Klein–Gordon equation with mean field interaction. Orbital and asymptotic stability of solitary waves

We investigate orbital and asymptotic stability of solitary wave solutions to the U(1)-invariant nonlinear Klein–Gordon equation with mean field interaction.


Introduction
We study U(1)-invariant nonlinear Klein-Gordon equation on a line with mean field selfinteraction: ψ(x, t) = ∂ 2 x ψ(x, t) − m 2 ψ(x, t) + ρ(x)F( ψ(·, t), ρ(·) ), ψ(x, t) ∈ C, ρ(x) ∈ R, x, t ∈ R, (1.1) where m > 0, and ψ, ρ = ψ(x)ρ(x)dx. We write the equation as the dynamical system: where the charge Q(Ψ) = Im ψ(x)π(x) dx is conserved for solutions to (1.2). We express the condition in term of nonlinearity (condition (4.13) below). In particular, in the case when F(z) = |z| 2κ z and ρ is close to δ-function, the condition holds for any κ < 0 and ω ∈ (−m, m), if 0 < κ < 1 it holds only for m √ κ < |ω| < m. In the second part of the paper, we prove the scattering asymptotics of type (1.4) where W(t) is the dynamical group of the free Klein-Gordon equation, Ξ ± ∈ E are the corresponding asymptotic scattering states, and the remainder decays to zero as O(|t| −1/2 ) in global norm of E. The asymptotics holds for the solutions with initial states close to the stable part of the solitary manifold, extending the results of [2,3,5,15,17] to equation (1.2). For the proof we develop the approach [2,3] to equation (1.2). This approach is essentially based on properties of linearized dynamics at a solitary wave e −iωt φ ω . Let us note that the model (1.2) allows us to explicitly check the most of them. Namely, we prove weighted energy decay for the solution of the linearized equation. For linearization operator (operator A(ω) in (3.6)) we prove the absence of virtual levels at the embedding threshold ±i(m + |ω|) and give the criterion for the absence of virtual levels at the endpoints ±i(m − |ω|) of the essential spectrum (condition (A.25)). Moreover, we prove the limiting absorption principle for the corresponding resolvent. We show also that under condition (1.3), the linearized operator A(ω) has no real nonzero eigenvalues, and zero eigenvalue is of multiplicity 2 in the case when ∂ ω Q(φ ω ) = 0.
The only assumption we postulate is the absence of pure imaginary eigenvalues of the linearized operator. In appendix A we provide examples when this assumption holds. Namely, in the case when F(z) = |z| 2κ z, and ρ(x) is close to δ(x), the condition holds for any κ −1/2 and ω ∈ (−m, m); if κ > −1/2, κ = 0, it holds only for m (1+2κ) 2 3+4κ < |ω| < m. The well-posedness of the model (1.2), as well as the global attraction to the set of all solitary waves in local seminorms of H 1−ε (R) ⊕ H −ε (R) were proved in [8]. Let us emphasize that such global attraction is completely different from asymptotics (1.4) with fixed solitary wave and asymptotic scattering state which happens in the global energy norm.

Linearization at a solitary wave
Substituting into (2.15), we obtain: Further, equation (2.10) leads tȯ Then the first order part of (3.2) is given bẏ Remark 3.1. We note that the above definition of κ is compatible with the pure power case a(τ ) = τ κ , κ = 0, τ > 0, when a (z 2 )z 2 = κa(z 2 ). Denote In terms of operators (3.5), the system (3.4) reads as followṡ Theorem 2.1 generalizes to equation (3.6): for every initial function X(x, 0) = X 0 (x) ∈ E, the equation admits a unique solution X(x, t) ∈ C b (R, E). Note that A(ω) is factored into where Σ is defined in (2.17) and We note that there is a standard relation

Orbital stability
Here we will study the orbital stability of solitary waves following the approach of Grillakis-Shatah-Strauss [7]. First, we investigate the spectral properties of the operator H(ω) from (3.8).
(a) The essential spectrum of H(ω) is given by The operator H(ω) has no negative eigenvalues for κ 0, and it has exactly one simple negative eigenvalue for κ > 0.
We start with the spectrum of the operators L κ (ω).
Further, we denote and consider the operator which is similar to H(ω): Thus, the spectral problem for H(ω) is reduced to studying the spectrum of
) if and only if κ = 0, and σ p (H 0 (ω)) = {0} ∪ {ω 2 + 1}, with the corresponding eigenvectors Remark 4.4. The lemma was proved in [4] in the case ρ(x) = δ(x). However, the proof it still valid in our case. Denote Due to (4.5) and (4.9), To complete the proof of theorem 4.1, it remains to note that Proof. We will follow the Grillakis-Shatah-Strauss theory [7]. Let us first consider the case κ < 0. In this case, by theorem 4.1, the operator H(ω) has simple eigenvalue λ = 0 with the corresponding eigenvector JΦ ω and the rest of its spectrum is positive and bounded away from zero. Then all assumption of [7, theorem 1] are satisfied, and that theorem shows that in this case the bound state e Jωt Φ ω is orbitally stable. Obviously, the stability of Φ ω -orbit of (2.15) implies the stability of φ ω -orbit of (1.2).
To treat the case κ = 0, we consider the linearization of equation (1.2) at a solitary wave The linearized equation on ζ, is C-linear, with the linearization operator A(ω) given by Due to (4.6), the operatorH 0 (ω) has the same spectrum. Namely, by lemma 4.3,  .14), the condition ∂ ω (ω ϕ ω 2 ) < 0 can be written as

Spectral stability
Recall that the solitary wave e −iωt φ ω is called spectrally stable if the linearization operator A has purely imaginary spectrum.
Step 1. We note that if λ belongs to σ p (A(ω)), then so doλ, −λ, and −λ, since the spectrum of A(ω) is symmetric with respect to both R and iR: indeed, A(ω) has real coefficients, while, by (3.7), To achieve this, we consider the eigenvalue problem for , and σ 1 is the first Pauli matrix.
If κ 0, thenH(ω) is nonnegative and selfadjoint, hence one can extract the square root which is also nonnegative and selfadjoint; therefore, sinceH 1/2 (ω)ΣH 1/2 (ω) is antiselfadjoint, Eliminating v, we get For λ = 0, one can see that u is orthogonal to Coupling this relation with u and taking into account that u, Step 2. To find whether −λ 2 can be negative in the case κ > 0 and ω = 0 (thus corresponding to linear instability), one considers the minimization problem which implies that u satisfies with μ, ν ∈ R the Lagrange multipliers. We note that if μ 0, then ν = 0, or else one would have μ = λ − κ (the only negative eigenvalue of H κ (ω)), which is not possible since u − (x) from (4.8) corresponding to eigenvalue λ − κ is not orthogonal to ker(σ 1 H 0 σ 1 ): one has The sign of μ could be found from the condition that u is orthogonal to ker(σ 1 H 0 (ω)σ 1 ): We consider Since h(z) is monotonically increasing on ρ(H κ ), the sign of μ is opposite to the sign of h(0), which is given by which in turn follows from (2.10).
Step 3. It remains to consider the case κ > 0 and ω = 0. In this case (5.4) reads By lemma 4.2, one has: showing that there is linear instability Below, we will need the following spectral property.

Asymptotic stability
In the remaining part of the paper, we will prove asymptotic stability of solitary waves and scattering asymptotics (1.4). In the appendix A.1 we show that the continuous spectrum of A(ω) coincides with We assume that the following spectral conditions hold Proof. First, note that ∂ ω (ω ϕ ω 2 ) ω=0 = ϕ 0 2 = 0 by (2.12). Further, (5.9) and lemma 4.2 imply that σ p (A(0)) = {0} if and only in κ −1/2. It remains to consider the endpoints ±im. Due to (3.5) and (3.6), the equation Theorem 6.6 (asymptotic stability). Assume that the spectral conditions (assumption with ω 0 ∈ I and θ 0 ∈ R mod 2π: Then for ν sufficiently small the solution admits the following asymptotics: where Ξ ± ∈ E are the corresponding asymptotic scattering states and Above, W(t) the dynamical group of the free Klein-Gordon equation Here and below we denote by a± any number a ± with an arbitrary small, but fixed > 0.

Invariant subspace of the discrete spectrum
The real version of the set S reads The tangent space to S at the point e Jθ Φ ω with parameters ω, θ is the linear span of the derivatives with respect to θ and ω: Notice that the operator A(ω) corresponds to θ = 0 since we have extracted the phase factors e iθ from the solution in the process of linearization (3.1). The tangent space to S at the point Φ ω with parameters (ω, 0) is spanned by the vectors We introduce the symplectic form Ω for the real vectors X, Y ∈ L 2 (R) ⊗ R 4 by the integral where '·' stands for the scalar product on R 4 . By assumption 6.1(a), It follows that Ω is a nondegenerate symplectic form on the tangent space T ω,0 S. There is a symplectic projection P 0 (ω) : Due to remark 2.6 the symplectic projection P 0 (ω) is well define for Ψ ∈ E −5/2− . Corollary 7.1.
is also a symplectic projector.
Remark 7.2. By lemma 5.2, on the generalized null space of A(ω), one has A 2 (ω) = 0, and so the semigroup e A(ω)t reduces to I + A(ω)t.

Time decay in the continuous spectrum
Here and below we will write A instead of A(ω), P 0 instead of P 0 (ω), etc. Due to remark 7.2, the solutions X(t) = e At X 0 , where A = A(ω), of the linearized equation (3.6) do not decay as t → ∞ if P 0 X 0 = 0. On the other hand, we do expect time decay of P c X(t), as a consequence of the Laplace representation for P c e At : Here the integral is taken over the continuous spectrum C of the operator A defined in (6.1), and R(λ ± 0), λ ∈ C, are the right and the left limits as ε → 0 of the resolvent R(λ ± ε) = (A − λ ∓ ε) −1 . We prove the existence these limits in appendix A.2.
Proposition 8.1. Assume that the conditions (b) and (c) of assumption 6.1 hold. Then We prove the proposition in appendix A.4.

Modulation equations
In this section we present the modulation equations which allow us to construct solutions Ψ(x, t) of equation (2.15) such that, at each time t, it remains close to a soliton, with time varying ('modulating') parameters (ω, θ) = (ω(t), θ(t)). It will be assumed that Ψ(x, t) is a given weak solution of (2.15) as provided by theorem 2.1, so that the map t → Ψ(·, t), R + → E, is continuous.
We look for a solution to (2.15) in the form where Z is small and with γ treated perturbatively. The choice of parameters (ω(t), θ(t)) is determined by restriction Z(t) to lie in the image of the projection operator onto the continuous spectrum P c t = P c (ω(t)) or equivalently that with the projection operators defined in (7.4). Now we give a system of modulation equations for ω(t) and γ(t) which ensure that the conditions (9.3) are preserved by the time evolution.
It remains to show that for the initial data sufficiently close to a soliton there exist solutions to (9.6) and (9.7), at least locally. To achieve this, we observe that if the spectral conditions from assumption 6.1(a) hold, then the denominator appearing on the right-hand side of (9.6) and (9.7) does not vanish for small Z E −5/2− by (7.3). This has the consequence that the orthogonality conditions can be satisfied for small Z because they are equivalent to a locally well-posed set of ordinary differential equations for t → (θ(t), ω(t)). This implies the following corollary: (a) In the situation of lemma 9.1(a) assume that assumption 6.1 hold. If Z(t) E −5/2− is sufficiently small, the right hand sides of (9.6) and (9.7) are smooth in θ, ω and there exists C(ω, Z) > 0 which depends continuously on ω and Z such that (b) Assume given Ψ, a solution of (2.15) as in theorem 2.1. If ω 0 satisfies (7.3) and Z(x, 0) = e −Jθ 0 Ψ(x, 0) − Φ ω 0 (x) is small in E 5/2+ norm and satisfies (9.3) there is a time interval on which there exist C 1 functions t → (ω(t), γ(t)) which satisfy (9.6) and (9.7).

Time decay for the transversal dynamics
We deduce theorem 6.6 from the following time decay of the transversal component Z(t) in the nonlinear setting, Theorem 10.1. Let all the assumptions of theorem 6.6 hold. There are ν 0 > 0 and c > 0 such that for ν ∈ (0, ν 0 ) there exist C 1 -functions t → (ω(t), γ(t)) defined for t 0 such that the solution Ψ(x, t) of (2.15) can be written as in (9.1)-(9.3) with (9.6)-(9.7) satisfied, and there exists M > 0, depending only on the initial data, such that and moreover M cν.
Similarly to [2, lemma 10.1], we can assume that (9.3) holds initially without loss of generality. Then the local existence asserted in corollary 9.2 implies the existence of an interval [0, t 1 ] on which are defined C 1 functions t → (ω(t), γ(t)) satisfying (9.6) and (9.7) and such that M(t 1 ) = δ for some t 1 > 0 and δ > 0. By continuity we can make δ as small as we like by making ν and t 1 small. In section 11 below we will prove the following proposition if ν = Z(0) E 5/2+ < ν 1 and δ ∈ (0, δ 1 ).
Assuming the truth of proposition 10.2 for now theorem 10.1 will follow from the next argument. Consider the set T of t 1 0 such that (ω(t), γ(t)) are defined on [0, t 1 ] and M(t 1 ) δ. This set is relatively closed by continuity. On the other hand, (10.2) and corollary 9.2 with sufficiently small δ and ν imply that this set is also relatively open, and hence sup T = +∞, completing the proof of theorem 10.1.

Frozen linearized equation
Note that linear part of (9.4) is non-autonomous. Therefore (following [3]) we introduce a small modification of (9.1), which leads to an autonomous linearized equation. Namely, we represent the solution in the form where θ(t) is defined in (9.2), θ 1 (t) = ω 1 t + γ(0) with ω 1 = ω(t 1 ). Thus, The matrices A and e J(θ−θ 1 ) do not commute: (11.4) and κ(t) = κ(ω(t)) is defined in (3.3). Hence, (11.3) implieṡ To obtain an autonomous equation we rewrite the first two terms on the right-hand side by freezing the coefficients at t = t 1 . Note that Here ω 1 = ω(t 1 ), A 1 = A(ω 1 ), and . Now equation (11.5) can be written in the following frozen forṁ The following lemma show that the additional terms in f 1 can be estimated as small uniformly in t 1 .
From these it follows that there exists c > 0 such that X 0 1 (t) E σ cΔ X(t) E σ and hence (11.13) follows as claimed.

Estimation of M
Here we show that both terms in M (see (10.1)) are bounded by δ/4, uniformly in t 1 . As in corollary 9.2, we have Taking δ 1 < 1/(4c 0 ), we bound the second term in M by δ/4. By lemma 11.3, to estimate the first term in M it is enough to estimate X c 1 (t). Let us apply the projection P c 1 to both sides of (11.7). Then the equation for X c 1 (t) readṡ Now to estimate X c 1 , we use the Duhamel representation: (11.18) Now (11.16)-(11.18) and (11.13) yield We multiply the above by (1 + t) 3/2 to deduce Then (11.19) where both these integrals are bounded uniformly in t. Thus (11.19) implies that there exist c 2 , c 3 , independent of t 1 , such that Recall that m(t 1 ) δ δ 1 by assumption. Therefore this inequality implies that m(t) c 4 ν for t t 1 if ν and δ are sufficiently small. The constant c 4 does not depend on t 1 . We choose ν in (6.2) small enough that ν < δ/(4c 4 ). Therefore, if ν and δ are sufficiently small. Since Z = e −J(θ−θ 1 ) X, the last inequality bounds the first term in M as <δ/4 and hence M(t 1 ) < δ/2, completing the proof of proposition 10.2.

Soliton asymptotics
Here we prove our main theorem 6.6 using the bounds (10.1) from theorem 10.1. For a solution Ψ(x, t) to (1.2), we define the accompanying soliton as S( + γ(t). Then for the difference D(x, t) = Ψ(x, t) − S(x, t) we obtain from equations (1.2) and (2.10) (S 1 (x, t)) . Then where W(t) is the dynamical group of the free Klein-Gordon equation (6.5). Since as t → ∞, to establish the asymptotic behaviour (6.3) it suffices to prove that E Due to (10.1), Hence, the 'unitarity' in E of the group W(t) implies (12.2).

A.1. Free resolvent
For ω ∈ (−m, m), denote We have Hence, Then in the case 0 < |ω| < m, (A.1) implies and , and The well-known properties of the Schrödinger resolvent R 0 (see [1,14]) imply: (a) For any s ∈ R the resolvents are holomorphic operator-valued functions of λ ∈ C\C ± . (b) For λ ∈ C ± the convergence (the limiting absorption principle) holds Recall also (see, for example, [10, theorem 16.1]) that for R (k) 0 the following decay holds: for k = 0, 1, 2, s = 0, 1 and l = −1, 0, 1. The decay immediately implies that for k = 0, 1, 2, Similarly to the case of free Klein-Gordon (6.5), the dynamical group e A 0 t of equationẊ = A 0 X admits the following representation (cf [16]): Here U(t) is an integral operator with the integral kernel J 0 is the Bessel function of order 0, and θ is the Heaviside function. Moreover, for X ∈ E 1/2+ , the following integral representation holds where the integrals converge in the sense of distributions of t ∈ R with the values in E −1/2− . Let ζ ∈ C ∞ 0 be an even function, such that (A.13) Applying [13, lemma 2.3], we obtain the following high energy decay, (A.14)

A.3. Virtual levels
Here we study virtual levels (also known as a threshold resonance) of the operator A(ω). Recall that in the case ω = 0, there are virtual levels at ±im if an only if κ = −1/2 (see lemma 6.3). Now we consider the case ω = 0. We prove that there are no virtual levels of the operator A at the embedded thresholds ±i(m + |ω|) and give the condition on absence virtual levels at the edge points points ±i(m − |ω|). For concreteness, we consider 0 < ω < m and the points λ ± = i(m ± ω). Formulae (A.4) and (A.5) imply with appropriate operators B ± j . Following [14, formula (3.1)], we introduce generalized eigenspaces where Ran(B ± 0 ) is the range of B ± 0 . By [14, theorem 7.2], the condition M ± = 0 is equivalent to the absence of virtual levels of A at the points λ ± . Lemma A.3. M + = 0 for any 0 < ω < m and for any κ ∈ R, while M − = 0 only for ω = ω κ , where ω κ is the solution to In particular, M − = 0 for κ −1/2.
This completes the proof of proposition 8.1.
Note that we cannot take as an example a purely power function a(τ ) = τ κ , κ = 0, since this function does not satisfy the condition of theorem 2.1. So we will change it a little.