bounded linear operators A ∈ L(L2(Rn)), n ≥ 1, such that ‖A‖Jq(L2(Rn)) = (tr (|A|q))1/q < ∞, q ≥ 1, where tr (·) denotes the trace of operators in L2(Rn). We recall that ‖A‖ ≤ ‖A‖Jq(L2(Rn)) for all q ≥ 1. Theorem 7. (see, e.g., [24, Theorem XI.20].) Suppose 2 ≤ q < ∞ and let f , g ∈ Lq(Rn). Then f(·)g(−i∇) ∈ Jq(L(R)), and ‖f(·)g(−i∇)‖Jq(L2(Rn)) ≤ (2π)−n/2‖f‖Lq(Rn)‖g‖Lq(Rn). We will apply Theorem 7 to the exponentially decaying Q = f ∈ Lq(Rn), q > 1, and an appropriate choice of g. Throughout, c is a generic constant. 230 F. GESZTESY, C. K. R. T. JONES, Y. LATUSHKIN & M. STANISLAVOVA Proof of Lemma 5. We proceed with the proof of (a) in Lemma 5 for any n ≥ 1, to demonstrate where the argument in the proof breaks down for n > 1 (that is why Lemma 6 is needed). We recall that β < 0 by (H3). First, let g(x) = (|x|2 − β)(ξ2 + (|x|2 − β)2)−1. Then, for n ≥ 1, one has g(−i∇) = [ξ2+ (−∆−β)2]−1(−∆−β) = [ξ2+D2]−1D. For r = |x|, one infers