1 Introduction

The main goal of our paper is global attraction to solitary manifold for 1D Dirac equation with point coupling to an \(\mathbf {U}(1)\)-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

$$\begin{aligned} i{\dot{\psi }}(x,t)=D_m\psi (x,t)-D_m^{-1}\delta (x)F(\psi (0,t)),\quad x\in {\mathbb R}. \end{aligned}$$
(1.1)

Here \(D_m\) is the Dirac operator \(D_m:=\alpha \partial _x+m\beta \), where \(m>0\), and

$$\begin{aligned} \alpha =\left( \begin{array}{c@{\quad }c} 0 &{} 1\\ -1&{} 0\end{array}\right) , \quad \beta =\left( \begin{array}{c@{\quad }c} 1 &{} 0 \\ 0&{} -1\end{array}\right) , \end{aligned}$$

\(\psi (x,t)=(\psi _1(x,t),\psi _2(x,t))\) is a continuous \({\mathbb C}^2\)-valued wave function, and \(F(\zeta )=(F_1(\zeta _1), F_2(\zeta _2))\), \(\zeta =(\zeta _1,\zeta _2)\in {\mathbb C}^2\), 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 \(\mathbf {U}(1)\)-invariant; that is,

$$\begin{aligned} F(e^{i\theta }\zeta )=e^{i\theta }F(\zeta ),\qquad \zeta \in {\mathbb C}^2,\quad \theta \in {\mathbb R}. \end{aligned}$$
(1.2)

This condition leads to the existence of two-frequency solitary wave solutions of type

$$\begin{aligned} \psi (x,t)=\phi _{\omega _1}(x)e^{-i\omega _1 t}+\phi _{\omega _2}(x)e^{-i\omega _2 t},\quad (\omega _1,\omega _2)\in {\mathbb R}^2. \end{aligned}$$
(1.3)

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 \(H^1({\mathbb R})\otimes {\mathbb C}^2\) converges to the set \(\mathscr {S}\) of all solitary waves:

$$\begin{aligned} \psi (\cdot ,t)\longrightarrow \mathscr {S}, \qquad t\rightarrow \pm \infty , \end{aligned}$$
(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 \(\mathscr {S}\) 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 \(\omega =0\) was established in [15,16,17,18,19,20,21,22,23,24] for a number nonlinear wave problems.

First results on the attraction to the set of all stationary orbits for nonlinear \(\mathbf {U}(1)\)-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 well-posedness 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 \(\psi (0)\) is not well defined for \(\psi \in L^2({\mathbb R})\). 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 \(s_j\rightarrow \infty \) the solutions \(\psi (x,t+s_j)\) 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\(\gamma (x,t)\). To prove the global convergence (1.4) is suffices to show that any omega-limiting trajectory lies on \(\mathscr {S}\).

The proof relies on the study of the Fourier transform in time \({\tilde{\psi }}(x,\omega )\) for each \(x\in {\mathbb R}\) and of its support \(\mathrm {supp}\,\psi (x,\cdot )\) which is the time-spectrum. The key role is played by the absolute continuity of the spectral densities \({\tilde{\psi }}(x,\cdot )\) outside the spectral gap \( [-m,m]\) for each \(x\in {\mathbb R}\). 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 of \({\tilde{\gamma }}(x,\cdot )=({\tilde{\gamma }}_1(x,\cdot ),{\tilde{\gamma }}_2(x,\cdot ))\) is contained in the spectral gap \([-m,m]\) for each \(x\in {\mathbb R}\). Finally, we apply the Titchmarsh convolution theorem (see [12, Theorem 4.3.3]) to conclude that time-spectrum of each components \(\gamma _j(x,\cdot )\) 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 \(\psi _S(x,t)\) to the Klein–Gordon equation with the source \(F(\psi (0,t))\delta (x)\) and with zero initial data. In this case the corresponding Fourier transform of \(\psi _S(x,t)\) has a simple structure. Namely,

$$\begin{aligned} {\hat{\psi }}_S(x,\omega )\!=\!{\hat{z}}(\omega )e^{ik(\omega )|x|},\quad \hat{z}(\omega )\!=\!-{\hat{f}}(\omega )/(2ik(\omega )), \quad k(\omega )\!=\!\sqrt{\omega ^2-m^2},\quad \omega \in R,\nonumber \\ \end{aligned}$$
(1.5)

where \(f(t)=F(\psi (0,t))\). Moreover, \(z(t)=\psi _S(0,t)\), and similar representation holds for bounded and dispersion parts of \(\psi _S(x,t)\). The key role in the proof plays the absolute continuity of \({\hat{z}}(\omega )\) on the continuous spectrum \(|\omega |>m\) of the Klein–Gordon generator.

In our case one need to prove a global attraction for the solution \(\psi _S(x,t)\) to the Dirac equation with the source \(D_m^{-1}F(\psi (0,t))\delta (x)\). The corresponding Fourier transform \({\hat{\psi }}_S(x,\omega )\) has more complicated structure than (1.5) [see formulas (4.3)–(4.4)]. Now \({\hat{z}}^{\pm }(\omega )e^{ik(\omega )|x|}\) with \(\hat{z}^{\,\pm }(\omega )=-\big (I+\frac{m}{\omega }\beta \pm \frac{ik(\omega )}{\omega }\alpha \big ) \frac{{\hat{f}}(\omega )}{2ik(\omega )}\) is only a part of \({\hat{\psi }}_S(x,\omega )\) for \(\pm x>0\). Moreover, \(z^{\pm }(t)\not \equiv \psi _S(0,t)\), and the representation (4.4) holds only for \(|\omega |>m\). To solve this difficulty, we derive a novel continuity properties for \({\hat{f}}(\omega )\) (see Lemma 4.3), Moreover, we use an alternative representation (3.14) for \(|\omega |<m\).

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 \(F(\psi )\).

2 Main Results

2.1 Model

We consider the Cauchy problem for the Dirac equation coupled to a nonlinear oscillator:

(2.1)

We will assume that the nonlinearity \(F=(F_1,F_2)\) admits a real-valued potential:

$$\begin{aligned} F_j(\zeta )=-\partial _{\overline{\zeta }\!_j} U(\zeta ),\quad \zeta _j\in {\mathbb C}, \quad j=1,2, \qquad U\in C^2({\mathbb C}^2). \end{aligned}$$
(2.2)

Then Eq. (2.1) formally can be written as a Hamiltonian system,

$$\begin{aligned} {\dot{\psi }}(t)=J\,D\mathscr {H}(\psi ),\qquad J=-iD_m^{-1}, \end{aligned}$$

where \(D\mathscr {H}\) is the variational derivative of the Hamilton functional

$$\begin{aligned} \mathscr {H}(\psi )=\frac{1}{2}\langle \psi , (-\partial _x^2+m^2)\psi \rangle +U(\psi (0)). \end{aligned}$$
(2.3)

2.2 Global well-posedness

To have a priori estimates available for the proof of the global well-posedness, we assume that

$$\begin{aligned} U(\zeta )\ge A-B|\zeta |^2, \qquad \zeta \in {\mathbb C}^2,\qquad A\in {\mathbb R},\qquad 0\le B<m. \end{aligned}$$
(2.4)

We will write \(L^2\) and \(H^1\) instead of \(L^2({\mathbb R})\otimes {\mathbb C}^2\) and instead of \(H^1({\mathbb R})\otimes {\mathbb C}^2\), respectively.

Theorem 2.1

Let conditions (2.2) and (2.4) hold. Then:

  1. 1.

    For every \(\psi _0\in H^1\) the Cauchy problem (2.1) has a unique solution \(\psi (t)\in C({\mathbb R},H^1)\cap C^1({\mathbb R},L^2)\).

  2. 2.

    The map \(W(t):\;\psi _0\mapsto \psi (t)\) is continuous in \(H^1\) for each \(t\in {\mathbb R}\).

  3. 3.

    The energy is conserved:

    $$\begin{aligned} \mathscr {H}(\psi (t))=\mathrm{const},\quad t\in {\mathbb R}. \end{aligned}$$
    (2.5)
  4. 4.

    The following a priori bound holds:

    $$\begin{aligned} \Vert \psi (t)\Vert _{H^1}\le C(\psi _0),\qquad t\in {\mathbb R}. \end{aligned}$$
    (2.6)

We prove this theorem in “Appendix A”.

2.3 Solitary waves and the main theorem

We assume that the nonlinearity is polynomial. More precisely,

$$\begin{aligned} U(\zeta )=U_1(\zeta _1)+U_2(\zeta _2),\qquad \zeta _j\in {\mathbb C}, \end{aligned}$$
(2.7)

where

$$\begin{aligned} U_j(\zeta _j)=\sum \limits _{n=0}^{N_j}u_{n,j}|\zeta _j|^{2n},\quad u_{n,j}\in {\mathbb R}, \quad u_{N_j,j}>0, \quad N_j\ge 2,\quad j=1,2. \end{aligned}$$
(2.8)

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

$$\begin{aligned}&F_j(\zeta _j)=-\partial _{\overline{\zeta }_j} U_j(\zeta _j)=a_j(|\zeta _j|^2)\zeta _j,\quad j=1,2, \end{aligned}$$
(2.9)
$$\begin{aligned}&a_j(|\zeta _j|^2):=-\sum \limits _{n=1}^{N_j}2nu_{n,j}|\zeta _j|^{2n-2}. \end{aligned}$$
(2.10)

Definition 2.2

(i) The solitary wave solution of Eq. (1.1) are solutions of the form

$$\begin{aligned} \psi (x,t)&=\phi _{\omega _1}(x)e^{-i\omega _1 t}+\phi _{\omega _2}(x)e^{-i\omega _2 t},\quad (\omega _1,\omega _2)\in {\mathbb R}^2,\quad \nonumber \\&\quad \phi _{\omega _k}\in H^1,~~k=1,2. \end{aligned}$$
(2.11)

(ii) The solitary manifold is the set: \(\mathcal{S}=\left\{ \phi _{\omega _1}+\phi _{\omega _2}\mathrm{:}\ ~~(\omega _1,\omega _2)\in {\mathbb R}^2\right\} \).

Note that for any \((\omega _1,\omega _2)\in {\mathbb R}^2\) there is a zero solitary wave with \(\phi _{\omega _1}=\phi _{\omega _2}\equiv 0\), since \(F(0)=0\). From (2.7) it follows that the set \(\mathcal{S}\) is invariant under multiplication by \(e^{i\theta }\), \(\theta \in {\mathbb R}\).

Denote \(\varkappa _j=\varkappa (\omega _j)=\sqrt{m^2-\omega _j^2}> 0\) for \(\omega _j\in (-m,m)\).

Proposition 2.3

(Existence of nonzero solitary waves). Assume that \(F(\zeta )\) satisfies (2.9). Then nonzero solitary waves may exist only for \(\omega _j\in (-m,m)\). The amplitudes of solitary waves are given by

$$\begin{aligned}&\phi _{\omega _1}(x)=C_1\left( \begin{array}{c@{\quad }c} e^{-\varkappa _1|x|}+\frac{me^{-\varkappa _1|x|}-\varkappa _1e^{-m|x|}}{\omega _1}\\ \varkappa _1\mathrm {sgn}x\frac{e^{-\varkappa _1|x|}-e^{-m|x|}}{\omega _1}\end{array}\!\!\right) ,\nonumber \\ {}&\phi _{\omega _2}(x)=C_2\left( \begin{array}{c@{\quad }c} -\varkappa _2\mathrm {sgn}x\frac{e^{-\varkappa _2|x|}-e^{-m|x|}}{\omega _2}\\ e^{-\varkappa _2|x|}-\frac{me^{-\varkappa _2|x|}-\varkappa _2e^{-m|x|}}{\omega _2} \end{array}\right) . \end{aligned}$$
(2.12)

where \(C_j\) are solutions to

$$\begin{aligned} 2C_j\varkappa _j=F_j\Big (C_j\Big [1+(-1)^{j+1}\frac{m-\varkappa (\omega _j)}{\omega _j}\Big ]\Big ),\quad j=1,2. \end{aligned}$$
(2.13)

Corollary 2.4

Substituting (2.12) into (2.11) we obtain the following representation for solitary wave solutions

$$\begin{aligned} \psi (x,t)=&{} C_1\left( \begin{array}{c@{\quad }c} e^{-\varkappa _1|x|}+\frac{me^{-\varkappa _1|x|}-\varkappa _1e^{-m|x|}}{\omega _1}\\ \varkappa _1\mathrm {sgn}x\frac{e^{-\varkappa _1|x|}-e^{-m|x|}}{\omega _1}\end{array}\right) e^{-i\omega _1 t}\nonumber \\ {}&\quad +C_2\left( \begin{array}{c@{\quad }c} -\varkappa _2\mathrm {sgn}x\frac{e^{-\varkappa _2|x|}-e^{-m|x|}}{\omega _2}\\ e^{-\varkappa _2|x|}-\frac{me^{-\varkappa _2|x|}-\varkappa _2e^{-m|x|}}{\omega _2} \end{array}\right) e^{-i\omega _2 t}. \end{aligned}$$
(2.14)

Proof of Proposition 2.3

We look for solution \(\psi (x,t)\) to (1.1) in the form (2.11). Consider the function

$$\begin{aligned} \chi (x,t):=\psi (x,t)-iD_m^{-1}{\dot{\psi }}(x,t)=\chi _{\omega _1}(x)e^{-i\omega _1 t}+\chi _{\omega _2}(x)e^{-i\omega _2 t}, \end{aligned}$$
(2.15)

where

$$\begin{aligned} \chi _{\omega _k}=\phi _{\omega _k}-\omega _k D_m^{-1}\phi _{\omega _k}=D_m^{-1}(D_m-\omega _k)\phi _{\omega _k},\qquad k=1,2. \end{aligned}$$

Hence,

$$\begin{aligned} \phi _{\omega _k}= & {} D_m (D_m-\omega _k)^{-1}\chi _{\omega _k}=D_m (D_m+\omega _k)(D_m^2-\omega _k^2)^{-1}\chi _{\omega _k}\nonumber \\= & {} \chi _{\omega _k}+(\omega _k^2+\omega _k D_m)(D_m^2-\omega _k^2)^{-1}\chi _{\omega _k}. \end{aligned}$$
(2.16)

Equation (1.1) implies, that

$$\begin{aligned} D_m \chi (x,t)=D_m\psi (x,t)-i{\dot{\psi }}(x,t)=D_m^{-1}F(\psi (0,t))\delta (x). \end{aligned}$$

Applying the operator \(D_m\), we obtain by (2.15)

$$\begin{aligned} e^{-i\omega _1 t}D_m^2 \chi _{\omega _1}(x)+e^{-i\omega _2 t}D_m^2 \chi _{\omega _2}(x)=F(\psi (0,t))\delta (x), \quad D_m^2=-\partial ^2_x+m^2. \end{aligned}$$

Therefore, in the case \(\omega _1\ne \omega _2\),

$$\begin{aligned} \chi _{\omega _k,j}(x)=C_{kj}\frac{e^{-m|x|}}{2m},\quad k,j=1,2, \end{aligned}$$
(2.17)

where \(C_{kj}\) are solutions to

$$\begin{aligned} e^{-i\omega _1 t}C_{1j}+e^{-i\omega _2 t}C_{2j} =F_j(\phi _{\omega _1,j}(0)e^{-i\omega _1 t}+\phi _{\omega _2,j}(0)e^{-i\omega _2 t}),\quad j=1,2.\nonumber \\ \end{aligned}$$
(2.18)

We can also assume this formulas in the case \(\omega _1=\omega _2\) setting \(\chi _{\omega _2}=0\). We will return to Eq. (2.18) later. First we derive the explicit formulas for \(\phi _{\omega _k}(x)\), using (2.16) and (2.17) only. Applying [10, Formula 1.2.(11)], we get

$$\begin{aligned} (D_m^2-\omega _k^2)^{-1}\frac{C_{kj}e^{-m|x|}}{2m}= & {} \frac{C_{kj}}{2\pi }\int _{{\mathbb R}}\frac{e^{-i\xi x} d\xi }{(\xi ^2+m^2)(\xi ^2+m^2-\omega _k^2)}\\= & {} \frac{C_{kj}}{\omega _k^2\pi }\int _0^{\infty }\Big (\frac{\cos \xi x}{\xi ^2+m^2-\omega _k^2}-\frac{\cos \xi x}{\xi ^2+m^2}\Big ) d\xi \\= & {} \frac{C_{kj}}{2\omega _k^2}\Big (\frac{e^{-\varkappa _k|x|}}{\varkappa _k}-\frac{e^{-m|x|}}{m}\Big ),\quad \varkappa _k=\varkappa (\omega _k). \end{aligned}$$

Substituting this into (2.16), we obtain

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} \phi _{\omega _k,1}(x)\\ \phi _{\omega _k,2}(x)\end{array}\right) =&\left( \begin{array}{c@{\quad }c} C_{k1}\\ C_{k2}\end{array}\right) \frac{e^{-\varkappa _k|x|}}{2\varkappa _k} +\left( \begin{array}{c@{\quad }c} C_{k1}\\ -C_{k2}\end{array}\right) \Big (\frac{me^{-\varkappa _k|x|}-\varkappa _ke^{-m|x|}}{2\omega _k\varkappa _k}\Big )\nonumber \\ {}&+\left( \begin{array}{c@{\quad }c} -C_{k2}\\ C_{k1}\end{array}\right) \mathrm {sgn}x\frac{e^{-\varkappa _k|x|}-e^{-m|x|}}{2\omega _k}. \end{aligned}$$
(2.19)

Hence,

$$\begin{aligned} \phi _{\omega _k,j}(0)=\frac{C_{kj}}{2\varkappa _k}\big (1+(-1)^{j+1}\frac{m-\varkappa _k}{\omega _k}\big ),\quad k,j=1,2. \end{aligned}$$
(2.20)

Now we turn to the study of the Eq. (2.18). First, consider the case when \(\omega _1=\omega _2=\omega \). We set \(C_{1j}=C_j\), \(C_{2j}=0\), \(j=1,2\), and Eq. (2.18) becomes

$$\begin{aligned} e^{-i\omega t}C_{j}=F_j(\phi _{\omega ,j}(0)e^{-i\omega t}) =a_j (|\phi _{\omega ,j}(0)e^{-i\omega t}|)\phi _{\omega ,j}(0)e^{-i\omega t},\quad j=1,2 \end{aligned}$$

by (2.9). Using (2.20), we get after cancelation of exponential

$$\begin{aligned} 2C'_{j}\varkappa =F_j(\phi _{\omega ,j}(0))=F_j(C'_{j}\big (1+(-1)^{j+1}\frac{m-\varkappa }{\omega }\big )),\quad \varkappa =\sqrt{m^2-\omega ^2},\quad j=1,2.\nonumber \\ \end{aligned}$$
(2.21)

Here we denote \(C_j'=C_j/(2\varkappa )\). Finally, in the case \(\omega _1=\omega _2=\omega \), Eq. (2.11) reads

$$\begin{aligned} \psi (x,t)=\phi _{\omega }(x,t)e^{-i\omega t}, \end{aligned}$$

where, in accordance with (2.19),

(2.22)

Now consider the case when \(\omega _1\not =\omega _2\). Taking into account (2.9), we rewrite (2.18) as

$$\begin{aligned} e^{-i\omega _1 t}C_{1j}+e^{-i\omega _2 t}C_{2j}&=a_j (|\phi _{\omega _1,j}(0)e^{-i\omega _1 t}+\phi _{\omega _2,j}(0)e^{-i\omega _2 t}|^2) (\phi _{\omega _1,j}(0)e^{-i\omega _1 t}\nonumber \\&\quad +\,\phi _{\omega _2,j}(0)e^{-i\omega _2 t}),\quad j=1,2. \end{aligned}$$
(2.23)

\(\square \)

Lemma 2.5

Let \(\omega _1\ne \omega _2\). Then for solutions to (2.18) we have either \(\phi _{\omega _1,j}(0)=0\) or \(\phi _{\omega _2,j}(0)=0\) for each \(j=1,2\).

Proof

It suffices to consider the case \(j=1\) and \(\omega _1<\omega _2\) only. Denote \(q_1:=\phi _{\omega _1,1}(0)\), \(q_2:=\phi _{\omega _2,1}(0)\). We should prove that either \(q_1=0\) or \(q_2=0\). Assume, to the contrary, that \(q_1\ne 0\) and \(q_2\ne 0\). Then

$$\begin{aligned}&|\phi _{\omega _1,1}(0)e^{-i\omega _1 t}+\phi _{\omega _2,1}(0)e^{-i\omega _2 t}|^2= |q_1|^2+|q_2|^2+{q}_1\overline{q}_2 e^{i\delta t}+\overline{q}_1 {q}_2e^{-i\delta t},\\&\quad \delta :=\omega _2-\omega _1>0, \end{aligned}$$

where \({q}_1\overline{q}_2\ne 0\) and \(\overline{q}_1 q_2\ne 0\). Hence, (2.10) implies

$$\begin{aligned} a_1 (|\phi _{\omega _1,1}(0)e^{-i\omega _1 t}+\phi _{\omega _2,1}(0)e^{-i\omega _2 t}|^2) =be^{i(N_j-1)\delta t}+\overline{b}e^{-i(N_j-1)\delta t}+\sum \limits _{|n|\le N_1-2}c_ne^{in\delta t}, \end{aligned}$$

where \(b\ne 0\) since \(a_1\) is a polynomial of degree \(N_1-1\ge 1\) due to (2.8) and (2.10). Now the right hand side of (2.18) contains the terms \(e^{-i[\omega _1t-(N_j-1)\delta ] t}\) and \(e^{-i[\omega _2t+(N_j-1)\delta ] t}\) with nonzero coefficients, which are absent on the left hand side. This contradiction proves the lemma.\(\quad \square \)

This lemma and (2.20) imply

Corollary 2.6

Let \(\omega _1\ne \omega _2\). Then for solutions to (2.18) we have either \(C_{1,j}=0\) or \(C_{2,j}=0\) for each \(j=1,2\).

Note that the cases \(C_{21}=C_{22}=0\) and \(C_{11}=C_{12}=0\) are exactly the case when \(\omega _1=\omega _2\).

Suppose now, that \(C_{12}=C_{21}=0\). Then (2.11) and (2.19) imply

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} \psi _{1}(x,t)\\ \psi _{2}(x,t)\end{array}\right)&=\frac{C_{1}}{2\varkappa _1}\left( \begin{array}{c@{\quad }c} e^{-\varkappa _1|x|}+\frac{me^{-\varkappa _1|x|}-\varkappa _1e^{-m|x|}}{\omega _1}\\ \varkappa _1\mathrm {sgn}x\frac{e^{-\varkappa _1|x|}-e^{-m|x|}}{\omega _1}\end{array}\right) e^{-i\omega _1 t}\nonumber \\ {}&+\frac{C_{2}}{2\varkappa _2}\left( \begin{array}{c@{\quad }c} -\varkappa _2\mathrm {sgn}x\frac{e^{-\varkappa _2|x|}-e^{-m|x|}}{\omega _2}\\ e^{-\varkappa _2|x|}-\frac{me^{-\varkappa _2|x|}-\varkappa _2e^{-m|x|}}{\omega _2} \end{array}\right) e^{-i\omega _2 t}, \end{aligned}$$
(2.24)

with \(C_1=C_{11}\), \(C_2=C_{22}\). Taking into account (2.20), we obtain equations for \(C_j\):

$$\begin{aligned} C_{j}=F_j(\frac{C_{j}}{2\varkappa _j}[1+(-1)^{j+1}\frac{m-\varkappa _j}{\omega _j}]),\quad j=1,2. \end{aligned}$$
(2.25)

Equations (2.24) and (2.25) will coincide with Eqs. (2.14) and (2.13) after the replacement \(C_{j}\) by \(2C_j\varkappa _j\).

It is easy to check that in the case \(C_{11}=C_{22}=0\), we obtain the same formulas, interchanging \(\omega _1\) and \(\omega _2\).

Proposition is completely proved. \(\quad \square \)

The following lemma gives a sufficient condition for the existence of nonzero solitary waves.

Lemma 2.7

Let F satisfies (2.9)–(2.10) with \(M_j=-u_{1,j}>0\), where \(j\in \{1;2\}\). Then there exists open subset \(I(M_j)\subset (-m,m)\) such that for any \(\omega _j\in I(M_j)\) the j-th equation of (2.13) has nonzero solutions \(C_j=C_j(\omega _j)\). Moreover, if \(M_j>(1+\sqrt{2})m\), then \(I(M_j)=(-m,m)\).

We prove this lemma in “Appendix C”.

Remark 2.8

(i):

Equation (2.13) has generally discrete set of solutions \(C_j\), while \(\omega _j\) belongs generally to an open set,

(ii):

In the linear case, when \(F_j(\psi _k)=a_j\psi _j\) with \(a_j\in {\mathbb R}\), the situation is contrary : we see from (2.13) that

$$\begin{aligned} 2\varkappa _j=a_j\Big (1+(-1)^{j+1}\frac{m-\varkappa _j}{\omega _j}\Big ),\quad \varkappa _j=\sqrt{m^2-\omega _j^2}, \end{aligned}$$

i.e., \(\omega _j\) generally belongs to a discrete set, while \(C_j\in {\mathbb C}\) is arbitrary.

Our main result is the following theorem.

Theorem 2.9

(Main Theorem). Let the nonlinearity \(F(\psi )\) satisfy (2.9)–(2.10). Then for any \(\psi _0\in H^1\) the solution \(\psi (t)\in C({\mathbb R}, H^1)\) to the Cauchy problem (2.1) with \(\psi (0)=\psi _0\) converges to \(\mathcal{S}\) in the space \(H^1_{loc}({\mathbb R})\otimes C^2\):

$$\begin{aligned} \psi (t)\rightarrow \mathcal{S},\quad t\rightarrow \pm \infty . \end{aligned}$$
(2.26)

3 Splitting of Solutions

It suffices to prove Theorem 2.9 for \(t\rightarrow +\infty \); We will only consider the solution \(\psi (x,t)\) restricted to \(t\ge 0\) and split it as

$$\begin{aligned} \psi (x,t)=\phi (x,t)+\psi _{S}(x,t),\quad t\ge 0. \end{aligned}$$

Here \(\phi (x,t)\) is a solution to the Cauchy problem for the free Dirac equation

$$\begin{aligned} i{\dot{\phi }}(x,t)=D_m\phi (x,t), \qquad \phi \vert _{_{t=0}}=\psi _{0}, \end{aligned}$$
(3.1)

and \(\psi _S(x,t)\) is a solution to the Cauchy problem for Dirac equation with the source

$$\begin{aligned} i{\dot{\psi }}_S(x,t)= D_m \psi _S(x,t) -D^{-1}_m F(\psi (0,t))\delta (x), \quad \psi _S(x,0) = 0. \end{aligned}$$
(3.2)

The following lemma states well known local decay for the free Dirac equation.

Lemma 3.1

(cf. [19, Proposition 4.3]). Let \(\psi _0\in H^1\). Then \(\phi \in C_b(\overline{{\mathbb R}^+}, H^1)\), and \(\forall R>0\),

$$\begin{aligned} \Vert \phi (t)\Vert _{H^1(-R,R)}\rightarrow 0,\qquad t\rightarrow \infty . \end{aligned}$$
(3.3)

Now (2.6) implies that

$$\begin{aligned} \psi _S=\psi (t)-\phi (t)\in C_b(\overline{{\mathbb R}^+}, H^1). \end{aligned}$$
(3.4)

Due to (3.3) it suffices to prove (2.26) for \(\psi _S\) only.

3.1 Complex Fourier-Laplace transform

Let us analyse the complex Fourier-Laplace transform of \(\psi _S(x,t)\):

$$\begin{aligned} \displaystyle {\tilde{\psi }}_S(x,\omega )=\int _0^\infty e^{i\omega t}\psi _S(x,t)\,dt,\quad \omega \in {\mathbb C}^{+}, \end{aligned}$$
(3.5)

where \({\mathbb C}^{+}:=\{z\in {\mathbb C}:\;\mathrm{Im\,}z>0\}\). Due to (3.4), \({\tilde{\psi }}_S(\cdot ,\omega )\) is an \(H^1\)-valued analytic function of \(\omega \in {\mathbb C}^{+}\).

Denote \(f(t)=F(\psi (0,t)\). Then Eq. (3.2) for \(\psi _S\) with zero initial data implies that

$$\begin{aligned} (D_m-\omega ){\tilde{\psi }}_S(x,\omega )-D_m^{-1}\delta (x){\tilde{f}}(\omega )=0,\quad \omega \in {\mathbb C}^{+}. \end{aligned}$$

It is easy to see that

$$\begin{aligned} {\tilde{\psi }}_S(x,\omega ):=(D_m+\omega )D_m^{-1}{\tilde{f}}(\omega )G(x,\omega ) =-{\tilde{f}}(\omega )G(x,\omega )-\omega D_m^{-1}{\tilde{f}}(\omega )G(x,\omega ), \end{aligned}$$

where \(G(\cdot ,\omega )\in H^1\) is the unique elementary solution to

$$\begin{aligned} G''(x,\omega )+(\omega ^2-m^2)G(x,\omega )=\delta (x),\qquad \omega \in {\mathbb C}^{+}. \end{aligned}$$

This solution is given by \(\displaystyle G(x,\omega )=\frac{e^{ik(\omega )|x|}}{2ik(\omega )}\), where \(k(\omega )\) stands for the analytic function

$$\begin{aligned} k(\omega ):=\sqrt{\omega ^2-m^2},\qquad \mathrm{Im\,}k(\omega )>0,\qquad \omega \in {\mathbb C}^{+}, \end{aligned}$$
(3.6)

which we extend to \(\omega \in \bar{{\mathbb C}^{+}}\) by continuity. Thus,

$$\begin{aligned} {\tilde{\psi }}_S(x,\omega )=-{\tilde{f}}(\omega )\frac{e^{i k(\omega )|x|}}{2ik(\omega )} -\omega D_m^{-1}\tilde{f}(\omega )\frac{e^{i k(\omega )|x|}}{2ik(\omega )},\qquad \omega \in {\mathbb C}^{+}. \end{aligned}$$
(3.7)

Note, that

$$\begin{aligned} D_m^{-2}\frac{e^{i k(\omega )|x|}}{2ik(\omega )}=\frac{1}{2\pi }\int \frac{e^{-ikx}dk}{(k^2+m^2)(-k^2-m^2+\omega ^2)} =\frac{1}{\omega ^2}\Big (\frac{e^{-m|x|}}{2m}+\frac{e^{ik(\omega )|x|}}{2ik(\omega )}\Big ). \end{aligned}$$

Therefore,

$$\begin{aligned} \omega D_m^{-1}{\tilde{f}}(\omega )\frac{e^{i k(\omega )|x|}}{2ik(\omega )} \!=\!m\beta \frac{\tilde{f}(\omega )}{\omega }\Big (\frac{e^{-m|x|}}{2m}\!+\!\frac{e^{ik(\omega )|x|}}{2ik(\omega )}\Big ) \!+\!\alpha \frac{{\tilde{f}}(\omega )}{2\omega }\mathrm{sgn}x(e^{ik(\omega )|x|}-e^{-m|x|}), \end{aligned}$$

and (3.7) becomes

$$\begin{aligned} {\tilde{\psi }}_S(x,\omega )&=-{\tilde{f}}(\omega )\frac{e^{i k(\omega )|x|}}{2ik(\omega )} -m\beta \frac{{\tilde{f}}(\omega )}{\omega }\Big (\frac{e^{-m|x|}}{2m}+\frac{e^{ik(\omega )|x|}}{2ik(\omega )}\Big )\nonumber \\ {}&\quad -\,\alpha \frac{{\tilde{f}}(\omega )}{2\omega }\mathrm {sgn}x(e^{ik(\omega )|x|}-e^{-m|x|}) =-\Big (I+\beta \,\frac{m+ik(\omega )}{\omega }\Big )\frac{\tilde{f}(\omega )}{2ik(\omega )}e^{i k(\omega )|x|} \nonumber \\ {}&\quad +\,(\beta -\alpha \, \mathrm {sgn}x)\tilde{f}(\omega )\frac{e^{ik(\omega )|x|}-e^{-m|x|}}{2\omega },\qquad \omega \in {\mathbb C}^{+}. \end{aligned}$$
(3.8)

Here the last term vanishes for \(x=0\). Denote \(y(t):=\psi _S(0,t)\in C_b({\mathbb R})\). Then (3.8) implies

$$\begin{aligned} {\tilde{y}}(\omega )={\tilde{\psi }}_S(0,\omega )=-\Big (I+\beta \,\frac{m+ik(\omega )}{\omega }\Big )\frac{\tilde{f}(\omega )}{2ik(\omega )},\quad \omega \in {\mathbb C}^{+}. \end{aligned}$$
(3.9)

Now (3.8) becomes

$$\begin{aligned} {\tilde{\psi }}_S(x,\omega )={\tilde{y}}(\omega )e^{i k(\omega )|x|} +(\beta -\alpha \, \mathrm {sgn}x)\tilde{f}(\omega )\frac{e^{ik(\omega )|x|}-e^{-m|x|}}{2\omega },\qquad \omega \in {\mathbb C}^{+}.\nonumber \\ \end{aligned}$$
(3.10)

Let us extend \(\psi _S(x,t)\) and f(t) by zero for \(t<0\). Then

$$\begin{aligned} \psi _S\in C_{b}({\mathbb R}, H^1) \end{aligned}$$
(3.11)

by (3.4). The Fourier transform \({\hat{\psi }}_S(\cdot ,\omega ):=\mathscr {F}_{t\rightarrow \omega }[\psi _S(\cdot ,t)]\) is a tempered \(H^1\)-valued distribution of \(\omega \in {\mathbb R}\). The distribution \({\hat{\psi }}_S(\cdot ,\omega )\) is the boundary value of the analytic \(H^1\)-valued function \({\tilde{\psi }}_S(\cdot ,\omega )\), in the following sense:

$$\begin{aligned} {\hat{\psi }}_S(\cdot ,\omega )=\lim \limits _{\varepsilon \rightarrow 0+}{\tilde{\psi }}_S(\cdot ,\omega +i\varepsilon ),\qquad \omega \in {\mathbb R}, \end{aligned}$$
(3.12)

where the convergence holds in the space of tempered distributions \(\mathscr {S}'({\mathbb R},H^1)\). Indeed,

$$\begin{aligned} {\tilde{\psi }}_S(\cdot ,\omega +i\varepsilon )=\mathscr {F}_{t\rightarrow \omega }[\psi _S(\cdot ,t)e^{-\varepsilon t}], \end{aligned}$$

and \(\psi _S(\cdot ,t)e^{-\varepsilon t}\mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0+}\psi _S(\cdot ,t)\), where the convergence holds in \(\mathscr {S}'({\mathbb R},H^1)\) by (3.11). Therefore, (3.12) holds by the continuity of the Fourier transform \(\mathscr {F}_{t\rightarrow \omega }\) in \(\mathscr {S}'({\mathbb R})\).

Similarly to (3.12), the distributions \({\hat{f}}(\omega )\) and \({\hat{y}}(\omega )\) of \(\omega \in {\mathbb R}\) are boundary values of analytic in \({\mathbb C}^{+}\) functions \({\tilde{f}}(\omega )\) and \({\tilde{y}}(\omega )\), \(\omega \in {\mathbb C}^{+}\), respectively:

(3.13)

since the function \(f(t)=F(\psi (0,t))\) is bounded for \(t\ge 0\) and vanishes for \(t<0\). The convergences hold in the space of tempered distributions \(\mathscr {S}'({\mathbb R})\). Let us justify that the representation (3.10) for \({\hat{\psi }}_S(x,\omega )\) is also valid when \(\omega \in {\mathbb R}\).

Lemma 3.2

For any fixed \(x\in {\mathbb R}\),

$$\begin{aligned} {\hat{\psi }}_S (x,\omega )={\hat{y}}(\omega )e^{i k(\omega )|x|}+(\beta -\alpha \, \mathrm {sgn}x){\hat{f}}(\omega )\frac{e^{ik(\omega )|x|}-e^{-m|x|}}{2\omega }, \quad \omega \in {\mathbb R}. \end{aligned}$$
(3.14)

Here the multiplications are understood in the sense of quasimeasures (see [17, Appendix B]).

The proof follows from (3.10) similarly to [17, Proposition 3.1]. Namely, the convergence (3.13) holds in the space of quasimeasures, while \(e^{ik(\omega )|x|}\) and \(\frac{e^{ik(\omega )|x|}-e^{-m|x|}}{2\omega }\) are multiplicators in the space of quasimeasures.

4 Absolutely Continuous Spectrum

Denote

$$\begin{aligned} \Omega _\delta :=(-\infty ,-m-\delta )\cup (m+\delta ,\infty ),\qquad \delta \ge 0. \end{aligned}$$
(4.1)

Consider the functions

$$\begin{aligned} \tilde{z}^{\,\pm }(\omega ):=-\big (I+\frac{m}{\omega }\beta \pm \frac{ik(\omega )}{\omega }\alpha \big ) \frac{{\tilde{f}}(\omega )}{2ik(\omega )},\quad \omega \in {\mathbb C}^+. \end{aligned}$$
(4.2)

From (3.13) it follows that for \(\omega \in \Omega _0\) there exist

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0+}\frac{\tilde{f}(\omega +i\varepsilon )}{\omega +i\varepsilon } =\frac{\hat{f}(\omega )}{\omega },\qquad \lim \limits _{\varepsilon \rightarrow 0+}\big (I+\beta \frac{m}{\omega +i\varepsilon }\big )\frac{\tilde{f}(\omega +i\varepsilon )}{2ik(\omega +i\varepsilon )} =\big (I+\beta \frac{m}{\omega }\big )\frac{{\hat{f}}(\omega )}{2ik(\omega )}. \end{aligned}$$

Hence, for \(\omega \in \Omega _0\) there exist boundary values \(\hat{z}^{\,\pm }(\omega )\) of \({\tilde{z}}^{\,\pm }(\omega )\) :

$$\begin{aligned} {\hat{z}}^{\,\pm }(\omega )=\lim _{\varepsilon \rightarrow 0+}\tilde{z}^{\,\pm }(\omega +i\varepsilon ) =-\big (I+\frac{m}{\omega }\beta \pm \frac{ik(\omega )}{\omega }\alpha \big ) \frac{{\hat{f}}(\omega )}{2ik(\omega )},\quad \omega \in \Omega _0. \end{aligned}$$
(4.3)

Now we rewrite (3.10) as

$$\begin{aligned} {\hat{\psi }}_S (x,\omega )=e^{i k(\omega )|x|}{\hat{z}}^{\,\pm }(\omega ) +e^{-m|x|}\big (\pm \alpha -\beta )\frac{{\hat{f}}(\omega )}{2\omega },\quad \pm x>0,\quad \omega \in \Omega _0. \end{aligned}$$
(4.4)

We study the regularity of \({\hat{z}}^{\,\pm }(\omega )\). Note that the function \(\omega k(\omega )\) is positive for \(\omega \in \Omega _{0}\).

Proposition 4.1

The distributions \({\hat{z}}^{\,\pm }(\omega )\) are absolutely continuous for \(\omega \in \Omega _ 0\), i.e. and \({\hat{z}}^{\,\pm }\in L^1_{loc}(\Omega _0)\). Moreover,

$$\begin{aligned} \int _{\Omega _0}\big (|{\hat{z}}^{\,+}(\omega )|^2+|\hat{z}^{\,-}(\omega )|^2\big ) \omega k(\omega )\,d\omega <\infty . \end{aligned}$$
(4.5)

Proof

We use the arguments of Paley-Wiener type. Namely, the Parseval identity and (3.4) imply that

$$\begin{aligned} \int \limits _{\mathbb R}\Vert {\tilde{\psi }}_S(\cdot ,\omega +i\varepsilon )\Vert _{H^1}^2\,d\omega =2\pi \int \limits _0^\infty e^{-2\varepsilon t} \Vert \psi _S(\cdot ,t)\Vert _{H^1}^2\,dt \le \frac{C}{\varepsilon },\quad \varepsilon >0. \end{aligned}$$
(4.6)

Then (4.6) gives

$$\begin{aligned} \int \limits _{{\mathbb R}}\Vert {\tilde{\psi }}_S(\cdot ,\omega +i\varepsilon )\Vert _{H^1}^2\,d\omega \le \frac{C}{\varepsilon },\quad \varepsilon >0. \end{aligned}$$
(4.7)

Evidently,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0+}\varepsilon \Vert e^{-m|x|}\Vert _{H^1}\rightarrow 0. \end{aligned}$$

Hence, (4.4) and (4.7) results in

$$\begin{aligned}&\varepsilon \int _{\Omega _0}\Big (|\tilde{z}^{\,+}(\omega +i\varepsilon )|^2 \Vert e^{ik(\omega +i\varepsilon )|x|}\Vert _{H^1(0,\infty )}^2 \nonumber \\&\quad ~~\qquad +|\tilde{z}^{\,-}(\omega +i\varepsilon )|^2 \Vert e^{ik(\omega +i\varepsilon )|x|}\Vert _{H^1(-\infty ,0)}^2\Big )\,d\omega \le C,\qquad \varepsilon >0. \end{aligned}$$
(4.8)

Here is a crucial observation about the norm of \(e^{ik(\omega +i\varepsilon )|x|}\).

Lemma 4.2

(cf. [17, Lemma 3.2]).

  1. 1.

    For \(\omega \in {\mathbb R}\),

    $$\begin{aligned} \lim _{\varepsilon \rightarrow 0+}\varepsilon \Vert e^{ik(\omega +i\varepsilon )|x|}\Vert _{H^1}^2=n(\omega ):= \left\{ \begin{array}{ll}\omega k(\omega ), &{}\quad |\omega |>m \\ 0, &{}\quad |\omega |<m\end{array} \right. \!\!, \end{aligned}$$
    (4.9)

    where the norm in \(H^1\) is chosen to be \(\Vert \psi \Vert _{H^1}=\left( \Vert \psi '|\Vert _{L^2}^2+m^2\Vert \psi \Vert _{L^2}^2\right) ^{1/2}.\)

  2. 2.

    For any \(\delta >0\) there exists \(\varepsilon _\delta >0\) such that

    $$\begin{aligned} \varepsilon \Vert e^{ik(\omega +i\varepsilon )|x|}\Vert _{H^1}^2\ge n(\omega )/2,\quad \omega \in {\Omega _\delta },\quad \varepsilon \in (0,\varepsilon _\delta ). \end{aligned}$$
    (4.10)

Substituting (4.10) into (4.8), we get:

$$\begin{aligned} \int _{\Omega _\delta }(|{\tilde{z}}^{\,+}(\omega +i\varepsilon )|^2+|\tilde{z}^{\,-}(\omega +i\varepsilon )|^2)\omega k(\omega )\,d\omega \le 2C, \qquad 0<\varepsilon <\varepsilon _\delta , \end{aligned}$$
(4.11)

with the same constant C as in (4.8), and the region \(\Omega _\delta \) defined in (4.1). We conclude that for each \(\delta >0\) the set of functions

$$\begin{aligned} g^{\pm }_{\delta ,\varepsilon }(\omega )=\tilde{z}^{\,\pm }(\omega +i\varepsilon )|\omega k(\omega )|^{1/2},\qquad \varepsilon \in (0,\varepsilon _\delta ), \end{aligned}$$

defined for \(\omega \in \Omega _\delta \), is bounded in the Hilbert space \(L^2(\Omega _\delta )\), and, by the Banach Theorem, is weakly compact. Hence, the convergence of the distributions (3.13) implies the following weak convergence in the Hilbert space \(L^2(\Omega _\delta )\):

$$\begin{aligned} g^{\pm }_{\delta ,\varepsilon }\rightharpoondown g^{\pm }_\delta ,\qquad \varepsilon \rightarrow 0+, \end{aligned}$$
(4.12)

where the limit function \(g^{\pm }_\delta (\omega )\) coincides with the distribution \({\hat{z}}^{\,\pm }(\omega )|\omega k(\omega )|^{1/2}\) restricted onto \({\Omega _\delta }\). It remains to note that the norms of all functions \(g^{\pm }_\delta \), \(\delta >0\), are bounded in \(L^2(\Omega _\delta )\) by (4.11), hence (4.5) follows. Finally, \({\hat{z}}^{\,\pm }(\omega )\in L^1_{loc}(\Omega _0)\) by (4.5) and the Cauchy-Schwarz inequality. \(\quad \square \)

Lemma 4.3

$$\begin{aligned} \int _{\Omega _0}|{\hat{f}}(\omega )|^2 \frac{k(\omega )}{\omega }\,d\omega <\infty . \end{aligned}$$
(4.13)

Proof

Denote the \(2\times 2\) matrix \(A^{\pm }(\omega )=I+\frac{m}{\omega }\beta \pm \frac{ik(\omega )}{\omega }\alpha \). Then \({\hat{z}}^{\,\pm }(\omega )=-A^{\pm }(\omega )\frac{\hat{f}(\omega )}{2ik(\omega )}\). For any \(\omega \in \Omega _0\), the matrix \(A^{\pm }(\omega )\) has two eigenvalues: \(\lambda =0\) and \(\lambda =2\) since

$$\begin{aligned} \mathrm{det} (A^{\pm }-\lambda I)=\left| \begin{array}{c@{\quad }c} (1+\frac{m}{\omega })-\lambda &{} \pm \frac{ik(\omega )}{\omega }\\ {\mp }\frac{ik(\omega )}{\omega } &{} (1-\frac{m}{\omega })-\lambda \end{array}\right| =-2\lambda +\lambda ^2=\lambda (\lambda -2). \end{aligned}$$

The unit eigenvectors \(\nu ^{\pm }(\omega )\) of operators \(A^{\pm }(\omega )\) corresponding to the eigenvalue \(\lambda =2\) read

$$\begin{aligned} \nu ^{\pm }(\omega )=\big (\sqrt{\frac{\omega +m}{2\omega }},~{\mp } i\sqrt{\frac{\omega -m}{2\omega }}\big ). \end{aligned}$$
(4.14)

Denote \(g(\omega ):=\frac{{\hat{f}}(\omega )}{2ik(\omega )}\), and \(a^{\pm }(\omega ):=\langle g(\omega ),\nu ^{\pm }(\omega )\rangle \). Then \(\hat{z}^{\,\pm }(\omega )=-A^{\pm }(\omega )g(\omega )=-2a^{\pm }(\omega )\nu ^{\pm }(\omega )\), and hence

$$\begin{aligned} a^{\pm }(\omega )=\langle g(\omega ),\nu ^{\pm }(\omega )\rangle =-\frac{1}{2}\langle {\hat{z}}^{\,\pm }(\omega ), \nu ^{\pm }(\omega )\rangle . \end{aligned}$$

Taking into account (4.14), we get the system of equations for \(g(\omega )=(g_1(\omega ),g_2(\omega ))\)

(4.15)

Therefore

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} {\hat{f}}_1(\omega )\\ \hat{f}_2(\omega )\end{array}\!\!\!\right) \!=\!2ik(\omega )\left( \begin{array}{c@{\quad }c} g_1(\omega )\\ g_2(\omega )\end{array}\!\!\!\right) \!=\!-|\omega | \left( \begin{array}{c@{\quad }c} i\sqrt{\frac{\omega -m}{2\omega }} &{} i\sqrt{\frac{\omega -m}{2\omega }} \\ \sqrt{\frac{\omega +m}{2\omega }}&{} \!-\!\sqrt{\frac{\omega +m}{2\omega }}\end{array}\right) \left( \begin{array}{c@{\quad }c} \langle {\hat{z}}^{\,+}(\omega ), \nu ^{+}(\omega )\rangle \\ \langle {\hat{z}}^{\,-}(\omega ), \nu ^{-}(\omega )\rangle \end{array}\!\!\!\right) . \end{aligned}$$

Hence,

$$\begin{aligned} |{\hat{f}}(\omega )|^2\le C|\omega |^2(|{\hat{z}}^{\,+}(\omega )|^2+|\hat{z}^{\,-}(\omega )|^2). \end{aligned}$$

Now (4.13) follows from (4.5). \(\quad \square \)

5 Further Decomposition of Solutions

Denote

$$\begin{aligned} {\hat{f}}_{d}(\omega ):=\left\{ \begin{array}{c@{\quad }c} {\hat{f}}(\omega ), &{}\quad \omega \in \Omega _0\\ 0, &{} \quad \omega \in {\mathbb R}\setminus \Omega _0 \end{array}\right. \end{aligned}$$
(5.1)

and set

$$\begin{aligned} {\hat{\psi }}_{d}(x,\omega )=e^{i k(\omega )|x|}\hat{z}_d(x,\omega )+e^{-m|x|}\big (\alpha \, \mathrm{sgn}x-\beta \big )\frac{\hat{f}_d(\omega )}{2\omega },\quad \omega \in {\mathbb R}, \end{aligned}$$
(5.2)

where

$$\begin{aligned} \hat{z}_d(x,\omega )=&{} -\big (I+\frac{m}{\omega }\beta +\frac{ik(\omega )}{\omega }\alpha \, \mathrm {sgn}x\big ) \frac{{\hat{f}}_d(\omega )}{2ik(\omega )}\nonumber \\=&{} -\,\big (I+\frac{m}{\omega }\beta \pm \frac{ik(\omega )}{\omega }\alpha \big ) \frac{{\hat{f}}_d(\omega )}{2ik(\omega )} ={\hat{z}}_d^{\,\pm }(\omega ),\quad \pm x>0. \end{aligned}$$
(5.3)

Consider

$$\begin{aligned} \psi _{d}(x,t):=\frac{1}{2\pi }\int _{{\mathbb R}}{\hat{\psi }}_{d}(x,\omega )e^{-i\omega t}\,d\omega ,\qquad x\in {\mathbb R},\quad t\in {\mathbb R}. \end{aligned}$$
(5.4)

We will show that \(\psi _d(x,t)\) is a dispersive component of the solution \(\psi (x,t)\), in the following sense.

Proposition 5.1

(i):

\(\psi _{d}(\cdot ,t)\) is a bounded continuous \(H^1\)-valued function:

$$\begin{aligned} \psi _{d}(\cdot ,t)\in C_{b}({\mathbb R},H^1). \end{aligned}$$
(5.5)
(ii):

The local energy decay holds for \(\psi _{d}(\cdot ,t)\): for any \(R>0\),

$$\begin{aligned} \Vert \psi _{d}(\cdot ,t)\Vert _{H^1(-R,R)}\rightarrow 0, \qquad t\rightarrow \infty . \end{aligned}$$
(5.6)

Proof

We split \(\psi _{d}(x,t)\) as \(\psi _{d}(x,t)=\varphi _{d}(x,t)+\chi _{d}(x,t)\), where

$$\begin{aligned}&\varphi _{d}(x,t)=\frac{1}{2\pi }\int _{{\mathbb R}}e^{-i\omega t}e^{i k(\omega )|x|}{\hat{z}}_d(x,\omega )\,d\omega , \nonumber \\ {}&\quad \chi _{d}(x,t)=\frac{1}{2\pi }e^{-m|x|}\big (\alpha \, \mathrm {sgn}x-\beta \big )\int _{{\mathbb R}}e^{-i\omega t}\frac{{\hat{f}}_d(\omega )}{2\omega }\,d\omega . \end{aligned}$$
(5.7)

First, consider \(\chi _{d}(x,t)\). Note that

$$\begin{aligned}&\int \limits _{{\mathbb R}}\big |\frac{{\hat{f}}_d(\omega )}{\omega }\big |\,d\omega = \int \limits _{\Omega _{0}}\big |\frac{{\hat{f}}_d(\omega )}{\sqrt{\omega k(\omega )}}\sqrt{\frac{k(\omega )}{\omega }}\big |\,d\omega \le \Big (\int \limits _{\Omega _{0}}|\hat{f}(\omega )|^2\frac{k(\omega )}{\omega }\,d\omega \Big )^{1/2}\nonumber \\&\quad \times \Big (\int \limits _{\Omega _{0}}\frac{d\omega }{\omega \sqrt{\omega ^2-m^2})}\Big )^{1/2}<\infty \end{aligned}$$
(5.8)

by Lemma 4.3. Hence,

$$\begin{aligned} \chi _{d}\in C_{b}({\mathbb R}, H^1({\mathbb R}\setminus 0)). \end{aligned}$$
(5.9)

Moreover,

$$\begin{aligned} \Vert \chi _{d}(\cdot ,t)\Vert _{H^1({\mathbb R}\setminus 0)}\rightarrow 0,\qquad t\rightarrow \infty \end{aligned}$$
(5.10)

by Riemann–Lebesgue Theorem. Now consider \(\varphi _{d}(x,t)\). Changing the variable \(\omega \rightarrow k(\omega )=\sqrt{\omega ^2-m^2}\), we rewrite \(\varphi _{d}(x,t)\) as follows:

$$\begin{aligned} \varphi _{d}(x,t)=\frac{1}{2\pi }\int _{{\mathbb R}}\hat{z}_d(x,\omega (k))e^{-i\omega (k) t}e^{i k|x|}\frac{k\,dk}{\omega (k)}. \end{aligned}$$
(5.11)

Here \(\omega (k)=\sqrt{k^2+m^2}\) is the branch analytic for \(\mathrm{Im\,}k>0\) and continuous for \(\mathrm{Im\,}k\ge 0\). Note that the function \(\omega (k)\), \(k\in {\mathbb R}\backslash 0\), is the inverse function to \(k(\omega )\) defined on \(\bar{{\mathbb C}^{+}}\) [see (3.6)] and restricted onto \(\Omega _0\). Let us introduce the functions

$$\begin{aligned} \varphi ^\pm (x,t)=\frac{1}{2\pi }\int _{\mathbb R}\hat{z}_d^{\,\pm }(\omega (k))e^{\pm ikx}e^{-i\omega (k) t}\frac{kdk}{\omega (k)}, \quad x\in {\mathbb R},\quad t\ge 0. \end{aligned}$$

Both functions \(\varphi ^{\pm }(x,t)\) are solutions to the free Dirac equation (3.1) on the whole real line (see “Appendix B”). Moreover,

$$\begin{aligned} \partial _x\varphi ^{\pm }(x,t)\!:=\frac{1}{2\pi }\int _{\mathbb R}\pm ik\, \hat{z}_d^{\,\pm }(\omega (k)) e^{\pm ikx}e^{-i\omega (k) t}\frac{kdk}{\omega (k)}, \quad x\in {\mathbb R},\quad t\ge 0.\qquad \end{aligned}$$
(5.12)

Hence, the Parseval identity implies

$$\begin{aligned} \Vert \varphi ^{\pm }(\cdot ,0)\Vert _{H^1}^2= & {} \int _{{\mathbb R}} (m^2+k^2)|\hat{z}_d^{\,\pm }(\omega (k))|^2\frac{k^2}{\omega ^2(k)}\,dk \nonumber \\= & {} \int _{\Omega _0} \omega ^2|\hat{z}^{\,\pm }(\omega )|^2\frac{k(\omega )}{\omega }\,d\omega =\int _{\Omega _0}|{\hat{z}}^{\,\pm }(\omega )|^2\omega k(\omega )\,d\omega <\infty \end{aligned}$$

by (4.11). Hence, both \(\varphi ^{-}\) and \(\varphi ^{+}\) are bounded continuous \(H^1\)-valued functions:

$$\begin{aligned} \varphi ^{\pm }\in C_{b}({\mathbb R}, H^1), \end{aligned}$$
(5.13)

and for any \(R>0\)

$$\begin{aligned} \Vert \varphi ^{\pm }(\cdot ,t)\Vert _{H^1(-R,R)}\rightarrow 0,\qquad t\rightarrow \infty \end{aligned}$$
(5.14)

by Lemma 3.1. The function \(\varphi _{d}(x,t)\) coincides with \(\varphi ^{+}(x,t)\) for \(x\ge 0\) and with \(\varphi ^{-}(x,t)\) for \(x\le 0\):

$$\begin{aligned} \varphi _{d}(x,t)=\varphi ^{\pm }(x,t),\qquad \pm x\ge 0. \end{aligned}$$
(5.15)

It remains to note that \(\psi _{d}(x,t)=\varphi _{d}(x,t)+\chi _{d}(x,t)\) has no jump at \(x=0\) and therefore \(\partial _x\psi _{d}(x,t)\) is square-integrable over the whole x-axis. Hence,

$$\begin{aligned} \Vert \psi _{d}(t)\Vert _{H^1}^2=\Vert \psi _{d}(t)\Vert _{H^1({\mathbb R}^-)}^2+\Vert \psi _{d}(t)\Vert _{H^1({\mathbb R}^+)}^2. \end{aligned}$$

Finally, (5.5) follows from (5.13) and (5.9), and (5.6) follows from (5.14) and (5.10). \(\quad \square \)

Denote \(y_d(t)=\psi _d(0,t)\in C_b(|{\mathbb R})\). Formulas (5.2) and (5.3) imply

$$\begin{aligned} \hat{y}_d(\omega )={\hat{\psi }}_d(0,\omega )=-\big (I+\frac{m+ik(\omega )}{\omega }\beta \big ) \frac{{\hat{f}}_{d}(\omega )}{2ik(\omega )},, \qquad \omega \in {\mathbb R}, \end{aligned}$$
(5.16)

and (5.2) becomes

$$\begin{aligned} {\hat{\psi }}_{d}(x,\omega )={\hat{y}}_de^{-\varkappa (\omega )|x|} +(\beta -\alpha \, \mathrm {sgn}x)\hat{f}(\omega )\frac{e^{ik(\omega )|x|}-e^{-m|x|}}{2\omega }, \qquad \omega \in {\mathbb R}.\nonumber \\ \end{aligned}$$
(5.17)

6 Bound Component

6.1 Spectral representation

We introduce the bound component of the solution \(\psi (x,t)\) by

$$\begin{aligned} \psi _{b}(x,t)=\psi _S(x,t)-\psi _{d}(x,t),\ \ x\in {\mathbb R},\ \ t\in {\mathbb R}. \end{aligned}$$
(6.1)

Then (3.11) and (5.5) imply that

$$\begin{aligned} \psi _{b}\in C_{b}({\mathbb R},H^1). \end{aligned}$$
(6.2)

In particular, \(y_b(t):=\psi _{b}(0,t)=\psi _S(0,t)-\psi _{d}(0,t)\in C_b({\mathbb R})\). Hence, \({\hat{y}}_b(\omega ):={\hat{\psi }}_{b}(0,\omega )\) is a quasimeasure. Moreover, formulas (3.13) and (5.16) yield

$$\begin{aligned} {\hat{y}}_b(\omega )={\hat{y}}(\omega )-{\hat{y}}_d(\omega ) =-\big (I+\frac{m+ik(\omega )}{\omega }\beta \big ) \frac{\hat{f}_{b}(\omega )}{2ik(\omega )}. \end{aligned}$$
(6.3)

Here we denote

$$\begin{aligned} {\hat{f}}_{b}(\omega ):={\hat{f}}(\omega )-{\hat{f}}_{d}(\omega ). \end{aligned}$$
(6.4)

Further, (5.1) implies, that

$$\begin{aligned} \mathrm {supp}\,{\hat{y}}_b(\omega )=\mathrm {supp\,}{\hat{\psi }}_b(0,\omega )\subset [-m,m]. \end{aligned}$$
(6.5)

Denote

$$\begin{aligned} \varkappa (\omega ):=-i k(\omega )=\sqrt{m^2-\omega ^2},\qquad \mathrm{Re\, }\varkappa (\omega )\ge 0\quad \mathrm{for}\quad \mathrm{Im\,}\omega \ge 0, \end{aligned}$$
(6.6)

where \(k(\omega )\) was introduced in (3.6). Let us note that \(\varkappa (\omega )>0\) for \(\omega \in (-m,m)\). Now we rewrite (6.3) as

$$\begin{aligned} {\hat{y}}_b(\omega )=\big (I+\frac{m-\varkappa (\omega )}{\omega }\beta \big ) \frac{{\hat{f}}_{b}(\omega )}{2\varkappa (\omega )}=\sigma (\omega )\hat{f}_b(\omega ), \end{aligned}$$
(6.7)

where

$$\begin{aligned} \sigma (\omega )=\frac{1}{2\varkappa (\omega )}\left( \begin{array}{c@{\quad }c} 1+\frac{m-\varkappa (\omega )}{\omega } &{} 0\\ 0 &{} 1-\frac{m-\varkappa (\omega )}{\omega }\end{array}\right) . \end{aligned}$$

Hence

$$\begin{aligned} {\hat{f}}_b(\omega )=\sigma ^{-1}(\omega )\hat{y}_b(\omega ),\qquad \sigma ^{-1}(\omega )=\left( \begin{array}{c@{\quad }c} \varkappa (\omega )+m-\omega &{} 0\\ 0 &{} \varkappa (\omega )+m+\omega \end{array}\right) . \end{aligned}$$
(6.8)

Now (3.14), (5.17), (6.1) and (6.8) imply the multiplicative relation

$$\begin{aligned} {\hat{\psi }}_{b}(x,\omega )={\hat{y}}_be^{-\varkappa (\omega )|x|} +\hat{h}_b(x,\omega )\frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega }, \qquad \omega \in {\mathbb R}, \end{aligned}$$
(6.9)

where we denote

$$\begin{aligned} {\hat{h}}_b(x,\omega )=(\beta -\alpha \, \mathrm {sgn}x)\sigma ^{-1}(\omega )\hat{y}_b(\omega ). \end{aligned}$$
(6.10)

From (6.8) and (6.10) it follows that \(\hat{h}_b(x,\omega )\) for any fixed \(x\in {\mathbb R}\setminus 0\) is a quasimeasure with the support \(\mathrm{supp}{\hat{h}}_b(x,\omega )\subset [-m,m]\). Moreover, \(e^{-\varkappa (\omega )|x|}\) and \((e^{-\varkappa (\omega )|x|}-e^{-m|x|})/\omega \) are multiplicators. Hence, function \({\hat{\psi }}_{b}(x,\omega )\) is quasimeasure for any fixed \(x\in {\mathbb R}\) with supports in \([-m,m]\). Finally,

$$\begin{aligned} \psi _{b}(x,t)=\frac{1}{2\pi }\langle {\hat{\psi }}_{b}(x,\omega ), e^{-i\omega t}\rangle ,\quad x\in {\mathbb R},\quad t\in {\mathbb R}, \end{aligned}$$
(6.11)

where \(\langle \cdot ,\cdot \rangle \) is an extension of the scalar product \(\langle f,g\rangle =\int f(\omega )\overline{g}(\omega )d\omega \).

6.2 Compactness

We are going to prove a compactness of the set of translations of the bound component, \(\{\psi _{b,n}(x,s+t)\mathrm{:}\ s\ge 0\}\), \(n=1,2\).

Lemma 6.1

  1. (i)

    The function \(\psi _{b}(x,t)\) is smooth for \(x\ne 0\) and \(t\in {\mathbb R}\). Moreover, for any fixed \(x\not =0\), \(t\in {\mathbb R}\), and any nonnegative integers j, k, the following representation holds

    $$\begin{aligned} \partial _x^j \partial _t^k \psi _{b}(x,t)=\frac{1}{2\pi }\langle \Lambda _j(x,\omega ), (-i\omega )^k e^{-i\omega t}\rangle , \end{aligned}$$
    (6.12)

    where

    $$\begin{aligned} \Lambda _j(x,\omega )= & {} \big (\!\!-\!\varkappa (\omega )\mathrm{sgn}x\big )^j e^{-\varkappa (\omega )|x|}{\hat{y}}_b(\omega )\\&+{\hat{h}}_b(x,\omega )\big [\big (\!\!-\!\varkappa (\omega )\mathrm{sgn}x\big )^j\,\frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega }\\&+\frac{\varkappa ^j(\omega )-m^j}{\omega }(-\mathrm{sgn}x)^je^{-m|x|}\big ]. \end{aligned}$$
  2. (ii)

    There is a constant \(C_{j,k}>0\) so that

    $$\begin{aligned} \sup \limits _{x\not =0}\,\,\sup \limits _{t\in {\mathbb R}}\big |\partial _x^j \partial _t^k\psi _{b}(x,t)\big |\le C_{j,k}. \end{aligned}$$
    (6.13)

The lemma follows similarly Proposition 4.1 from [17], since the factors \(e^{-\varkappa (\omega )|x|}\zeta (\omega )\), \(\frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega }\zeta (\omega )\), and \(\frac{\varkappa ^j(\omega )-m^j}{\omega }\zeta (\omega )\) are multiplicators in the space of quasimeasures. Here \(\zeta (\omega )\in C_{0}^\infty ({\mathbb R})\) is any cutoff function satisfying

$$\begin{aligned} \zeta |_{[-m-1,m+1]}=1. \end{aligned}$$

Corollary 6.2

By the Ascoli-Arzelà Theorem, for any sequence \(s_{l}\rightarrow \infty \) there exists a subsequence (which we also denote by \(s_l\)) such that for any nonnegative integers j and k,

$$\begin{aligned} \partial _x^j \partial _t^k \psi _{b}(x,s_{l}+t)\rightarrow \partial _x^j \partial _t^k \gamma (x,t),\qquad x\ne 0,\quad t\in {\mathbb R}. \end{aligned}$$
(6.14)

for some \(\gamma \in C_{b}({\mathbb R},H^1)\). The convergence in (6.14) is uniform in x and t as long as \(|x|+|t|\le R\), for any \(R>0\).

We call omega-limit trajectory any function \(\gamma (x,t)\) that can appear as a limit in (6.14). Previous analysis demonstrates that the long-time asymptotics of the solution \(\psi (x,t)\) in \(H^1_{loc}\) depends only on the bound component \(\psi _{b}(x,t)\). 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,

$$\begin{aligned} \gamma (x,t)=&{} C_1\left( \begin{array}{c@{\quad }c} e^{-\varkappa _1^{+}|x|}+\frac{me^{-\varkappa _1^{+}|x|}-\varkappa _1e^{-m|x|}}{\omega _1^+}\\ \varkappa _1^{+}\mathrm {sgn}x\frac{e^{-\varkappa _1^{+}|x|}-e^{-m|x|}}{\omega _1^+}\end{array}\right) e^{-i\omega _1^+t} \nonumber \\ {}&\quad +C_2\left( \begin{array}{c@{\quad }c} -\varkappa _2^{+}\mathrm {sgn}x\frac{e^{-\varkappa _2^{+}|x|}-e^{-m|x|}}{\omega _2^+}\\ e^{-\varkappa _2^{+}|x|}-\frac{me^{-\varkappa _2^{+}|x|}-\varkappa _2^{+}e^{-m|x|}}{\omega _2^+} \end{array}\right) e^{-i\omega _2^+t},~~~ \varkappa _j^{+}=\sqrt{m^2-(\omega _j^+)^2}\nonumber \\ \end{aligned}$$
(6.15)

with some \(\omega _1^{+},\omega _2^+\in [-m,m]\).

6.3 Spectral identity for omega-limit trajectories

Here we study the time spectrum of the omega-limit trajectories.

Definition 6.3

Let \(\mu \) be a tempered distribution. By \(\mathrm{Spec}\mu \) we denote the support of its Fourier transform:

$$\begin{aligned} \mathrm{Spec}\mu :=\mathrm{supp}{\tilde{\mu }}. \end{aligned}$$

Proposition 6.4

  1. 1.

    For any omega-limit trajectory \(\gamma (x,t)\), the following spectral representation holds:

    $$\begin{aligned} \gamma (x,t)= & {} \frac{1}{2\pi }\langle \hat{p}(\omega )e^{-\varkappa (\omega )|x|},e^{-i\omega t}\rangle \nonumber \\&+\frac{1}{2\pi }\langle \hat{q}(x,\omega )\frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega },e^{-i\omega t}\rangle , \qquad x\in {\mathbb R},\qquad t\in {\mathbb R},\nonumber \\ \end{aligned}$$
    (6.16)

    where \({\hat{p}}(\omega )\) and \({\hat{q}}(x,\omega )=(\beta -\alpha \, \mathrm {sgn}x)\sigma ^{-1}(\omega ){\hat{p}}(\omega )\) are quasimeasures for all \(x\in {\mathbb R}\), and

    $$\begin{aligned} \mathrm {supp}\,{\hat{p}}\subset [-m,m],\qquad \mathrm {supp}\,{\hat{q}}(x)\subset [-m,m]. \end{aligned}$$
    (6.17)
  2. 2.

    The following bound holds:

    $$\begin{aligned} \sup \limits _{t\in {\mathbb R}} \Vert \gamma (\cdot ,t)\Vert _{H^1}<\infty . \end{aligned}$$
    (6.18)

Note that, according to (6.16), \({\hat{p}}(\omega )\) is the Fourier transform of the function \(p(t):= \gamma (0,t)\), \(t\in {\mathbb R}\).

Proof

Formula (6.9) and representation (6.11) imply that

$$\begin{aligned}&\psi _{b}(x,s_{l}+t)=\frac{1}{2\pi }\langle {\hat{y}}_b(\omega )e^{ -\varkappa (\omega )|x|}e^{-i\omega s_{l}},e^{-i\omega t}\rangle \nonumber \\&\quad +\frac{1}{2\pi }\langle \hat{h}_b(x,\omega )\frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega }e^{-i\omega s_{l}},e^{-i\omega t}\rangle , \quad x\ne 0,\quad t\in {\mathbb R}. \end{aligned}$$
(6.19)

Further, the convergence (6.14) and the bound (6.13) with \(j=k=0\) imply that

$$\begin{aligned} y_b(s_{l}+t)\rightarrow p(t),\qquad s_{l}\rightarrow \infty , \end{aligned}$$
(6.20)

where \(p\in C_b({\mathbb R})\). The convergence is uniform on \([-T,T]\) for any \(T>0\). Hence,

$$\begin{aligned} {\hat{y}}_b(\omega )e^{-i\omega s_{l}}\rightarrow {\hat{p}}(\omega ), \qquad s_{l}\rightarrow \infty \end{aligned}$$
(6.21)

in the space of quasimeasures. Therefore,

$$\begin{aligned} {\hat{y}}_b(\omega )e^{ -\varkappa (\omega )|x|}e^{-i\omega s_{l}}\rightarrow \hat{p}(\omega )e^{-\varkappa (\omega )|x|}, \qquad s_{l}\rightarrow \infty \end{aligned}$$
(6.22)

in the space of quasimeasures. Similarly,

$$\begin{aligned}&\hat{h}_b(x,\omega )\frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega }e^{-i\omega s_{l}} =(\beta -\alpha \, \mathrm {sgn}x)\sigma ^{-1}(\omega ){\hat{y}}_b(\omega )\frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega }e^{-i\omega s_{l}} \nonumber \\ {}&\rightarrow (\beta -\alpha \, \mathrm {sgn}x)\sigma ^{-1}(\omega ){\hat{p}}(\omega )\frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega } ={\hat{q}}(x,\omega ) \frac{e^{-\varkappa (\omega )|x|}-e^{-m|x|}}{2\omega },\qquad s_{l}\rightarrow \infty .\nonumber \\ \end{aligned}$$
(6.23)

Hence, the representation (6.16) follows from (6.19), (6.22) and (6.23); and (6.17) follows from (6.5). Finally, the bound (6.18) follows from (6.2) and (6.14). \(\quad \square \)

The relation (6.16) implies the basic spectral identity:

Corollary 6.5

For any omega-limit trajectory \(\gamma (x,t)\),

$$\begin{aligned} \mathrm {Spec}\,\gamma (x,\cdot )=\mathrm {Spec}\,p,\qquad x\in {\mathbb R}. \end{aligned}$$
(6.24)

7 Nonlinear Spectral Analysis

Here we will derive (6.15) from the following identity:

$$\begin{aligned} p_j(t)=C_je^{-i\omega _j^{+}t},\qquad j=1,2,\qquad t\in {\mathbb R}, \end{aligned}$$
(7.1)

which will be proved in three steps.

7.1 Step 1

The identity (7.1) is equivalent to \(\hat{p}_j(\omega )\sim \delta (\omega -\omega _j^{+})\), so we start with an investigation of \(\mathrm {Spec}\,p_j:=\mathrm {supp}\,{\hat{p}}_j\).

Lemma 7.1

For omega-limit trajectories the following spectral inclusion holds:

$$\begin{aligned} \mathrm{Spec}F_j(p_j(\cdot ))\subset \mathrm{Spec}p_j,\qquad j=1,2. \end{aligned}$$
(7.2)

Proof

The convergence (6.14), Lemma 3.1 and Proposition 5.1 (ii) imply that the limiting trajectory \(\gamma (x,t)\) is a solution to Eq. (1.1):

$$\begin{aligned} i{\dot{\gamma }}(x,t)=D_m\gamma (x,t)-D_m^{-1}\delta (x)F(\gamma (0,t)), \qquad (x,t)\in {\mathbb R}^2. \end{aligned}$$
(7.3)

Applying to both side operator \(D_m\), we get

$$\begin{aligned} iD_m{\dot{\gamma }}(x,t)=D_m^2\gamma (x,t)-\delta (x)F(\gamma (0,t)), \qquad (x,t)\in {\mathbb R}^2. \end{aligned}$$

Since \(\gamma (x,t)\) is smooth function for \(x\le 0\) and \(x\ge 0\), we get the following algebraic identities :

$$\begin{aligned} \gamma '_j(0+,t)-\gamma '_j(0-,t)=-F_j(p_j(t)), \quad t\in {\mathbb R},\quad j=1,2. \end{aligned}$$
(7.4)

The identities imply the spectral inclusion

$$\begin{aligned} \mathrm {Spec}F_j(p_j(\cdot ))\subset \mathrm {Spec}\,\gamma '_j(0+,\cdot )\cup \mathrm {Spec}\,\gamma '_j(0-,\cdot ). \end{aligned}$$
(7.5)

On the other hand, \(\mathrm {Spec}\,\gamma '_j(0+,\cdot )\cup \mathrm {Spec}\,\gamma '_k(0-,\cdot )\subset \mathrm {Spec}\,p_j\) by (6.24). Therefore, (7.5) implies (7.2). \(\quad \square \)

7.2 Step 2

Proposition 7.2

For any omega-limit trajectory, the following identity holds:

$$\begin{aligned} |p_j(t)|=\mathrm{const},\qquad j=1,2, \qquad t\in {\mathbb R}. \end{aligned}$$
(7.6)

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)]

$$\begin{aligned} F_j(t)=a_j(t)p_j(t),\quad j=1,2, \end{aligned}$$
(7.7)

where, according to (2.10),

$$\begin{aligned} a_j(t)=-\sum \limits _{n=1}^{N_j} 2n_j u_{n,j}|p_j(t)|^{2n-2},\qquad N_j\ge 2;\quad u_{N_j,j} > 0. \end{aligned}$$
(7.8)

Both functions \(p_j(t)\) and \(a_j(t)\) are bounded continuous functions in \({\mathbb R}\) by Proposition 6.4 (ii). Hence, \(p_j(t)\) and \(a_j(t)\) are tempered distributions. Furthermore, \(\hat{p}_j\) and \(\hat{{\overline{p}}}_j\) 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 many \(\hat{p}_j\) and \(\hat{{\overline{p}}}_j\) by (7.8). Then the relation (7.7) translates into a convolution in the Fourier space, \({\hat{F}}_j={\hat{a}}_j*{\hat{p}}_j/(2\pi ),\) and the spectral inclusion (7.2) takes the following form:

$$\begin{aligned} \mathrm {supp}{\hat{F}}_j= \mathrm {supp}\,{\hat{a}}_j*{\hat{p}}_j\subset \mathrm {supp}\,{\hat{p}}_j. \end{aligned}$$
(7.9)

Let us denote \(\mathbf{F}_j=\mathrm{supp}{\hat{F}}_j\), \(\mathbf {A}_j=\mathrm {supp}\,\hat{a}_j\), and \(\mathbf {P}_j=\mathrm {supp}\,{\hat{p}}_j\). Then the spectral inclusion (7.9) reads as

$$\begin{aligned} \mathbf{F}_j\subset \mathbf{P}_j. \end{aligned}$$
(7.10)

On the other hand, it is well-known that \(\mathrm {supp}\,{\hat{a}}_j*\hat{p}_j\subset \mathrm {supp}\,{\hat{a}}_j+\mathrm {supp}\,{\hat{p}}_j\), or \(\mathbf{F}_j \subset \mathbf{A}_j+\mathbf{P}_j.\) Moreover, the Titchmarsh convolution theorem (see [12, Theorem 4.3.3]) imply that

$$\begin{aligned} \inf \mathbf{F}_j=\inf \mathbf{A}_j+\inf \mathbf{P}_j,\qquad \sup \mathbf{F}_j=\sup \mathbf{A}_j+\sup \mathbf{P}_j. \end{aligned}$$
(7.11)

Now (7.10) and (7.11) result in

$$\begin{aligned} \inf \mathbf{F}_j=\inf \mathbf{A}_j+\inf \mathbf{P}_j\ge \inf \mathbf{P}_j,\qquad \sup \mathbf{F}_j=\sup \mathbf{A}_j+\sup \mathbf{P}_j\le \sup \mathbf{P}_j, \end{aligned}$$
(7.12)

so that \(\inf \mathbf{A}_j\ge 0\ge \sup \mathbf{A}_j\). Thus, we conclude that \(\mathrm{supp}{\hat{a}}_j=\mathbf{A}_j\subset \{0\}\), therefore the distribution \({\hat{a}}_j(\omega )\) is a finite linear combination of \(\delta (\omega )\) 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). \(\quad \square \)

7.3 Step 3

Now the same Titchmarsh arguments imply that \(P_j:=\mathrm{Spec}p_j\) is a point \(\omega _j^{+}\in [-m,m]\). Indeed, (7.6) means that \(p_j(t) {\overline{p}}_j(t)\equiv C_j\), hence in the Fourier transform \(\hat{p}_j *\hat{{\overline{p}}_j}=2\pi C_j\delta (\omega )\). Therefore, if \(p_j\) is not identically zero, the Titchmarsh Theorem implies that

$$\begin{aligned} 0=\sup P_j+\sup (-P_j)=\sup P_j-\inf P_j. \end{aligned}$$

Hence \(\inf P_j=\sup P_j\) and therefore \(P_j=\omega _k^{+}\in [-m,m]\), so that \({\hat{p}}_j(\omega )\) is a finite linear combination of \(\delta (\omega -\omega _j^{+})\) and its derivatives. As the matter of fact, the derivatives could not be present because of the boundedness of \(p_j(t)=\gamma _j(0,t)\) that follows from Proposition 6.4 (ii). Thus, \(\hat{p}_j\sim \delta (\omega -\omega _j^{+})\), which implies (7.1).

7.3.1 Conclusion of the proof of Theorem 2.9

According to (7.1) and (6.23)

$$\begin{aligned} {\hat{q}}(\omega ,x)&=(\beta -\alpha \, \mathrm {sgn}x)\sigma ^{-1}(\omega )\hat{p}(\omega ) \\ {}&=2\pi (\beta -\alpha \, \mathrm {sgn}x) \left( \begin{array}{c@{\quad }c} C_1(\varkappa _1^++m-\omega _1^+ )\delta (\omega -\omega _1^{+})\\ C_2( \varkappa _2^++\omega _2^++m)\delta (\omega -\omega _2^{+})\end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} \varkappa _j^+=\sqrt{m^2-(\omega _j^+)^2}. \end{aligned}$$

Then the representation (6.16) implies

(7.13)

After simple evaluation, (7.13) becomes

where we denote

$$\begin{aligned} C_1'=C_1\frac{\varkappa _1^++m-\omega _1^+}{2\varkappa _1^+},\quad C_2'=C_2\frac{\varkappa _2^++m+\omega _2^+}{2\varkappa _2^+}. \end{aligned}$$

Therefore, \(\gamma (x,t)\) is a solitary wave (2.14). Due to Lemma 3.1 and Proposition 5.6 it remains to prove that

$$\begin{aligned} \lim _{t\rightarrow \infty } \mathrm{dist}_{H^1_{loc}}(\psi _b(t),\mathcal{S})=0. \end{aligned}$$
(7.14)

Assume by contradiction that there exists a sequence \(s_l\rightarrow \infty \) such that

$$\begin{aligned} \mathrm{dist}_{H^1_{loc}}(\psi _b(s_l),\mathcal{S})\ge \delta ,\quad \forall l\in {\mathbb N}\end{aligned}$$
(7.15)

for some \(\delta >0\). According to Corollary 6.2, there exist a subsequence \(s_{l_n}\) of the sequence \(s_l\), \(\omega _1^+,\omega _2^+\in {\mathbb R}\) and vector-function \(\gamma (x,t)\), defined in (7.13) such that the following convergence hold

$$\begin{aligned} \psi _b(x,t+s_{l_n})\rightarrow \gamma (x,t),\quad l_n\rightarrow \infty ,\quad t\in {\mathbb R}. \end{aligned}$$

This implies that

$$\begin{aligned} \psi _b(x,s_{l_n})\rightarrow \gamma (x,0)=\phi _{\omega _1^+}(x)+\phi _{\omega _2^+}(x),\quad l_n\rightarrow \infty , \end{aligned}$$
(7.16)

where

$$\begin{aligned}&\phi _{\omega _1^+}(x)=C_1\!\left( \!\begin{array}{cc} e^{-\varkappa _1^{+}|x|}+\frac{me^{-\varkappa _1^{+}|x|}-\varkappa _1e^{-m|x|}}{\omega _1^+}\\ \varkappa _1^{+}\mathrm {sgn}x\frac{e^{-\varkappa _1^{+}|x|}-e^{-m|x|}}{\omega _1^+}\end{array}\right) , \\ {}&\phi _{\omega _2^+}(x)=C_2\!\left( \!\begin{array}{cc} -\varkappa _2^{+}\mathrm {sgn}x\frac{e^{-\varkappa _2^{+}|x|}-e^{-m|x|}}{\omega _2^+}\\ e^{-\varkappa _2^{+}|x|}-\frac{me^{-\varkappa _2^{+}|x|}-\varkappa _2^{+}e^{-m|x|}}{\omega _2^+} \end{array}\right) . \end{aligned}$$

The convergence (7.16) contradicts to (7.15). This completes the proof of Theorem 2.9. \(\quad \square \)