A Degenerate Edge Bifurcation in the 1D Linearized Nonlinear Schrodinger Equation

This work deals with the focusing Nonlinear Schrodinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.


Introduction
The focusing, pure-power, Nonlinear Schrödinger Equation for ψ(x, t) ∈ C, x ∈ R n , t ∈ R, famously exhibits solitary waves (sometimes called solitons), solutions which maintain a fixed spatial profile, and which are observed to play a key role in the dynamics of general solutions. One naturally asks about the stability of these waves, which leads immediately to an investigation of the spectrum of the linearized operator governing the dynamics close to the solitary wave solution. Systematic spectral analysis of the linearized operator has a long history (eg. [26,12], and for more recent studies [9,4,24,25]).
The principle motivation for the present work comes from [4] where resonance eigenvalues (with explicit resonance eigenfunctions) were observed to sit at the edges (or thresholds) of the spectrum for the 1D linearized NLS problem with focusing cubic nonlinearity. Numerically, it was observed that the same problem with power nonlinearity close to p = 3 (on both sides) has a true eigenvalue close to the threshold. In this paper we establish analytically the observed qualitative behaviour. Stated roughly, our main result is: for p ≈ 3, p = 3, the linearization of the 1D (NLS p ) about its soliton has purely imaginary eigenvalues, bifurcating from resonances at the edges of the essential spectrum of linearized (NLS 3 ), whose distance from the thresholds is of order (p − 3) 4 .
The exact statement is given as Theorem 4 in Section 4, and includes the precise leading order behaviour of the eigenvalues.
The eigenvalues obtained here, being on the imaginary axis, correspond to stable behaviour at the linear level. A further motivation for obtaining detailed information about the spectra of linearized operators is that such information is a key ingredient in studying the asymptotic stability of solitary waves: see [2,5,6,11,20,21,1,7] for some results of this type. Such results typically assume the absence of threshold eigenvalues or resonances. The presence of a resonance is an exceptional case which complicates the stability analysis by retarding the time-decay of perturbations. Nevertheless, the asymptotic stability of solitons in the 1D cubic focusing NLS was recently proved in [10]. The proof relies on integrable systems technology and so is only available for the cubic equation. The solitons are known to be stable in the (weaker) orbital sense for all p < 5 (the so-called mass subcritical range) while for p ≥ 5 they are unstable [13,27], but the question of asymptotic stability for p < 5 and p = 3 seems to be open. The existence (and location) of eigenvalues on the imaginary axis, which is shown here, should play a role in any attempt on this problem.
The generic bifurcation of resonances and eigenvalues from the edge of the essential spectrum was studied by [8] and [23] in three dimensions. Edge bifurcations have also been studied in one dimensional systems using the Evans function in [18] and [19]. We do not follow that route, but rather adopt the approach of [8,23] (going back also to [17], and in turn to the classical work [16]), using a Birman-Schwinger formulation, resolvent expansion, and Lyapunov-Schmidt reduction.
Our work is distinct from [8,23] due to the unique challenges of working in one dimension, in particular the strong singularity of the free resolvent at zero energy, which among other things necessitates a double Lyapunov-Schmidt reduction procedure.
Moreover, our work is distinct from all of [18,19,8,23] in that we study the particular (and as it turns out non-generic) resonance and perturbation corresponding to the near-cubic pure-power NLS problem. Generically, a resonance is associated with the birth or death of an eigenvalue, and such is the picture obtained in [8,23,18,19]: an eigenvalue approaches the essential spectrum, becomes a resonance on the threshold and then disappears. In our setting, the eigenvalue approaches the essential spectrum, sits on the threshold as a resonance, then returns as an eigenvalue. The bifurcation is degenerate in the sense that the expansion of the eigenvalue begins at higher order, and the analysis we develop to locate this eigenvalue is thus considerably more delicate.
The paper is organized as follows. The problem is set up in Section 2. In Section 3 we collect some results that are necessary for the bifurcation analysis. Section 4 is devoted to the statement and proof of the main result. The positivity of a certain (explicit) coefficient, which is crucial to the proof, is verified numerically; details of this computation are given in Section 5.

Mathematical Setup
We consider (NLS p ) in one space dimension: In one dimension the explicit solutions of (2.3) for each p ∈ (1, ∞) are classically known to be the unique H 1 solutions of (2.3) up to spatial translation and phase rotation (see e.g. [3]). In what follows we study the linearized NLS problem. That is, linearize (2.1) about the solitary wave solutions (2.2) by considering solutions of the form Then h solves, to leading order (i.e. neglecting terms nonlinear in h) We write the above as a matrix equation The above JĤ is the linearized operator as it appears in [4]. We now consider the system rotated i∂ t h = iJĤ h and find U unitary so that, UiJĤU * = σ 3 H, where σ 3 is one of the Pauli matrices and with H self-adjoint: In this way we are consistent with the formulation of [8,23]. We can also arrive at this system, i∂ t h = σ 3 H h, by letting h = hh T from the start.
Thus we are interested in the spectrum of and so in what follows we consider the eigenvalue problem That the essential spectrum of L p is and 0 is an eigenvalue of L p are standard facts [4]. When p = 3 we have the following resonance at the threshold z = 1 [4] in the sense that Our main interest is how this resonance bifurcates when p = 3 but |p − 3| is small. As is natural we seek an eigenvalue of (2.5) in the following form We note that the spectrum of L p for the soliton (2.4) may only be located on the Real or Imaginary axes [4], and so any eigenvalues in the neighbourhood of z = 1 must be real. There is also a resonance at z = −1 which we do not mention further; symmetry of the spectrum of L p ensures the two resonances bifurcate in the same way. We now recast the problem in accordance with the Birman-Schwinger formulation (pp. 85 of [14]), as in [8,23]. For (2.8), (2.5) becomes The constant-coefficient operator on the left is now invertible so we can write (2.10) We now seek solutions (α, w) of (2.9) which correspond to eigenvalues 1 − α 2 and eigenfunctions |V 0 | − 1 2 w of (2.5). The decay of the potential V (p) and hence |V 0 | 1 2 now allows us to work in the space L 2 = L 2 (R, C 2 ), whose standard inner product we denote by ·, · .
The resolvent R (α) has integral kernel These operators have the following integral kernels and for α > 0 the remainder term R R is continuous in α and uniformly bounded as an operator from a weighted L 2 space (with sufficiently strong polynomial weight) to its dual, and the entries of R R (x, y) grow at most quadratically in |x − y|. We also expand the potential V (p) in ε where ε := p − 3 and Here we have expanded and the computation gives By Taylor's theorem, the remainder term q R (x) satisfies an estimate of the form |q R (x)| ≤ C(1 + |x| 3 ) sech 2 (x/2) for some constant C which is uniform in x and ε ∈ (−1, 1). We will henceforth write Q for Q 3 and K α,ε for K α,p .

Some Preliminaries
We study (2.9), that is: Using the expansions (2.11) and (2.12) for R (α) and V (p) we make the following expansion where K R4 is uniformly bounded and continuous in α > 0 and ε in a neighbourhood of 0, as an operator on L 2 (R, C 2 ). Before stating the main theorem we assemble some necessary facts about the above operators. Lemma 1. Each operator appearing in the expansion (3.2) for K α,ε is a Hilbert-Schmidt (so in particular bounded and compact) operator from L 2 (R, C 2 ) to itself.
Proof. This is a straightforward consequence of the spatial decay of the weights which surround the resolvent. The facts that |V 0 | − 1 2 ≤ C/ sech(x), and that |V 0 | 1/2 decays like sech(x), while each of V 0 , V 1 , V 2 , decay like sech 2 (y), and V R decays at worst like sech 2 (3x/4) (say if we restrict to |ε| < 1 2 ) imply easily that these operators all have square integrable integral kernels.
We will also need the projections P and P which are defined as follows: as well as the complementary P := 1 − P . A direct computation shows that for any f ∈ L 2 we have Note that all operators in the expansion containing R −1 return outputs in the direction of v.
Lemma 2. The operator P (K 00 + 1)P has a one dimensional kernel spanned by as an operator from Ran(P ) to Ran(P ).
Proof. First note that by (2.7) from which it follows that Then a direct computation using (3.4), the expansion (3.2), the expression for R 0 , and integration by parts, shows that (K 00 + 1)w 0 = 2v and so indeed P (K 00 + 1)P w 0 = 0. Theorem 5.2 in [17] shows that the kernel of the analogous scalar operator can be at most one dimensional. We will use this argument, adapted to the vector structure, to show that any two non-zero elements of the kernel must be multiples of each other. Take w ∈ L 2 with w, v = 0 and P (K 00 + 1)w = 0. That is (K 00 + 1)w = cv for some constant c. This means We then obtain, after rearranging and expanding We now rearrange the first component. Expand and rewrite the first term as − x 2 x −∞ Q 2 (y)(2u 1 (y) + u 2 (y))dy + 1 2 x −∞ yQ 2 (y)(2u 1 (y) + u 2 (y))dy x yQ 2 (y)(2u 1 (y) + u 2 (y))dy where b := 1 2 R yQ 2 (y)(2u 1 (y) + u 2 (y))dy and where we used R 2Q 2 u 1 +Q 2 u 2 = 0 since w, v = 0. So putting everything back together we see We claim that as x → ∞ Here we have used the fact that w ∈ L 2 implies Q|2u 1 + u 2 | ∈ L 2 and that |y|Q ∈ L 2 . As well, in the second component x −∞ e √ 2y Q 2 (y)|u 1 (y) + 2u 2 (y)|dy 2y Q 2 (y)(u 1 (y) + 2u 2 (y))dy → 0 as x → ∞ which addresses the claim.
Next we claim that if c+b = 0 in (3.5) then u ≡ 0. To address the claim we first note that if c + b = 0 then u ≡ 0 for all x ≥ X for some X, by estimates similar to those just done. Finally, we appeal to ODE theory. Differentiating (3.5) in x twice returns the system Any solution u to the above with u ≡ 0 for all large enough x must be identically zero.
With the claim in hand we finish the argument. Given two non-zero elements of the kernel, say u andũ with limits as x → ∞ (written as above) c + b andc +b respectively, the combination satisfies (3.5) but with u * (x) → 0 as x → ∞, and so u * ≡ 0. Therefore, u and u are linearly dependent, as required.
Remark 3. Arguments similar to the estimates in the above Lemma 2 show that for α > 0 and w ∈ L 2 solving (2.9) the corresponding eigenfunction of (2.5) u = |V 0 | − 1 2 w is in L 2 and so the eigenvalue z = 1 − α 2 is in fact a true eigenvalue.
Note that K 00 , and hence P (K 00 + 1)P , is self-adjoint. Indeed, a direct computation shows that V 0 = −|V 0 | and so As we have seen above in Lemma 1, thanks to the decay of the potential, P K 00 P is a compact operator. Therefore, the simple eigenvalue −1 of P K 00 P is isolated and so exists and is bounded. Finally, with the above facts assembled, we are now in a position to state the main theorem.
Remark 5. This theorem confirms the behaviour observed numerically in [4]: for p = 3 but close to 3, the linearized operator JĤ (which is unitarily equivalent to iL p ) has true, purely imaginary eigenvalues in the gap between the branches of essential spectrum, which approach the thresholds as p → 3. Note Remark 3 to see that u = |V 0 | − 1 2 w is a true L 2 eigenfunction of (2.5). In addition, the eigenfunction approaches the resonance eigenfunction in some weighted L 2 space. Furthermore, we have found that α 2 , the distance of the eigenvalues from the thresholds, is to leading order proportional to (p − 3) 4 . Finally, note that α = ε 2 α 2 + O(ε 3 ) with α 2 > 0 gives α > 0 for both ε > 0 and ε < 0, ensuring the eigenvalues appear on both sides of p = 3.
Note that the functions on which P (K 00 + 1)P is being inverted in the expressions for P w 1 and P w 2 are orthogonal to both w 0 and v, and so these quantities are well-defined by (3.8). The identity has been verified analytically. It is because of this identity that the O(ε) term is absent in the expansion of α in (4.1). The fact that comes from our definition of α 2 . The above definitions, along with (3.3), imply the relationships below 0 = K −10 w 0 (4.2) 0 = K −11 w 0 + K −10 w 1 (4.3) 0 = K −10 w 2 + K −11 w 1 + K −12 w 0 + α 2 (K 00 + 1)w 0 (4.4) 0 = P (K 00 + 1)w 1 + P K 01 w 0 (4.5) 0 = P (K 00 + 1)w 2 + P K 01 w 1 + P K 02 w 0 + α 2 P K 10 w 0 (4.6) which we will use in what follows. Using the expression for α in (4.1), our expansion (3.2) for K α,ε now takes the form where K 1 is a bounded (uniformly in ε) operator depending on ε but notα, while K 2 is a bounded (uniformly in ε andα) operator depending on both ε andα.
Proof. We prove this by means of Banach Fixed Point Theorem. We must show that G(W ) maps the closed ball of radius R|ε| 3 into itself and that G(W ) is a contraction mapping. Taking W ∈ L 2 orthogonal to v and w 0 such that W L 2 ≤ R|ε| 3 and given M > 0 where |α| ≤ M|ε| 3 and |β| ≤ M|ε| 3 , we have, using the boundedness of −P (K 00 + 1)P −1 P 0 , G L 2 ≤ C|β| P K 00 v + P εK 1 +αK 2 v L 2 + C|α| P (K 10 w 0 + εf 2 +αh 1 ) L 2 for some appropriately chosen R with |ε| small enough. Here C is a positive, finite constant whose value changes at each appearance. Next consider with 0 < κ < 1 by taking |ε| sufficiently small. Hence G(W ) is a contraction, and we obtain the desired result.
Lemma 7 provides W as a function ofα and β, which we may then substi-tute into (4.8) to get which is the first of two equations relatingα and β.
Functions f 4 and f 5 depend on ε (and are uniformly L 2 -bounded) but notα, while the function h 2 depends on both ε andα (and is uniformly L 2 -bounded).
To make the relationship betweenα and β more explicit we take inner product with v (4.12) Now let ζ = α β and rewrite (4.10) and (4.12) in the following way which we recast as a fixed point problem Proof. The operators and function in question are compositions of continuous functions ofα.
Lemma 9. The W given by Lemma 7 is continuous in ζ for sufficiently small |ε|.
Proof. We prove this by means of the Brouwer Fixed Point Theorem. We show that F maps a closed square into itself and that F is a continuous function. Take |α|, |β| ≤ M|ε| 3 and and so by Lemma 7 we have W L 2 ≤ |ε| 3 R for some R > 0. Consider now
To reduce the double integral to a single integral we recall some facts about the integral kernels. Let Then h solves the equation Notice that −4Q 2 +3Q 4 = −2Q 2 u 1 −Q 2 u 2 where u 1 and u 2 are the components of the resonance u 0 (2.6). Observing the equation (3.6) we see that h = u 1 +c = 2 − Q 2 + c for some constant c. We can directly compute h(0) = −2 to find c = −2 and so h = −Q 2 . A similar argument involving (3.7) gives Computing inner products containing P w 1 is harder. We have an explicit expression for P w 1 but lack an explicit expression for P w 1 . Therefore we approximate P w 1 by numerically inverting P (K 00 + 1)P in P (K 00 + 1)P w 1 = − 1 4 P K 00 K −11 w 0 + P K 01 w 0 =: g.
Note that g, v = g, w 0 = 0. We represent P (K 00 + 1)P as a matrix with respect to a basis {φ j } N j=1 . The basis is formed by taking terms from the typical Fourier basis and projecting out the components of each function in the direction of v and w 0 . Some basis functions were removed to ensure linear independence of the basis. Let P w 1 = N j=1 a j φ j . Then where B j,k = φ j , (K 00 + 1)φ k and b j = φ j , g . So we can solve for a by inverting the matrix B. Once we have an approximation for P w 1 we can compute P (K 00 + 1)P w 1 directly to observe agreement with the function g. With an approximation for P w 1 in hand we can compute inner products containing P w 1 in the same way as the previous inner product containing w 0 . In this way we establish that α 2 > 0.