Global attractor for 1D Dirac field coupled to nonlinear oscillator

The long-time asymptotics is analyzed for all finite energy solutions to a model $\mathbf{U}(1)$-invariant nonlinear Dirac equation in one dimension, coupled to a nonlinear oscillator: {\it each finite energy solution} converges as $t\to\pm\infty$ to the set of all `nonlinear eigenfunctions' of the form $(\psi_1(x)e^{-i\omega_1 t},\psi_2(x)e^{-i\omega_2 t})$. The {\it global attraction} is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. We justify this mechanism by the strategy based on \emph{inflation of spectrum by the nonlinearity}. We show that any {\it omega-limit trajectory} has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. This equation implies the key {\it spectral inclusion} for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of $j$-th component of the omega-limit trajectory to a single harmonic $\omega_j\in[-m,m]$, $j=1,2$.


Introduction
We prove a global attraction for a model 1D nonlinear Dirac equation iψ(x,t) = D m ψ(x,t) − D −1 m δ (x)F(ψ(0,t)), x ∈ R. (1.1) Here D m is the Dirac operator D m := α∂ x + mβ , where m > 0, and ψ(x,t) = (ψ 1 (x,t), ψ 2 (x,t)) is a continuous C 2 -valued wave function, and F(ζ ) = (F 1 (ζ 1 ), F 2 (ζ 2 )), ζ = (ζ 1 , ζ 2 ) ∈ C 2 , is a nonlinear function. The dots stand for the derivatives in t. All derivatives and the equation are understood in the sense of distributions. Equation (1.1) describes the linear Dirac equation coupled to the nonlinear oscillator with the interaction concentrated in one point. We assume that equation (1.1) is U(1)-invariant; that is, This condition leads to the existence of two-frequency solitary wave solutions of type the scattering theory [26,22,9,8]. In this case, the attractor S consists of the zero solution only, and the asymptotics means well-known local energy decay. The attraction to the set of all static stationary states with ω = 0 was established in [10]- [19] for a number nonlinear wave problems.
First results on the attraction to the set of all stationary orbits for nonlinear U(1)-invariant Schrödinger equations were obtained in the context of asymptotic stability. This establishes asymptotics of type (1.4) but only for solutions with initial date close to some stationary orbit, proving the existence of a local attractor. This was first done in [3,23,24], and then developed in [2,4,5,6,16] and other papers.
The global attraction of type (1.4) to the solitary waves was established i) in [12,15] for 1D Klein-Gordon equations coupled to nonlinear oscillators; ii) in [13,14] for nD Klein-Gordon and Dirac equations with mean field interaction; iii) in [20,21] for 3D wave and Klein-Gordon equations with concentrated nonlinearity.
The global attraction (1.4) for Dirac equation with concentrated nonlinearity was not considered previously as well as the attraction to solitary waves with two frequencies.
Let us comment on our methods. First we prove the omega-limit compactness. This means that for each sequence s j → ∞ the solutions ψ(x,t + s j ) contain an infinite subsequence which converges in energy seminorms for |x| < R and |t| < T for any R, T > 0. Any limit function is called as the omega-limiting trajectory γ(x,t). To prove the global convergence (1.4) is suffices to show that any omega-limiting trajectory lies on S .
The proof relies on the study of the Fourier transform in timeψ(x, ω) and of its support which is the time-spectrum. The key role is played by the absolute continuity of the spectral densitiesψ(x, ω) outside the spectral gap [−m, m] for each x ∈ R. The absolute continuity is a nonlinear version of Kato's theorem on the absence of the embedded eigenvalues and provides the dispersion decay for the high energy component. Any omega-limit trajectory is the solution to (1.1).
This absolute continuity provides that the time-spectrum ofγ(x, ω) = (γ 1 (x, ω),γ 2 (x, ω)) is contained in the spectral gap [−m, m] for each x ∈ R. Finally, we apply the Titchmarsh convolution theorem (see [7,Theorem 4.3.3]) to conclude that each componentsγ j (x, ω) of omega-limit trajectory is a singleton, i.e. it's time-spectrum consists of a single frequency. The Titchmarsh theorem controls the inflation of spectrum by the nonlinearity. Physically, these arguments justify the following binary mechanism of the energy radiation, which is responsible for the attraction to the solitary waves: (i) nonlinear energy transfer from the lower to higher harmonics, and (ii) subsequent dispersion decay caused by the energy radiation to infinity.
The general scheme of the proof bring to mind the approach of [12,13]. Nevertheless, the Dirac equation with nonlinear point interaction requires new ideas due to a more singular character. In particular, the equation with nonlinearity F(ψ(0,t))δ (x) seems not well posed. Indeed, the solution of the free 1D Dirac equation with the source f (t)δ (x) does not belong to H 1 (R), so ψ(0,t) is not well defined.
We found the novel model of Dirac equation with non-standard Hamilton structure, which provides good a priori estimates needed for the global well-posedness. As a consequence, the formulation of the problem and the techniques used are not a straightforward generalization of the one-dimensional result [12].
The plan of the paper is as follows. In Section 2 we state the main assumptions and results. In Section 3 we eliminate a dispersive component of the solution and construct spectral representation for the remaining part. In Section 4 we prove absolute continuity of high frequency spectrum of the remaining part. In Section 5 we exclude the second dispersive component corresponding to the high frequencies. In Section 6 we establish compactness for the remaining component with the bounded spectrum. In Section 7 we state the spectral properties of all omega-limit trajectories and apply the Titchmarsh Convolution Theorem. In Appendices we establish the global well-posedness for equation (1.1) and prove global attraction (1.4) in the case on linear F(ψ).

Model
We consider the Cauchy problem for the Dirac equation coupled to a nonlinear oscillator: x ∈ R. (2.1) We will assume that the nonlinearity F = (F 1 , F 2 ) admits a real-valued potential: Then equation (2.1) formally can be written as a Hamiltonian system, where DH is the variational derivative of the Hamilton functional

Global well-posedness
To have a priori estimates available for the proof of the global well-posedness, we assume that We will write L 2 and H 1 instead of L 2 (R) ⊗ C 2 and instead of H 1 (R) ⊗ C 2 , respectively.
3. The energy is conserved: 4. The following a priori bound holds: We prove this theorem in Appendix A.

Solitary waves and the main theorem
We assume that the nonlinearity is polynomial. More precisely, This assumption guarantees the bound (2.4) and it is crucial in our argument: it allow to apply the Titchmarsh convolution theorem. Equality (2.8) implies that The solitary manifold is the set: Note that for any (ω 1 , ω 2 ) ∈ R 2 there is a zero solitary wave with φ ω 1 = φ ω 2 ≡ 0, since F(0) = 0. From (2.7) it follows that the set S is invariant under multiplication by e iθ , θ ∈ R.
Note that the cases C 21 = C 22 = 0 and C 11 = C 12 = 0 are exactly the case when ω 1 = ω 2 . Suppose now, that C 12 = C 21 = 0. Then (2.11) and (2.19) imply Taking into account (2.20), we obtain equations for C j : Equations (2.24) and (2.24) will coincide with equations (2.14) and (2.13) after the replacement C j by 2C j κ j . It is easy to check that in the case C 11 = C 22 = 0, we obtain the same formulas, interchanging ω 1 and ω 2 . Proposition is completely proved.
Remark 2.7. (i) Equation (2.13) has generally discrete set of solutions C j , while ω j belongs generally to an open set.
(i) In the linear case, when F j (ψ k ) = a j ψ j with a j ∈ R, the situation is contrary : we see from (2.13) that i.e., ω j generally belongs to a discrete set, while C j ∈ C is arbitrary.
Our main result is the following theorem.

Splitting of solutions
It suffices to prove Theorem 2.8 for t → +∞; We will only consider the solution ψ(x,t) restricted to t ≥ 0 and split it as Here φ (x,t) is a solution to the Cauchy problem for the free Dirac equation and ψ S (x,t) is a solution to the Cauchy problem for Dirac equation with the source The following lemma states well known local decay for the free Dirac equation.
Due to (3.3) it suffices to prove (2.26) for ψ S only.

Complex Fourier-Laplace transform
Let us analyse the complex Fourier-Laplace transform of ψ S (x,t): where It is easy to see thatψ where G(·, ω) ∈ H 1 is the unique elementary solution to

This solution is given by
, where k(ω) stands for the analytic function which we extend to ω ∈ C + by continuity. Thus, Note, that Therefore, , Here the last term vanishes for Let us extend ψ S (x,t) and f (t) by zero for t < 0. Then The distribution ψ S (·, ω) is the boundary value of the analytic H 1 -valued functionψ S (·, ω), in the following sense: where the convergence holds in the space of tempered distributions S ′ (R, H 1 ). Indeed, , where the convergence holds in S ′ (R, H 1 ) by (3.11). Therefore, (3.12) holds by the continuity of the Fourier transform F t→ω in S ′ (R). Similarly to (3.12), the distributionsf (ω) andŷ(ω) of ω ∈ R are boundary values of analytic in C + functionsf (ω) andỹ(ω), ω ∈ C + , respectively: since the function f (t) = F(ψ(0,t)) is bounded for t ≥ 0 and vanishes for t < 0. The convergences hold in the space of tempered distributions S ′ (R). Let us justify that the representation (3.10) forψ S (x, ω) is also valid when ω ∈ R.

Further decomposition of solutions
We will show that ψ d (x,t) is a dispersive component of the solution ψ(x,t), in the following sense.

Compactness
We are going to prove a compactness of the set of translations of the bound component, {ψ b,n (x, s + t): s ≥ 0}, n = 1, 2.
is smooth for x = 0 and t ∈ R. Moreover, for any fixed x = 0, t ∈ R, and any nonnegative integers j, k, the following representation holds (ii) There is a constant C j,k > 0 so that sup The lemma follows similarly Proposition 4.1 from [12], since the factors e −κ(ω)|x| ζ (ω), e −κ(ω)|x| −e −m|x| 2ω ζ (ω), and Corollary 6.2. By the Ascoli-Arzelà Theorem, for any sequence s l → ∞ there exists a subsequence (which we also denote by s l ) such that for any nonnegative integers j and k, (6.14) for some γ ∈ C b (R, H 1 ). The convergence in (6.14) is uniform in x and t as long as |x| + |t| ≤ R, for any R > 0.
We call omega-limit trajectory any function γ(x,t) that can appear as a limit in (6.14). Previous analysis demonstrates that the long-time asymptotics of the solution ψ(x,t) in H 1 loc depends only on the bound component ψ b (x,t). By Corollary 6.2, to conclude the proof of Theorem 2.8, it suffices to check that every omega-limit trajectory belongs to the set of solitary waves; that is, γ(x,t) = ψ Ω + (x,t) + iD −1

Spectral identity for omega-limit trajectories
Here we study the time spectrum of the omega-limit trajectories. Definition 6.3. Let µ be a tempered distribution. By Spec µ we denote the support of its Fourier transform: Proposition 6.4.

Nonlinear spectral analysis
Here we will derive (6.15) from the following identity: which will be proved in three steps.
Step 2 Proposition 7.2. For any omega-limit trajectory, the following identity holds: Proof. We are going to show that (7.6) follows from the key spectral relations (6.17), (7.2). Our main assumption (??) implies that the function F j (t) := F j (p j (t)) admits the representation (cf. (2.9)) F j (t) = a j (t)p j (t), j = 1, 2, (7.7) where, according to (??), 2n j u n, j |p j (t)| 2n−2 , N j ≥ 2; u N j , j > 0. (7.8) Both functions p j (t) and a j (t) are bounded continuous functions in R by Proposition 6.4 (ii). Hence, p j (t) and a j (t) are tempered distributions. Furthermore,p j andp j have the supports contained in [−m, m] by (6.17). Hence, a j also has a bounded support since it is a sum of convolutions of finitely manyp j andp j by (7.8). Then the relation (7.7) translates into a convolution in the Fourier space,F j =â j * p j /(2π), and the spectral inclusion (7.2) takes the following form: Let us denote F j = suppF j , A j = suppâ j , and P j = suppp j . Then the spectral inclusion (7.9) reads as On the other hand, it is well-known that suppâ j * p j ⊂ suppâ j + suppp j , or F j ⊂ A j + P j . Moreover, the Titchmarsh convolution theorem (see [7,Theorem 4.3.3]) imply that Now (7.10) and (7.11) result in so that inf A j ≥ 0 ≥ sup A j . Thus, we conclude that suppâ j = A j ⊂ {0}, therefore the distributionâ j (ω) is a finite linear combination of δ (ω) and its derivatives. Then a j (t) are polynomial in t; since a j (t) is bounded by Proposition 6.4 (ii), we conclude that a j (t) is constant. Now the relation (7.6) follows since a j (t) is a polynomial in |p j (t)|, and its degree is strictly positive by (7.8).
Step 3 Now the same Titchmarsh arguments imply that P j := Spec p j is a point ω + j ∈ [−m, m]. Indeed, (7.6) means that p j (t)p j (t) ≡ C j , hence in the Fourier transformp j * p j = 2πC j δ (ω). Therefore, if p j is not identically zero, the Titchmarsh Theorem implies that 0 = sup P j + sup(−P j ) = sup P j − inf P j .
Hence inf P j = sup P j and therefore P j = ω + k ∈ [−m, m], so thatp j (ω) is a finite linear combination of δ (ω − ω + j ) and its derivatives. As the matter of fact, the derivatives could not be present because of the boundedness of p j (t) = γ j (0,t) that follows from Proposition 6.4 (ii). Thus,p j ∼ δ (ω − ω + j ), which implies (7.1).

Conclusion of the proof of Theorem 2.8
According to (7.1) and (6.23) After simple evaluation, (7.13) becomes where we denote Therefore, γ(x,t) is a solitary wave (2.14). Due to Lemma 3.1 and Proposition 5.6 it remains to prove that Assume by contradiction that there exists a sequence s l → ∞ such that for some δ > 0. According to Corollary 6.2, there exist a subsequence s l n of the sequence s l , ω + 1 , ω + 2 ∈ R and vectorfunction γ(x,t), defined in (7.13) such that the following convergence hold The convergence (7.16) contradict to (7.15). This completes the proof of Theorem 2.8. ✷

A Global well-posedness
Here we prove Theorem 2.1. We first need to adjust the nonlinearity F so that it becomes bounded, together with its derivatives. Define where ψ 0 ∈ H 1 is the initial data from Theorem 2.1 and A, B are constants from (2.4). Then we may pick a modified potential function U ∈ C 2 (C 2 ), so that U(ζ ) satisfies (2.4) with the same constants A, B as U(ζ ) does: and so that | U(ζ )|, | U ′ (ζ )|, and | U ′′ (ζ )| are bounded for ζ ∈ C 2 . We define and consider the Cauchy problem of type (1.1) with the modified nonlinearity, This is a Hamiltonian system, with the Hamilton functional which is Fréchet differentiable in the space H 1 . By the Sobolev embedding theorem, ψ 2 where | ψ | 2 := ψ ′ 2 L 2 + m 2 ψ 2 L 2 . Indeed, the Cauchy-Schwarz inequality and the Parseval identity imply

Thus (A.3) leads to
Taking into account (A.5), we obtain the inequality Lemma A.1.

Local well-posedness
Denote by e −iD m t the dynamical group of the free Dirac equation. Then the solution to the Cauchy problem (A.4) can be represented by The next lemma establishes the contraction principle for the integral equation (A.8).
Lemma A.3. There exists a constant C > 0 so that for any two functions ψ k (·,t) ∈ C([−1, 1], H 1 ), k = 1, 2, one has: Proof. It suffices to consider t ≥ 0. In this case, where G (t) is the integral operator with the integral kernel Here J 0 is the Bessel function. According to (A.8) and (A.9), where First we prove the L 2 estimate for I j (x,t). By the Sobolev embedding theorem, where we took into account that |∇ F(z)| is bounded due to the choice of U. Similarly, Now, we derive the L 2 estimates for the derivatives ∂ x I 1 (x,t) and ∂ x I 2 (x,t). We have where Hence, Further, Hence, The L 2 norm of J 1 (x,t) is estimated similarly to the L 2 norm of ∂ x I 1 (x,t). Further, similarly to (A.10), we get Corollary A.4. Cancelling the nonzero factors √ m + ω 1 and √ m − ω 2 , we obtain
which are exactly equations (C.3) for soliton parameters ω j . Finally, we obtain that for 0 < a j < 2m where ω j are given in (C.4). Hence, the inclusion (C.8) follows. This finishes the proof of Theorem C.1.