Global Attractor for 1D Dirac Field Coupled to Nonlinear Oscillator

Global attraction to solitary waves is proved for a model U(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {U}(1)$$\end{document}-invariant nonlinear 1D Dirac equation coupled to a nonlinear oscillator: each finite energy solution converges as t→±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \pm \infty $$\end{document} to a set of all “nonlinear eigenfunctions” of the form ψ1(x)e-iω1t+ψ2(x)e-iω2t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _1(x)e^{-i\omega _1 t}+\psi _2(x)e^{-i\omega _2 t}$$\end{document}. The global attraction is caused by nonlinear energy transfer from lower harmonics to continuous spectrum and subsequent dispersive radiation. We justify this mechanism by a strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap [-m,m]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-m,m]$$\end{document} and satisfies the original equation.Then the application of the Titchmarsh convolution theorem reduces the spectrum of the omega-limit trajectory to two harmonics ωj∈[-m,m]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _j\in [-m,m]$$\end{document}, j=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j =1,2$$\end{document}.


Introduction
The main goal of our paper is global attraction to solitary manifold for 1D Dirac equation with point coupling to an U(1)-invariant nonlinear oscillator. This goal is inspired by fundamental mathematical problem of an interaction between fields and point particles. Point interaction models were first considered since 1933 in the papers of Wigner, Bethe and Peierls, Fermi and others (see [2] for a detailed survey) and of Dirac [9]. Rigorous mathematical results were obtained since 1960 by Zeldovich, Berezin, Faddev, Cornish, Yafaev, Zeidler and others [3,7,11,33,35], and since 2000 by Noja, Posilicano, Yafaev and others [1,34].
In the case of the Maxwell field its coupling to a point particle is not well defined because the Hamilton functional is not bounded from below. This problem was resolved by Abraham by introduction of "extended electron" [31]. In the case of the Dirac equation the Hamilton functional also is not bounded from below even for the extended particle. So one need to find an appropriate type of point interaction to the 1D Dirac equation which guarantees a priori estimates sufficient for the global attraction. We have found a novel model of such coupling which provides the Hamilton structure and needed a priori estimates. Namely, we consider the following Dirac equation Here D m is the Dirac operator D m := α∂ x + mβ, where m > 0, and 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 vector function. The dots stand for the derivatives in t. All derivatives and Eq. (1.1) are understood in the sense of distributions. We assume that Eq. (1.1) is U(1)-invariant; that is, This condition leads to the existence of two-frequency solitary wave solutions of type We prove that indeed they form the global attractor for all finite energy solutions to (1.1). Namely, our main result is the following long-time asymptotics: In the case when polynomials F j are strictly nonlinear, any solution with initial data from H 1 (R) ⊗ C 2 converges to the set S of all solitary waves: ψ(·, t) −→ S , t → ±∞, (1.4) where the convergence holds in local H 1 -seminorms. The asymptotics of type (1.4) was discovered first for linear wave and Klein-Gordon equations with external potential in the scattering theory [13,14,27,32]. In this case, the attractor S consists of the zero solution only, and the asymptotics means well-known local energy decay.
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 [28,29], and then developed in [1,[4][5][6]8,21] and other papers.
The global attraction of type (1.4) to the solitary waves was established (i) in [17,20] for 1D Klein-Gordon equations coupled to nonlinear oscillators; (ii) in [18,19] for nD Klein-Gordon and Dirac equations with mean field interaction; (iii) in [25,26] for 3D wave and Klein-Gordon equations with concentrated nonlinearity. The global wellposedness and the global attraction (1.4) for the Dirac equationits with concentrated nonlinearity was not considered previously as well as the attraction to solitary waves with two frequencies.
In previous works [17,18,20,25,26] for the wave and Klein-Gordon fields the Hamilton functionals are bounded from below under appropriate assumptions. In the case of distributed interaction of the Dirac field with a nonlinear oscillator [19] the Hamilton functional is not bounded from below, and the global well-posedness is derived from the charge conservation.
In our case with the point interaction the charge conservation is not sufficient since ψ(0) is not well defined for ψ ∈ L 2 (R). That's why we suggest a novel model of 1D Dirac equation with a nonlinear point interaction (1.1) providing the Hamilton structure and strong a priori estimates.
Let us comment on our approach. 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) for each x ∈ R and of its support supp ψ(x, ·) which is the time-spectrum. The key role is played by the absolute continuity of the spectral densitiesψ(x, ·) 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, ·) = (γ 1 (x, ·), γ 2 (x, ·)) is contained in the spectral gap [−m, m] for each x ∈ R. Finally, we apply the Titchmarsh convolution theorem (see [12,Theorem 4.3.3]) to conclude that time-spectrum of each components γ j (x, ·) of omega-limit trajectory consists of two frequencies. 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 [17]. Nevertheless the formulation of the problem and the techniques used are not a straightforward generalization of the one-dimensional result [17].
In [17] the problem reduces to proving a global attraction for the solution ψ S (x, t) to the Klein-Gordon equation with the source F(ψ(0, t))δ(x) and with zero initial data. In this case the corresponding Fourier transform of ψ S (x, t) has a simple structure. Namely, , and similar representation holds for bounded and dispersion parts of ψ S (x, t). The key role in the proof plays the absolute continuity ofẑ(ω) on the continuous spectrum |ω| > m of the Klein-Gordon generator.
The plan of the paper is as follows. In Sect. 2 we state the main assumptions and results. In Sect. 3 we eliminate a dispersive component of the solution and construct spectral representation for the remaining part. In Sect. 4 we prove absolute continuity of high frequency spectrum of the remaining part. In Sect. 5 we exclude the second dispersive component corresponding to the high frequencies. In Sect. 6 we establish compactness for the remaining component with the bounded spectrum. In Sect. 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 Eq. (1.1) and prove global attraction (1.4) in the case on linear F(ψ).

Main Results
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 Eq. (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: (2.5) 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. (2.12) where C j are solutions to (2.14) Proof of Proposition 2.3. We look for solution ψ(x, t) to (1.1) in the form (2.11). Consider the function where Hence, Applying the operator D m , we obtain by (2.15) Therefore, in the case ω 1 = ω 2 , where C k j are solutions to We can also assume this formulas in the case ω 1 = ω 2 setting χ ω 2 = 0. We will return to Eq.
Proposition is completely proved.
The following lemma gives a sufficient condition for the existence of nonzero solitary waves.
We prove this lemma in "Appendix C". Remark 2.8. (i) Equation (2.13) has generally discrete set of solutions C j , while ω j belongs generally to an open set, (ii) 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. Theorem 2.9 (Main Theorem). Let the nonlinearity F(ψ) satisfy (2.9)-(2.10). Then for any ψ 0 ∈ H 1 the solution ψ(t) ∈ C(R, H 1 ) to the Cauchy problem (2.1) with (2.26)

Splitting of Solutions
It suffices to prove Theorem 2.9 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.
Complex Fourier-Laplace transform. Let us analyse the complex Fourier-Laplace trans- It is easy to see that This solution is given by which we extend to ω ∈C + by continuity. Thus, Note, that Therefore, and (3.7) becomes Here the last term vanishes for (3.10) Let us extend ψ S (x, t) and f (t) by zero for t < 0. Then 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).

Further Decomposition of Solutions
Denotef 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. function ψ b (x, t) 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 The lemma follows similarly Proposition 4.1 from [17], since the factors e −κ(ω)|x| ζ(ω), 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 2.9, it suffices to check that every omega-limit trajectory belongs to the set of solitary waves; that is,  For any omega-limit trajectory γ (x, t), the following spectral representation holds: Proof. Formula (6.9) and representation (6.11) imply that Further, the convergence (6.14) and the bound (6.13) with j = k = 0 imply that where p ∈ C b (R). The convergence is uniform on [−T, T ] for any T > 0. Hence, in the space of quasimeasures. Therefore, in the space of quasimeasures. Similarly, =q(x, ω) e −κ(ω)|x| −e −m|x| 2ω , s l → ∞.
The relation (6.16) implies the basic spectral identity:

Nonlinear Spectral Analysis
Here we will derive (6.15) from the following identity: which will be proved in three steps.
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). Recall that the function F j (t) := F j ( p j (t)) admits the representation [cf. (2.9)] where, according to (2.10), 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 [12,Theorem 4.3.3]) imply that Now (7.10) and (7.11) result in (7.12) 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 transform p j * p j = 2πC j δ(ω). Therefore, if p j is not identically zero, the Titchmarsh Theorem implies that 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 2.9 According to (7.1) and (6.23) Then the representation (6.16) implies 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 vector-function γ (x, t), defined in (7.13) such that the following convergence hold This implies that where The convergence (7.16) contradicts to (7.15). This completes the proof of Theorem 2.9. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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, where | ψ | 2 := ψ 2 L 2 + m 2 ψ 2 L 2 . Indeed, the Cauchy-Schwarz inequality and the Parseval identity imply
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 of zero order, and θ is the Heaviside function. According to (A.8) and (A.9), where 0, s)) ds, 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 Hence, Further, (A.13) 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 Moreover, t)). Hence,  , m). Taking into account (2.9)-(2.10), we rewrite this equation as Equation (C.1) has nonzero solutions C 1 = C 1 (ω 1 ) in the case when 2κ 1 (ω 1 ) < Mμ 1 (ω 1 ), i.e. Dividing by √ m + ω > 0, we arrive at the inequality Obviously, (C.2) holds for any M > 0 and ω 1 sufficiently close to m. It is not difficult to verify that this holds for any ω 1 ∈ (−m, m) in the case when M > (1 + √ 2)m. Indeed,