1 Introduction

In the last decades equations with point interactions became an intensively developing field of research, and this interest is driven by the possibility of investigating nonlinear problems in the context of solvable models. These equations are useful mathematical tool for modeling many phenomena in theoretical physics, (see introduction in [11]).

The first rigorous mathematical results for equations with point interaction were obtained since 1960 by F. Berezin, L. Faddeev, D. Yafaev, E. Zeidler and others [6, 16, 37], and since 2000 by S. Albeverio, R. Høegh-Krohn, D. Noja, D. Yafaev and others [2, 4, 5, 32, 38]. A comprehensive overview of the results can be found in [3, 15].

Our paper concerns 3D Dirac equation with nonlinear point interaction. Namely, we consider the system governed by the following equations

$$\begin{aligned} \left\{ \begin{array}{c} i{\dot{\psi }}(x,t)=D_m\psi (x,t)-D^{-1}_m \zeta (t)\delta (x)\\ \lim \limits _{\varepsilon \rightarrow 0+}\lim \limits _{x\rightarrow 0}K_m^{\varepsilon }\Big (\psi (x,t)-\zeta (t) g(x)\Big )= F(\zeta (t)) \end{array}\right| \quad x\in {{\mathbb {R}}}^3,\quad t\in {{\mathbb {R}}}. \end{aligned}$$
(1.1)

Here \(D_m\) is the Dirac operator \(D_m:=-i\alpha \cdot \nabla +m\beta \), where \(m>0\), \(\alpha _k\) with \(k=1,2,3\) and \(\beta \) are \(4\times 4\) Dirac matrices; \(\psi (x,t)\), \(\zeta (t)\) are vector functions with values in \({{\mathbb {C}}}^4\); g(x) is the Green function of the operator \(D_m^2=-\Delta +m^2\) in \({{\mathbb {R}}}^3\),

$$\begin{aligned} g(x):=\frac{e^{-m|x|}}{4\pi |x|}, \end{aligned}$$
(1.2)

and \(K_m^\varepsilon =(-\Delta +m^2)^{-\varepsilon }\) is a smoothing operator, defined as and \(K_m^\varepsilon =(-\Delta +m^2)^{-\varepsilon }\) is a smoothing operator, defined as

$$\begin{aligned} (K_m^{\varepsilon } \psi )(x)=\frac{1}{(4\pi )^3}\int e^{-i\xi \cdot x}\frac{{\widehat{\psi }}(\xi )\,d\xi }{(\xi ^2+m^2)^\varepsilon },\quad \varepsilon \ge 0, \end{aligned}$$

where \({\widehat{\psi }}(\xi )\) is the Fourier transform of \(\psi (x)\). Obviously, \((K_m^{\varepsilon }\psi )(x)\rightarrow \psi (x)\) as \(\varepsilon \rightarrow 0\) in \(H^s({{\mathbb {R}}}^3)\) for any \(\psi \in H^s({{\mathbb {R}}}^3)\) and any \(s\in {{\mathbb {R}}}\). Hence, in the limit \(\varepsilon \rightarrow 0\), the coupling in (1.1) formally depends on the value of \(\psi (x,t)-\zeta (t)g(x)\) at one point \(x=0\).

We assume that the nonlinearity \(F_j(\zeta )=F_j(\zeta _j)\), \(j=1, \ldots ,4\), admits a real-valued potential:

$$\begin{aligned} F_j(\zeta _j)=\partial _{\overline{\zeta }_j} U(\zeta ),\qquad U(\zeta )=\sum \limits _{j=1}^4 U_j(|\zeta _j|) \in C^2({{\mathbb {C}}}^4), \end{aligned}$$
(1.3)

where \(\partial _{\zeta _j}:=\frac{\partial U}{\partial \zeta _{j1}}+i\frac{\partial U}{\partial \zeta _{j2}}\) with \(\zeta _{j1}:={\mathrm{Re\, }}\zeta _j\) and \(\zeta _{j2}:={\mathrm{Im\,}}\zeta _j\), and

$$\begin{aligned} U(\zeta )\ge b|\zeta |^2-a, \quad {\mathrm{for}}\ \zeta \in {{\mathbb {C}}}^4,\quad {\mathrm{where}}\ b> 0~~~\mathrm{and}~~~a\in {{\mathbb {R}}}. \end{aligned}$$
(1.4)

The system (1.1) is \({\mathbf {U}}(1)\)-invariant; that is,

$$\begin{aligned} F_j(e^{i\theta }\zeta _j)=e^{i\theta }F_j(\zeta _j),\quad j=1, \ldots ,4, \quad \zeta _j\in {{\mathbb {C}}},\quad \theta \in {{\mathbb {R}}}. \end{aligned}$$
(1.5)

Our main results are as follows. First, for initial data of type

$$\begin{aligned} \psi (x,0)=f(x)+\zeta _{0} g(x), \quad f\in H^{\frac{5}{2}+}({{\mathbb {R}}}^3)\otimes {{\mathbb {C}}}^4, \quad \zeta _0\in {{\mathbb {C}}}^4, \end{aligned}$$
(1.6)

we prove a global well-posedness of the Cauchy problem for the system (1.1) (Theorem 2.1 below).

Further, we show that the system admits four-frequencies stationary orbits (or solitary wave solutions) of the type

$$\begin{aligned} \psi (x,t)=\sum \limits _{k=1}^4\psi _{\omega _k}(x)e^{-i\omega _k t},\quad \omega _k \in {{\mathbb {R}}},\quad k=1, \ldots ,4. \end{aligned}$$
(1.7)

We obtain explicit formulas for the amplitudes \(\psi _{\omega _k}(x)\).

Finally, we prove that solitary waves form a global attractor in the case when all polynomials \(F_j\) are strictly nonlinear [see. conditions (3.2)–(3.3)]. Namely, in this case any solution with initial data (1.6) converges to the set \({\mathscr {S}}\) of all solitary wave solutions:

$$\begin{aligned} \psi (\cdot ,t)\longrightarrow {\mathscr {S}}, \qquad t\rightarrow \pm \infty , \end{aligned}$$
(1.8)

where the convergence holds in local \(L^{2}\)- seminorms.

Let us comment on previous results on the attraction to the set of solitary waves for nonlinear \({\mathbf {U}}(1)\)-invariant equations. The first results on asymptotic stability of solitary waves for nonlinear Schrödinger equation were obtained in [8, 35, 36], and then developed in [9, 12, 23] and other papers. The asymptotic stability means the asymptotics of type (1.8) for solutions with initial data close to \(\mathscr {S}\!.\) Such local attraction for equations with nonlinear point interaction was proved in [2, 7, 23,24,25, 31]. These models allow an efficient analysis of the corresponding linearized dynamics.

Global attraction of type (1.8) to the set of all stationary orbits was established

  1. (i)

    in [20] for 1D Klein–Gordon equation coupled to nonlinear oscillator :

    $$\begin{aligned} \ddot{\psi }(x,t)=(\partial ^2_x-m^2)\psi (x,t)+\delta (x)F(\psi (0,t)); \end{aligned}$$
    (1.9)
  2. (ii)

    in [30] for 1D Dirac equations with more regular nonlinearity \(D_m^{-1}\delta (x)F(\psi (0,t))\);

  3. (iii)

    in [21, 22] for nD Klein–Gordon and Dirac equations with nonlinearity of type \(\rho (x)F(\langle \psi ,\rho \rangle )\);

  4. (iv)

    in [26, 27, 29] for 3D wave and Klein–Gordon equations with concentrated nonlinearity.

Global attraction of type (1.8) for 3D Dirac equation with nonlinear point interaction was not considered previously.

Remark 1.1

The nonlinearity in (1.1) is more singular than the nonlinearities considered in [26, 27, 30]. That’s why we introduced the smoothing operator \(K_m^{\varepsilon }\). In Sect. C.3.1, we show that without the operator \(K_m^{\varepsilon }\), the limit as \(x\rightarrow 0\) in the second equation of (1.1) generally does not exist.

We note also that the 3D Schrödinger equation with concentrated nonlinearity was justified in [10] as a scaling limit of a regularized nonlinear Schrödinger dynamics. We suppose that for the Dirac equation a justification can be done by suitable modification of methods [10], but it still remains an open question.

Let us comment on our approach. For the proof of global well-posedness we develop the approach which was introduced in [26, 32] in the context of the Klein–Gordon and wave equations. First, we obtain some regularity properties i) of solutions \(\varphi _{g}(x,t)\) to the free Dirac equation with initial function \(\zeta _0g(x)\), and ii) of solutions \(\psi _{S}(x,t)\) to the Dirac equation with zero initial function and with source \(D_m^{-1}\zeta (t)\delta (x)\) (Lemma 2.2, and Propositions 2.4 and 2.5 ). We use these regularity properties to prove the existence of a local solution to (1.1) of the type

$$\begin{aligned} \psi (x,t)=\psi _{\mathrm{free}}(x,t)+\psi _{S}(x,t), \quad \psi _{\mathrm{free}}=\psi _{f}+\varphi _{g}, \end{aligned}$$

where \(\psi _{f}(x,t)\) is a solution to the free Dirac equation with initial function f(x). We show that \(\zeta (t)\) is a solution to a first-order nonlinear integro-differential equation driven by \(\psi _{\mathrm{free}}(0,t)\). Then we prove that conditions (1.3)–(1.4) provide the conservation law (2.2). Finally, we use the conservation law to obtain the global existence theorem.

Note that our system (1.1) gives a novel model of nonlinear point interaction which provides a conservation law and a priori estimates. The introduced smoothing operator \(K_m^{\varepsilon }\) leads to justification of numerous limit permutation. We justify these limits by subtle arguments using properties of special functions (generalized hypergeometric functions \(_1F_2\), modified Struve functions \(L_{\nu }\), modified Bessel functions \(I_{\nu }\) and others) [14, 33].

To prove the global attraction, we split \(\psi (x,t)\) as

$$\begin{aligned} \psi (x,t)= & {} \psi _{f}(x,t)+\varphi _{g}^{+}(x,t)+\psi _{S}^{-}(x,t), \quad \varphi _{g}^{+}=\varphi _{g}+iD_m^{-1}\zeta _0{\dot{\gamma }},\\ \psi _{S}^{-}= & {} \psi _{S}-iD_m^{-1}\zeta _0{\dot{\gamma }}, \end{aligned}$$

where \({\dot{\gamma }}(x,t)\) is defined in (2.11). We show that \(\psi _f(\cdot ,t)\) and \(\varphi _{g}^{+}(\cdot ,t)\) converge to zero as \(t\rightarrow \pm \infty \) in local \(H^1\)-seminorms. Hence, it remains to prove (1.8) for \(\psi _{S}^{-}(\cdot ,t)\) only. The proof relies on the study of the Fourier transform in time \({\widetilde{\psi }}_{S}^{-}(x,\omega )\) and \({\widetilde{\zeta }}(\omega )\) and of their supports. First, we establish absolute continuity of the spectral density \({\widetilde{\zeta }}(\cdot )\) outside spectral gap \( [-m,m]\). The absolute continuity is a nonlinear version of Kato’s theorem on absence of embedded eigenvalues in the context of the nonlinear system (1.1).

Then we prove the omega-limit compactness. This means that for each sequence \(s_k\rightarrow \infty \) there exists an infinite subsequence \(s_{k_l}\rightarrow \infty \) such that the functions \(\zeta (t + s_{k_l})\) converge to some function \(\eta (t)\in {{\mathbb {C}}}^4\) uniformly in \( |t | < T\) for any \(T > 0\). The absolute continuity of \({\widetilde{\zeta }}(\cdot )\) provides that the time-spectrum of \({\widetilde{\eta }}(\cdot )\) is contained in the spectral gap \([-m,m]\). The convergence of \(\zeta (t + s_{k_l})\) implies the convergence of \(\psi _S^{-}( x,t+s_{k_l})\) to some function \(\phi _S(x,t)\) in in the topology of \(C_b([-T,T],L^{2}_{loc}({{\mathbb {R}}}^3))\).

Further, we apply the Titchmarsh convolution theorem ( [19, Theorem 4.3.3]) to conclude that the time-spectrum of each component \(\eta _j\), \(j=1, \ldots 4\), of function \(\eta \) consists of a single frequency, \({\widetilde{\eta }}_j(\omega )=C_j\delta (\omega -\omega _j)\). The Titchmarsh theorem controls the inflation of spectrum by the nonlinearity. Physically, these arguments justify the following binary mechanism of energy radiation, which is responsible for the attraction to solitary waves: (i) nonlinear energy transfer from lower to higher harmonics, and (ii) subsequent dispersion decay caused by energy radiation to infinity. We finish the proof using an integral representation of \(\phi _S(x,t)\) via \(\eta (t)\).

Remark 1.2

Our approach is also applicable for other interpretation of 3D Dirac equation with concentrated nonlinearities. Namely, the source \(D_m^{-1}\zeta (t)\delta (x)\) in the first equation of (1.1) can be replaced by more singular delta-like source \(\zeta (t)\delta (x)\). In this case, the function \(\psi (x,t)\) in the second equation of (1.1) should be replaced by the function \(D^{-1}_m \psi (x,t)\). For such a system, the convergence (1.8) holds in local \(H^{-1}\)-seminorms.

2 Global well-posedness

We fix a nonlinear function \(F:{{\mathbb {C}}}^4\rightarrow {{\mathbb {C}}}^4\) and define the domain

$$\begin{aligned} {\mathscr {D}}_F= & {} \{\psi \in L^2({{\mathbb {R}}}^3)\otimes {{\mathbb {C}}}^4: \psi (x)=\psi _{reg}(x)+\zeta g(x), ~~ \zeta \in {{\mathbb {C}}}^4,\\&\psi _{reg}\in H^{\frac{3}{2}-}({{\mathbb {R}}}^3)\otimes {{\mathbb {C}}}^4,\exists \lim _{\varepsilon \rightarrow 0+}\lim _{x\rightarrow 0} K_m^\varepsilon \psi _{reg}(x)=F(\zeta )\}, \end{aligned}$$

which generally is not a linear space. Note that the first equation of (1.1) can be written in the other form

$$\begin{aligned} i{\dot{\psi }}(x,t)=D_m^{F}\psi (x,t),\qquad D_m^{F}\psi (x,t):=D_m\psi _{reg}(x,t) \end{aligned}$$
(2.1)

(cf. equation (1.2) in [1], equation (7) in [2]).

Everywhere below we will write \(L^2\) and \(H^s\) instead of \(L^2({{\mathbb {R}}}^3)\otimes {{\mathbb {C}}}^4\) and \(H^{s}({{\mathbb {R}}}^3)\otimes {{\mathbb {C}}}^4\). Denote \(\Vert \cdot \Vert =\Vert \cdot \Vert _{L^2}\). In this section we will prove the following result.

Theorem 2.1

Let conditions (1.3) and (1.4) hold. Then

  1. (i)

    For every initial function \(\psi (x,0)=f(x)+\zeta _0 g(x)\in {\mathscr {D}}_F\) with \(f\in H^{\frac{5}{2}+}\) the equation (1.1) has a unique solution \(\psi (x,t)=\psi _{reg}(x,t)+\zeta (t) g(x)\in C({{\mathbb {R}}},{\mathscr {D}}_F)\), such that \(\zeta (t)\in C^1[0,\infty )\).

  2. (ii)

    The following conservation law holds:

    $$\begin{aligned} {\mathscr {H}}_F(\psi (\cdot ,t)):= \frac{1}{2}\Vert D_m\psi _{reg}(\cdot ,t)\Vert ^2+U(\zeta (t))={\mathrm{const}}, \quad t\in {{\mathbb {R}}}. \end{aligned}$$
    (2.2)
  3. (iii)

    The following a priori bound holds:

    $$\begin{aligned} |\zeta (t)|\le C(\psi (\cdot ,0)),\quad t\in {{\mathbb {R}}}. \end{aligned}$$
    (2.3)
  4. (iv)

    The map \(W: (f(\cdot ),\zeta _0)\mapsto (\psi _{reg}(\cdot ,\cdot ),\zeta (\cdot ))\) is continuous \(H^{\frac{5}{2}+}\oplus {{\mathbb {C}}}^4\rightarrow C({{\mathbb {R}}},H^{\frac{3}{2}-})\oplus (C^1({{\mathbb {R}}})\otimes {{\mathbb {C}}}^4)\).

Obviously, it suffices to prove Theorem 2.1 for \(t\ge 0\).

We split solutions to (1.1) as

$$\begin{aligned} \psi (x,t)=\psi _{free}(x.t)+\psi _S(x,t)=\psi _{f}+\varphi _{g}+\psi _S(x,t), \end{aligned}$$
(2.4)

where \(\psi _f(x,t)\) and \(\varphi _{g}\) are the unique solutions to the free Dirac equation with initial functions f and \(\zeta _0g\):

$$\begin{aligned}&i{\dot{\psi }}_{f}(x,t)=D_m \psi _{f}(x,t), \qquad \psi _f(x,0)=f(x),\\&i{\dot{\varphi }}_{g}(x,t)=D_m \varphi _{g}(x,t), \quad \varphi _{g}(x,0)=\zeta _0g(x), \end{aligned}$$

and \(\psi _S(x,t)\) is the solution to

$$\begin{aligned} \left\{ \begin{array}{l} i{\dot{\psi }}_S(x,t)=D_m\psi _S(x,t)-D_m^{-1}\zeta (t)\delta (x),\\ \lambda (t)+\lim \limits _{\varepsilon \rightarrow 0+}\lim \limits _{x\rightarrow 0}K_m^{\varepsilon }\Big (\varphi _{g}(x,t)+\psi _S(x,t)-\zeta (t) g(x)\Big )= F(\zeta (t)),\\ \psi _S(x,0)=0,\quad \zeta (0)=\zeta _0. \end{array}\right. \end{aligned}$$
(2.5)

Evidently,

$$\begin{aligned} \psi _f(\cdot ,t)\in C_b([0,\infty ), H^{\frac{5}{2}+}), \quad {\dot{\psi }}_f(\cdot ,t)\in C_b([0,\infty ), H^{\frac{3}{2}+}). \end{aligned}$$
(2.6)

Hence,

$$\begin{aligned} \lambda (t):=\psi _f(0,t)\in C^1_b[0,\infty )\otimes {{\mathbb {C}}}^4. \end{aligned}$$
(2.7)

Moreover, the linear map \(f(\cdot )\rightarrow \lambda (\cdot )\) is continuous \(H^{\frac{5}{2}+}\rightarrow C^1_b[0,\infty )\otimes {{\mathbb {C}}}^4\) since

$$\begin{aligned} \Vert \lambda \Vert _{C^1_b[0,\infty )\otimes {{\mathbb {C}}}^4}\le C(\varepsilon )\Vert f\Vert _{H^{\frac{5}{2}+\varepsilon }}, \quad \varepsilon >0. \end{aligned}$$
(2.8)

Now the existence and uniqueness of the solution \(\psi (\cdot ,t)\in C([0,\infty ),{\mathscr {D}}_F\) of the system (1.1) is equivalent to the existence and uniqueness of the solution \((\psi _S(\cdot ,t),\zeta (t))\) to (2.5) such that \(\psi _S(\cdot ,t)+\varphi _{g}(\cdot ,t)\in C([0,\infty ),{\mathscr {D}}_F)\) and \(\zeta \in C^1[0,\infty )\).

Let us obtain an explicit formula for \(\varphi _{g}(x,t)\). Note that the function

$$\begin{aligned} \phi (x,t):=\varphi _{g}(x,t)-\zeta _0g(x) \end{aligned}$$
(2.9)

satisfies

$$\begin{aligned} i{\dot{\phi }}(x,t)=D_m \phi (x,t)+D_m^{-1}\zeta _0\delta (x),\quad \phi (x,0)=0. \end{aligned}$$

Hence,

$$\begin{aligned} \phi (x,t)= (-i\partial _t-D_m)D_m^{-1}\zeta _0\gamma (x,t)=-iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)-\zeta _0\gamma (x,t), \end{aligned}$$
(2.10)

where

$$\begin{aligned} \gamma (x,t)=\frac{\theta (t-|x|)}{4\pi |x|} -\frac{m}{4\pi }\int _0^t\frac{\theta (s-|x|)J_1(m\sqrt{s^2-|x|^2})}{\sqrt{s^2-|x|^2}}ds, \end{aligned}$$
(2.11)

is the solution to

$$\begin{aligned} \ddot{\gamma }(x,t)=(\Delta -m^2)\gamma (x,t)+\delta (x),\quad \gamma (x,0)=0,\quad {\dot{\gamma }}(x,0)=0. \end{aligned}$$
(2.12)

Here \(J_1\) is the Bessel function of the first order and \(\theta \) is the Heaviside function. Finally, (2.9) and (2.10) imply

$$\begin{aligned} \varphi _{g}(x,t)=\zeta _0g(x)-\zeta _0\gamma (x,t)-iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)=\varphi _{g}^{+}(x,t)-iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t),\nonumber \\ \end{aligned}$$
(2.13)

where \(\varphi _{g}^{+}(x,t):=\zeta _0(g(x)-\gamma (x,t))\).

Lemma 2.2

For any \(t>0\) there exists

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0+}\lim _{x \rightarrow 0} K_m^{\varepsilon }(g(x)-\gamma (x,t))=\mu (t):=-\frac{m}{4\pi }+\frac{m}{4\pi }\int _0^t\frac{J_1(ms)}{s}ds.\qquad \end{aligned}$$
(2.14)

We prove this lemma in Sect. A. Note that the function \(\mu (t)\) is continuous for \(t>0\), and there exists

$$\begin{aligned} \mu (0)=\lim _{t \rightarrow 0}\,\mu (t)=-\frac{m}{4\pi }. \end{aligned}$$

Moreover,

$$\begin{aligned} \mu (t)\rightarrow 0,\quad t\rightarrow \infty , \end{aligned}$$
(2.15)

since \(\int \nolimits _0^{\infty }\frac{J_1(ms)}{s}ds=1\) by [33, Formula 10.22.43].

2.1 Reduction to integro-differential equation

Here we consider the first equation of (2.5) for \(\psi _S\) with some given function \(\zeta (t)\in C^1[0,\infty )\otimes C^4\). We construct the solution and formulate its properties which will be proved later. Further, we substitute the constructed solution into the second equation of (2.5) and obtain an integro-differential equation for \(\zeta \).

Lemma 2.3

Let \(\zeta (t)\in C^1[0,\infty )\otimes C^4\). Then the unique solution \(\psi _S(x,t)\) to the Dirac equation

$$\begin{aligned} i{\dot{\psi }}_S(x,t)= D_m \psi _S(x,t) -D_m^{-1}\zeta (t)\delta (x), \quad \psi _S(x,0) = 0 \end{aligned}$$
(2.16)

is given by

$$\begin{aligned} \psi _S(x,t):=\varphi _S(x,t)+iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)+ iD_m^{-1}p_S(x,t),\quad \zeta _0:=\zeta (0),\qquad \end{aligned}$$
(2.17)

where \(\gamma \) is defined in (2.11), and

$$\begin{aligned} \varphi _S(x,t)= & {} \frac{\theta (t-|x|)}{4\pi |x|}\zeta (t-|x|)\nonumber \\&-\frac{m}{4\pi } \int _0^t\frac{\theta (s-|x|)J_1(m\sqrt{s^2-|x|^2})}{\sqrt{s^2-|x|^2}}\zeta (t-s)ds, \end{aligned}$$
(2.18)
$$\begin{aligned} p_S(x,t)= & {} \frac{\theta (t-|x|)}{4\pi |x|}{\dot{\zeta }}(t-|x|)\nonumber \\&-\frac{m}{4\pi } \int _0^t\frac{\theta (s-|x|)J_1(m\sqrt{s^2-|x|^2})}{\sqrt{s^2-|x|^2}}{\dot{\zeta }}(t-s)ds. \end{aligned}$$
(2.19)

Proof

It is easy to verify that the function \(\varphi _S(x,t)\) is the unique solution to the Klein–Gordon with \(\delta \)-like source

$$\begin{aligned} \ddot{\varphi }_S(x,t)=(\Delta -m^2)\varphi _S(x,t)+\zeta (t)\delta (x), \quad \varphi _S(x,0) = 0,\quad {\dot{\varphi }}_S(x,0)=0.\nonumber \\ \end{aligned}$$
(2.20)

In the case \(m=0\) this is well-known formula [14, Section 175]. Hence,

$$\begin{aligned} \psi _S(x,t)= (i\partial _t+D_m)D_m^{-1}\varphi _S(x,t)=\varphi _S(x,t)+iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)+ iD_m^{-1}p_S(x,t).\nonumber \\ \end{aligned}$$
(2.21)

\(\square \)

In Sects. B and C , we justify the following limits

Proposition 2.4

For any \(\zeta (t) \in C^1 [0,\infty )\otimes {{\mathbb {C}}}^4\) there exists

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0+}\lim _{x\rightarrow 0}K_m^{\varepsilon }\left( \varphi _S(x,t)-\zeta (t)g(x)\right) \nonumber \\&\quad =\frac{1}{4\pi }\Big (m\zeta (t)-{\dot{\zeta }}(t)-m\int _0^t\frac{J_1(ms)}{s}\zeta (t-s)ds\Big ),\quad t>0. \end{aligned}$$
(2.22)

Proposition 2.5

For any \(\zeta (t)\in C^1[0,\infty )\otimes {{\mathbb {C}}}^4\) there exists

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0+}\lim _{x\rightarrow 0}K_m^{\varepsilon }D_m^{-1}p_S(x,t)= & {} \frac{m\beta }{4\pi }\Bigg (\zeta _0\Bigg [mt\int \nolimits _{mt}^{\infty }\frac{J_1(u)du}{u}-J_0(mt)\Bigg ] \nonumber \\&+\,\zeta (t)-m\int \nolimits _0^t \Bigg (\int \nolimits _{ms}^{\infty }\frac{J_1(u)du}{u}\Bigg )\zeta (t-s)ds\Bigg ),\quad t> 0.\nonumber \\ \end{aligned}$$
(2.23)

Substituting these limits into the second equation of (2.5) and taking into accunt (2.14), we obtain the equation for \(\zeta (t)\):

$$\begin{aligned}&\lambda (t)+\zeta _0\mu (t)+\frac{1}{4\pi }\Bigg (m\zeta (t)-{\dot{\zeta }}(t)-m\int \nolimits _0^t\frac{J_1(ms)}{s}\zeta (t-s)ds\Bigg )\nonumber \\&\quad +\,\frac{im\beta }{4\pi }\Bigg (\zeta _0\Bigg [tm\int \nolimits _{mt}^{\infty }\frac{J_1(u)du}{u}-J_0(mt)\Bigg ] +\zeta (t)\nonumber \\&\quad -\,m\int \nolimits _0^t \Bigg (\int \nolimits _{ms}^{\infty }\frac{J_1(u)du}{u}\Bigg )\zeta (t-s)ds\Bigg )=F(\zeta (t)),\quad \zeta (0)=\zeta _{0}. \end{aligned}$$
(2.24)

In the next two sections we will solve system (2.5) in reverse order: first we solve the equation (2.24) for \(\zeta (t)\) and then we solve the first equation of (2.21) for \(\psi _S(x,t)\) with this \(\zeta (t)\).

2.2 Local well-posedness

Here we prove the local well-posedness for the system (2.5). To do this, we modify the nonlinearity F so that it becomes Lipschitz continuous. Define

$$\begin{aligned} \Lambda (\psi (0))=\sqrt{({\mathscr {H}}_F(\psi (0))+a)/b}, \end{aligned}$$
(2.25)

where \(\psi (0)=\psi (\cdot ,0)\in {\mathscr {D}}_F\) is the initial function from Theorem 2.1 and a, b are constants from (1.4). Then we may pick a modified potential function \({{\widetilde{U}}}(\zeta )\in C^2({{\mathbb {C}}}^4,{{\mathbb {R}}})\), so that

  1. (i)

    the identity holds

    $$\begin{aligned} {{\widetilde{U}}}(\zeta )= U(\zeta ),\quad |\zeta |\le \Lambda (\psi (0)), \end{aligned}$$
    (2.26)
  2. (ii)

    \({\widetilde{U}}(\zeta )\) satisfies (1.4) with the same constant a, b as \(U(\zeta )\) does:

    $$\begin{aligned} {\widetilde{U}}(\zeta )\ge b|\zeta |^2 -a,\quad \zeta \in {{\mathbb {C}}}^4, \end{aligned}$$
    (2.27)
  3. (iii)

    the functions \({\widetilde{F}}_j(\zeta _j)=\partial _{\overline{\zeta }_j}{\widetilde{U}}(\zeta )\) are Lipschitz continuous:

    $$\begin{aligned} |{\widetilde{F}}_j(\zeta _j)-{\widetilde{F}}_j(\eta _j)|\le C|\zeta _j-\eta _j|,\quad \zeta _j,\eta _j\in {{\mathbb {C}}}. \end{aligned}$$
    (2.28)

First, we establish local well-posedness for system (2.5) with the modified nonlinearity \({\widetilde{F}}\).

Proposition 2.6

(Local well-posedness). Let the conditions (2.26)–(2.28) hold. Then

  1. (i)

    there exists a unique solution \((\psi _S(x,t),\zeta (t))\) to (2.5) such that

    $$\begin{aligned} \psi _{reg}^-(\cdot ,t):= \psi _S(\cdot ,t)+\varphi _{g}(\cdot ,t)-\zeta (t)g(\cdot )\in C([0,\tau ], H^{\frac{3}{2}-}), \quad \zeta \in C^2[0,\tau ]\otimes {{\mathbb {C}}}^4; \end{aligned}$$
  2. (ii)

    the map \(\zeta (\cdot )\rightarrow \psi _{reg}^{-}(\cdot ,\cdot )\) is continuous \(C^2[0,\tau ]\otimes {{\mathbb {C}}}^4\rightarrow C([0,\tau ], H^{\frac{3}{2}-})\).

Proof

(i) First, we solve integro-differential equation (2.24) with \({\widetilde{F}}\) instead of F:

$$\begin{aligned}&\lambda (t)+\zeta _0\mu (t)+\frac{1}{4\pi }\Bigg (m\zeta (t)-{\dot{\zeta }}(t)-m\int \nolimits _0^t\frac{J_1(ms)}{s}\zeta (t-s)ds\Bigg )\nonumber \\&\quad +\frac{im\beta }{4\pi }\Big (\zeta _0\Bigg [tm\int \nolimits _{mt}^{\infty }\frac{J_1(u)du}{u}-J_0(mt)\Bigg ] +\zeta (t)\nonumber \\&\quad -m\int \nolimits _0^t \Bigg (\int \nolimits _{ms}^{\infty }\frac{J_1(u)du}{u}\Bigg )\zeta (t-s)ds\Bigg )={\widetilde{F}}(\zeta (t)),\quad \zeta (0)=\zeta _{0}, \end{aligned}$$
(2.29)

where \(\lambda _j\), \(\mu \in C^1[0,\infty )\) by (2.7), (2.14). The next lemma follows by standard contraction mapping principle.

Lemma 2.7

Let conditions (2.26)–(2.28) be satisfied. Then

  1. (i)

    for sufficiently small \(\tau >0\) the Cauchy problem (2.29) has a unique solution \(\zeta \in C^1[0,\tau ]\otimes {{\mathbb {C}}}^4\);

  2. (ii)

    the map \(\lambda _j(\cdot )\rightarrow \zeta _j(\cdot )\) is continuous \(C^1[0,\tau ]\rightarrow C^2[0,\tau ]\) for every \(j=1, \ldots ,4\).

Now we define the function

$$\begin{aligned} \psi _S(x,t):=\varphi _S(x,t)+iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)+ iD_m^{-1}p_S(x,t), \quad t\in [0,\tau ], \end{aligned}$$

where \(\varphi _S(x,t)\) and \(p_S(x,t)\) are given by (2.18) and (2.19) with \(\zeta (t)\) the solution to (2.29).

Let us show that \((\psi _S(x,t),\zeta (t)\) is the solution to (2.5). Indeed, \(\psi _S\) satisfies the first equation of (2.5). Moreover, (2.14), (2.22) and (2.23) imply for \(t\in [0,\tau ]\)

$$\begin{aligned}&\lim \limits _{\varepsilon \rightarrow 0+}\lim \limits _{x\rightarrow 0}K_m^{\varepsilon }\Bigg (\psi _S(x,t)+\varphi _g(x,t)-\zeta (t) g(x)\Bigg )\nonumber \\&\quad =\zeta _0\mu (t)+\lim _{\varepsilon \rightarrow 0+}\lim _{x \rightarrow 0}\, K_m^{\varepsilon }\left( \varphi _S(x,t)+ iD_m^{-1}p_S(x,t)-\zeta (t) g(x)\right) \nonumber \\&\quad =\zeta _0\mu (t)+\frac{1}{4\pi }\bigg (m\zeta (t)-{\dot{\zeta }}(t)-m\int \nolimits _0^t\frac{J_1(ms)}{s}\zeta (t-s)ds\bigg ) \nonumber \\&\qquad +\,\frac{im\beta }{4\pi }\Bigg (\zeta _0\Bigg [tm\int \nolimits _{mt}^{\infty }\frac{J_1(u)du}{u}-J_0(mt)\Bigg ] +\zeta (t)\nonumber \\&\qquad -\,m\int \nolimits _0^t \Bigg (\int \nolimits _{ms}^{\infty }\frac{J_1(u)du}{u}\Bigg )\zeta (t-s)ds\Bigg ) ={\widetilde{F}}(\zeta (t))-\lambda (t), \end{aligned}$$
(2.30)

since \(\zeta (t)\) solves (2.29). Hence, the second equation of (2.5) with \({\widetilde{F}}\) holds.

Let us prove the uniqueness of this solution. Suppose that \(({\widetilde{\psi }}_{S}(\cdot ,t),{\widetilde{\zeta }}(t))\) with \(\psi _S(\cdot ,t)+\varphi _g(\cdot ,t)\in C([0,\tau ],{\mathscr {D}}_{{\widetilde{F}}})\) and \(\zeta \in C^1[0,\tau ]\otimes {{\mathbb {C}}}^4\) is another solution to (2.5). Then \({\widetilde{\psi }}_{S}(x,t)\) satisfies the first equation of (2.5) with the source \(D_m^{-1}{\widetilde{\zeta }}(t)\delta (x)\) and is given by formulas (2.17)–(2.19) with \({\widetilde{\zeta }}(t)\) instead of \(\zeta (t)\). Hence, Propositions 2.4 and 2.5 and the second equation of (2.5) imply that \({\widetilde{\zeta }}(t)\) solves the Cauchy problem (2.29). The uniqueness of the solution of (2.29) implies that \({\widetilde{\zeta }}(t)=\zeta (t)\). Hence, \({\widetilde{\psi }}_S=\psi _S\).

It remains to show that the function

$$\begin{aligned} \psi _{reg}^{-}(x,t)= & {} \psi _S(x,t)+\varphi _{g}(x,t)-\zeta (t)g(x)\\= & {} \psi (x,t)-\psi _f(x,t)-\zeta (t)g(x)=\psi _{reg}(x,t)-\psi _f(x,t) \end{aligned}$$

satisfies

$$\begin{aligned} \psi _{reg}^{-}(\cdot ,t)\in C([0,\tau ], H^{\frac{3}{2}-}({{\mathbb {R}}}^3)). \end{aligned}$$
(2.31)

Indeed, \(\psi _{reg}^{-}(x,t)\) is a solution to

$$\begin{aligned} i{\dot{\psi }}_{reg}^{-}(x,t)=D_m\psi _{reg}^{-}(x,t)-i{\dot{\zeta }}(t) g(x) \end{aligned}$$
(2.32)

with zero initial data. Hence, \(\psi _{reg}^{-}(x,t)=(-i\partial _t -D_m){\mathrm{w}}(x,t)\), where where \({\mathrm{w}}(x,t)\) is the solution to

$$\begin{aligned} \ddot{\mathrm{w}}(x,t) = (\Delta -m^2){\mathrm{w}}(x,t)-i{\dot{\zeta }}(t) g(x),\quad {\mathrm{w}}(x,0)=0,\quad \dot{\mathrm{w}}(x,0)=0. \end{aligned}$$

Then, for (2.31) we need to show that

$$\begin{aligned} {\mathrm{w}}(\cdot ,t)\in C([0,\tau ], H^{\frac{5}{2}-\varepsilon }({{\mathbb {R}}}^3)),\quad \dot{\mathrm{w}}(\cdot ,t)\in C([0,\tau ], H^{\frac{3}{2}-\varepsilon }({{\mathbb {R}}}^3)),\quad {\mathrm{for}}~ \mathrm{any}\quad \varepsilon >0.\nonumber \\ \end{aligned}$$
(2.33)

Applying the Fourier transform, we obtain

$$\begin{aligned} \widehat{{\mathrm{w}}} (\xi ,t)= & {} -i\int \nolimits _0^t\frac{\sin (s\sqrt{\xi ^2+m^2})}{(\xi ^2+m^2)^{\frac{3}{2}}} ~{\dot{\zeta }}(t-s)ds,\\ \widehat{\dot{\mathrm{w}}}(\xi ,t)= & {} -i\int \nolimits _0^t\frac{\cos (s\sqrt{\xi ^2+m^2})}{\xi ^2+m^2} ~{\dot{\zeta }}(t-s)ds. \end{aligned}$$

Hence, integration by parts gives

$$\begin{aligned} (\xi ^2+m^2)^{\frac{5}{4}-\frac{\varepsilon }{2}}\,\widehat{{\mathrm{w}}} (\xi ,t)= & {} -i\int \nolimits _0^t\frac{\sin (s\sqrt{\xi ^2+m^2})}{(\xi ^2+m^2)^{\frac{1}{4}+\frac{\varepsilon }{2}}} ~{\dot{\zeta }}(t-s)ds\\= & {} i\Big (\frac{{\dot{\zeta }}(0)\cos (s\sqrt{\xi ^2+m^2})}{(\xi ^2+m^2)^{\frac{3}{4}+\frac{\varepsilon }{2}}}\\&-\,\frac{{\dot{\zeta }}(t)}{(\xi ^2+m^2)^{\frac{3}{4}+\frac{\varepsilon }{2}}} -\int \nolimits _0^t\frac{\cos (s\sqrt{\xi ^2+m^2})}{(\xi ^2+m^2)^{\frac{3}{4}+\frac{\varepsilon }{2}}} ~\ddot{\zeta }(t-s)ds\Big ), \end{aligned}$$

where \(\ddot{\zeta }\in C[0,\tau ]\otimes {{\mathbb {C}}}^4\) by (2.7), (2.14) and (2.29). Therefore,

$$\begin{aligned} |(\xi ^2+m^2)^{\frac{5}{4}-\frac{\varepsilon }{2}}\,\widehat{{\mathrm{w}}_j} (\xi ,t)|\le \frac{C(1+\tau )\Vert \zeta _j\Vert _{C^2[0,\tau ]}}{(\xi ^2+m^2)^{\frac{3}{4}+\frac{\varepsilon }{2}}}, \quad t\in [0,\tau ],\quad j=1, \ldots ,4. \end{aligned}$$

Similarly,

$$\begin{aligned} |(\xi ^2+m^2)^{\frac{3}{4}-\frac{\varepsilon }{2}}\,\widehat{\dot{\mathrm{w}}_j} (\xi ,t)|\le \frac{{C(1+\tau )\Vert \zeta _j\Vert _{C^2[0,\tau ]}}}{(\xi ^2+m^2)^{\frac{3}{4}+\frac{\varepsilon }{2}}}, \quad t\in [0,\tau ],\quad j=1, \ldots ,4. \end{aligned}$$

Hence, (2.33) follows.

(ii) Evidently, the linear map \(\zeta (\cdot )\rightarrow \psi _{reg}^{-}(\cdot ,\cdot )\) is continuous \(C^2[0,\tau ]\otimes {{\mathbb {C}}}^4\rightarrow C([0,\tau ], H^{\frac{3}{2}-})\). \(\square \)

Corollary 2.8

It is obvious that (2.30) can be rewritten as

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0+}\lim \limits _{x\rightarrow 0}K_m^{\varepsilon }\psi _{reg}(x,t)={\widetilde{F}}(\zeta (t)),\quad t\in [0,\tau ]. \end{aligned}$$
(2.34)

2.3 Conservation law and a priori bound

Now we prove the conservation law (2.2) on the interval \([0,\tau ]\).

Lemma 2.9

Let conditions (2.26)–(2.28) hold, and let \(\psi (t)\in {\mathscr {D}}_{{\widetilde{F}}}\), \(t\in [0,\tau ]\), be a solution to (1.1). Then

$$\begin{aligned} {\mathscr {H}}_{{\widetilde{F}}}(\psi (\cdot ,t))=\Vert D_m\psi _{reg}(\cdot ,t)\Vert ^2+{\widetilde{U}}(\zeta (t))={\mathrm{const}},\quad t\in [0,\tau ]. \end{aligned}$$
(2.35)

Proof

Equations (2.32) and (2.34) imply for any \(t\in (0,\tau ]\)

$$\begin{aligned}&\lim \limits _{\varepsilon \rightarrow 0}\frac{d}{dt}\Vert K_m^{\varepsilon }D_m\psi _{reg}\Vert ^2\nonumber \\&\quad =\lim \limits _{\varepsilon \rightarrow 0}\Big [\langle K_m^{\varepsilon }D_m{\dot{\psi }}_{reg},\, K_m^{\varepsilon }D_m\psi _{reg}\rangle +\langle K_m^{\varepsilon }D_m\psi _{reg}, \,K_m^{\varepsilon }D_m{\dot{\psi }}_{reg}\rangle \Big ]\nonumber \\&\quad =\lim \limits _{\varepsilon \rightarrow 0}\Big [\langle -iK_m^{\varepsilon }D_m^2\psi _{reg}-K_m^{\varepsilon }D_m{\dot{\zeta }} g,\,K_m^{\varepsilon }D_m\psi _{reg}\rangle \nonumber \\&\qquad +\langle K_m^{\varepsilon }D_m\psi _{reg}, \,-iK_m^{\varepsilon }D_m^2\psi _{reg}-K_m^{\varepsilon }D_m {\dot{\zeta }} g\rangle \Big ]\nonumber \\&\quad =\lim \limits _{\varepsilon \rightarrow 0}\Big [-\langle D_m^2{\dot{\zeta }} g,\,K_m^{2\varepsilon }\psi _{reg}\rangle -\langle K_m^{2\varepsilon }\psi _{reg},\, D_m^2{\dot{\zeta }} g\rangle \Big ]\nonumber \\&\quad =\lim \limits _{\varepsilon \rightarrow 0}\Big [-{\dot{\zeta }}\cdot \langle \delta (x),K_m^{2\varepsilon }\psi _{reg}\rangle -{\overline{{\dot{\zeta }}}}\cdot \langle K_m^{2\varepsilon }\psi _{reg}, \,\delta (x)\rangle \Big ]\nonumber \\&\quad =-{\dot{\zeta }} \cdot \overline{{\widetilde{F}}}(\zeta )-{\overline{{\dot{\zeta }}}}\cdot {{\widetilde{F}}}(\zeta )=-2\frac{d}{dt}{\widetilde{U}}(\zeta ). \end{aligned}$$
(2.36)

Here the scalar product \(\langle K_m^{\varepsilon }D_m^2\psi _{reg},K_m^{\varepsilon }D_m\psi _{reg}\rangle \) exists since \(K_m^{\varepsilon }\psi _{reg}(\cdot ,t)=K_m^{\varepsilon }\psi _{reg}^{-}(\cdot ,t)+K_m^{\varepsilon }\psi _{f}(\cdot ,t)\in C([0,\infty ),H^{3/2})\) for any \(\varepsilon >0\) due to (2.6) and (2.31). Moreover, for any \(\nu >0\) and \(\varepsilon \ge 0\)

$$\begin{aligned} \sup \limits _{t\in [0,\tau ]}\Vert K_m^{\varepsilon }D_m\psi _{reg}(\cdot ,t)\Vert _{H^{1/2-\nu }}<\infty . \end{aligned}$$

Hence, uniformly in \(t\in [0,\tau ]\), we have

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0}\Vert K_m^{\varepsilon }D_m\psi _{reg}(t)\Vert =\Vert D_m\psi _{reg}(t)\Vert . \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{d}{dt}\Vert D_m\psi _{reg}(\cdot ,t)\Vert ^2= & {} \frac{d}{dt}\lim \limits _{\varepsilon \rightarrow 0}\Vert K_m^{\varepsilon }D_m\psi _{reg}(\cdot ,t)\Vert ^2 \\= & {} \lim \limits _{\varepsilon \rightarrow 0}\frac{d}{dt}\Vert K_m^{\varepsilon }D_m\psi _{reg}(\cdot ,t)\Vert ^2=-2\frac{d}{dt}{\widetilde{U}}(\zeta ),\quad \tau \in [0,\tau ]. \end{aligned}$$

in the sense of distributions. Then (2.35) follows. \(\square \)

Corollary 2.10

The following identity holds

$$\begin{aligned} {\widetilde{U}}(\zeta (t))=U(\zeta (t)), \quad t\in [0,\tau ]. \end{aligned}$$
(2.37)

Proof

First note that

$$\begin{aligned} {\mathscr {H}}_{F}(\psi (0))\ge U(\zeta _{0})\ge b|\zeta _{0}|^2-a. \end{aligned}$$

Therefore, \(|\zeta _0|\le \Lambda (\psi (0))\) and then \({\widetilde{U}}(\zeta _0)=U(\zeta _0)\), \({\mathscr {H}}_{{\widetilde{F}}}(\psi (0))={\mathscr {H}}_{F}(\psi (0))\). Further,

$$\begin{aligned} {\mathscr {H}}_{{\widetilde{F}}}(\psi (t))\ge {\widetilde{U}}(\zeta (t))\ge b|\zeta (t)|^2-a,\quad t\in [0,\tau ]. \end{aligned}$$

Hence, (2.35) implies the a priory bound

$$\begin{aligned} |\zeta (t)|\le & {} \sqrt{({\mathscr {H}}_{{\widetilde{F}}}(\psi (t))+a)/b}=\sqrt{({\mathscr {H}}_{{\widetilde{F}}}(\psi (0))+a)/b} \nonumber \\= & {} \sqrt{({\mathscr {H}}_{F}(\psi (0))+a)/b}=\Lambda (\psi (0)),\quad t\in [0,\tau ]. \end{aligned}$$
(2.38)

Therefore, (2.37) follows by (2.26). \(\square \)

2.4 Bootstrap argument

Identity (2.37) implies that we can replace \({\widetilde{F}}\) by F in Proposition 2.6 and in Lemma 2.9.

Now we can finish the proof of Theorem 2.1. The unique solution \(\psi _{free}(x,t)\) to the free Dirac equation with initial function \(f(x)+\zeta _0 g(x)\) exists for \(t\in [0,\infty )\) (see Formula (2.13)). At the same time, the solution \(\zeta (t)\) to equation (2.24) exists for \(0\le t\le \tau \), where the time span \(\tau \) in Lemma 2.7 depends only on \(\Lambda (\psi (0))\). This solution defines the function \(\psi _S(x,t)\) by formulas (2.17)–(2.19) so that \((\psi _S(x,t),\zeta (t)\) is the unique solution to the system (2.5) on the interval \([0,\tau ]\). The bound (2.38) at \(t=\tau \) allows us to extend the solution \(\zeta (t)\) to the time interval \([\tau , 2\tau ]\), and formulas (2.17)–(2.19) define \(\psi _{S}(x,t)\) on the interval \([0,2\tau ]\) then. We proceed by induction to obtain the solution for all \(t\ge 0\).

3 Solitary waves and main theorem

We assume that

$$\begin{aligned} U(\zeta )= & {} \sum \limits _{j=1}^4 U_j(\zeta _j),~~{\mathrm{where}} \quad U_j(\zeta _j)=\sum \limits _{n=0}^{N_j}u_{n,j}|\zeta _j|^{2n},\quad \nonumber \\ u_{n,j}\in & {} {{\mathbb {R}}}, \quad u_{N_j,j}>0, \quad N_j\ge 2,\quad j=1, \ldots ,4. \end{aligned}$$
(3.1)

This assumption guarantees the bound (1.4), and it is crucial in our argument: it allow us to apply the Titchmarsh convolution theorem. Equality (3.1) implies that

$$\begin{aligned} F_j(\zeta _j)=\partial _{{\overline{\zeta }}_j} U_j(\zeta _j)=a_j(|\zeta _j|^2)\zeta _j,\quad j=1, \ldots ,4, \end{aligned}$$
(3.2)

where

$$\begin{aligned} a_j(|\zeta _j|^2):\,=\sum \limits _{n=1}^{N_j}2nu_{n,j}|\zeta _j|^{2n-2}. \end{aligned}$$
(3.3)

Definition 3.1

  1. (i)

    The solitary waves of equation (1.1) are solutions of the form

    $$\begin{aligned} \psi (x,t)=\sum \limits _k\psi _{\omega _k}(x)e^{-i\omega _k t},\quad \omega _k\in {{\mathbb {R}}},\quad \omega _{l}\not =\omega _{j},\quad l\not =j, \quad \psi _{\omega _k}\in L^2({{\mathbb {R}}}^3),\nonumber \\ \end{aligned}$$
    (3.4)

    where the sum has a finite number of terms.

  2. (ii)

    The solitary manifold is the set: \({\mathscr {S}}=\left\{ \sum \limits _k\psi _{\omega _k}{\mathrm{:}}\ ~~\omega _k\in {{\mathbb {R}}},~~\omega _{l}\not =\omega _{j},~~ l\not =j\right\} \).

Below we show that the number of nonzero terms in (3.4) does not exceed 4. From (3.2) it follows that the set \({\mathscr {S}}\) is invariant under multiplication by \(e^{i\theta }\), \(\theta \in {{\mathbb {R}}}\). Note that there is a zero solitary wave, since \(F(0)=0\).

Now we derive more precise representation for solitary waves.

Proposition 3.2

Assume that \(F(\zeta )\) satisfies (3.2). Then nonzero solitary waves are given by

$$\begin{aligned} \psi (x,t)=\phi _\Omega (x,t)+iD_m^{-1}{\dot{\phi }}_\Omega (x,t),\quad \end{aligned}$$
(3.5)

where \(\Omega =(\omega _1, \ldots ,\omega _4)\) with \(|\omega _j|<m\),

$$\begin{aligned} \phi _\Omega (x,t)= & {} (\phi _{\omega _1}(x)e^{-i\omega _1t}\, \ldots ,\phi _{\omega _4}(x)e^{-i\omega _4 t}), \end{aligned}$$
(3.6)
$$\begin{aligned} \phi _{\omega _j}(x)= & {} C_j\frac{e^{-\sqrt{m^2-\omega _j^2}|x|}}{4\pi |x|},\quad j=1, \ldots ,4, \end{aligned}$$
(3.7)

and \(C_j=C_j(\omega _j)\in {{\mathbb {R}}}\) are solutions to

$$\begin{aligned} (m-\sqrt{m^2-\omega _j^2})(1+\sigma _j\frac{m}{\omega _j})=4\pi a_j(|C_j|^2),\quad j=1, \ldots ,4 \end{aligned}$$
(3.8)

with

$$\begin{aligned} \sigma _j=\left\{ \begin{array} {rr} 1,\quad j=1,2, \\ -1,\quad j=3,4 .\end{array}\right. \end{aligned}$$
(3.9)

Remark 3.3

In (3.5) some \(\omega _j\) may be identical in contrast to (3.4).

Proof

We look for a solution \(\psi (x,t)\) to (1.1) in the form (3.4):

$$\begin{aligned} \psi (x,t)=\sum \limits _k\psi _{\omega _k}(x)e^{-i\omega _k t},\quad {\mathrm{where}}\quad \omega _k<\omega _{k+1}. \end{aligned}$$
(3.10)

Consider the function

$$\begin{aligned} \chi (x,t):=\psi (x,t)-iD_m^{-1}{\dot{\psi }}(x,t)=\sum \limits _k\chi _{\omega _k}(x)e^{-i\omega _k t}, \end{aligned}$$
(3.11)

where

$$\begin{aligned} \chi _{\omega _k}=\psi _{\omega _k}-\omega _k D_m^{-1}\psi _{\omega _k}=D_m^{-1}(D_m-\omega _k)\psi _{\omega _k}. \end{aligned}$$

Hence,

$$\begin{aligned} \psi _{\omega _k}= & {} D_m (D_m-\omega _k)^{-1}\chi _{\omega _k}=D_m (D_m+\omega _k)(D_m^2-\omega _k^2)^{-1}\chi _{\omega _k}\nonumber \\= & {} \chi _{\omega _k}+(\omega _k^2+\omega _k D_m)(D_m^2-\omega _k^2)^{-1}\chi _{\omega _k}. \end{aligned}$$
(3.12)

Further, (3.11) implies that

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

by the first equation of (1.1). Hence,

$$\begin{aligned} \sum \limits _k e^{-i\omega _k t}D_m^2 \chi _{\omega _k}(x)=\zeta (t)\delta (x). \end{aligned}$$

by (3.11). Therefore,

$$\begin{aligned} \chi _{\omega _k}(x)=\overline{C}_{k}\frac{e^{-m|x|}}{4\pi |x|},\qquad \zeta (t)=\sum \limits _{k}\overline{C}_{k}e^{-i\omega _k t}. \end{aligned}$$
(3.13)

where \(\overline{C}_k:=(C_{k1}, \ldots , C_{k4})\). Now we derive the explicit formulas for \(\psi _{\omega _k}(x)\), using (3.12) and (3.13) only. One has

$$\begin{aligned} (D_m^2-\omega _k^2)^{-1}\,\frac{e^{-m|x|}}{4\pi |x|}= & {} \frac{1}{(2\pi )^3}\int \nolimits _{{{\mathbb {R}}}^3}\,\frac{e^{-i\xi x} d^3\xi }{(\xi ^2+m^2)(\xi ^2+m^2-\omega _k^2)}\\= & {} \frac{1}{(2\pi )^3\omega _k^2}\int \nolimits _{R^3}\Big (\frac{e^{-i\xi x}}{\xi ^2+m^2-\omega _k^2}-\frac{ e^{-i\xi x} }{\xi ^2+m^2}\Big ) d^3\xi \\= & {} \frac{1}{\omega _k^2}\Big (\frac{e^{-\sqrt{m^2-\omega _k^2}|x|}}{4\pi |x|}-\frac{e^{-m|x|}}{4\pi |x|}\Big ). \end{aligned}$$

Moreover,

$$\begin{aligned} D_m(D_m^2-\omega _k^2)^{-1}\,\frac{e^{-m|x|}}{4\pi |x|}= & {} D_m^{-1}(D_m^2-\omega _k^2)^{-1}D_m^2\,\frac{e^{-m|x|}}{4\pi |x|} =D_m^{-1}(D_m^2-\omega _k^2)^{-1}\delta (x)\\= & {} D_m^{-1}\,\frac{e^{-\sqrt{m^2-\omega _k^2}|x|}}{4\pi |x|}. \end{aligned}$$

Substituting this into (3.12), we obtain by (3.13)

$$\begin{aligned} \psi _{\omega _k}(x)=\varphi _{\omega _k}(x)+\omega _k D_m^{-1}\varphi _{\omega _k}(x), \quad \end{aligned}$$
(3.14)

where we denote

$$\begin{aligned} \varphi _{\omega _k}(x):=\overline{C}_{k}\frac{e^{-\sqrt{m^2-\omega _k^2}|x|}}{4\pi |x|}. \end{aligned}$$

Now we are able to find coefficients \(C_{kj}\). The second equation of (1.1) together with (3.4) and (3.14) imply

$$\begin{aligned}&\lim \limits _{\epsilon \rightarrow 0+} \lim \limits _{x\rightarrow 0}K_m^{\epsilon }\sum \limits _{k}e^{-i\omega _k t}\Big ( \varphi _{\omega _k}(x) +\omega _kD_m^{-1}\varphi _{\omega _k}(x) -\overline{C}_kg(x)\Big )\nonumber \\&\quad = \lim \limits _{\epsilon \rightarrow 0+} \lim \limits _{x\rightarrow 0}K_m^{\epsilon }\sum \limits _{k}e^{-i\omega _k t}\Big (\varphi _{\omega _k}(x) -\overline{C}_kg(x) +\omega _km\beta D_m^{-2}\varphi _{\omega _k}(x)\nonumber \\&\qquad -\,i\omega _k\alpha \cdot \nabla D_m^{-2}\varphi _{\omega _k}(x)\Big ) = F(\sum \limits _{k}\overline{C}_ke^{-i\omega _k t}). \end{aligned}$$
(3.15)

Note, that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0+}\lim _{x\rightarrow 0}K_m^{\epsilon }\Big (\varphi _{\omega _{k}}(x)-\overline{C}_{k}g(x)\Big )= & {} \overline{C}_{k}\lim _{\epsilon \rightarrow 0+}\lim _{x\rightarrow 0}K_m^{\epsilon }\Big (\frac{e^{-\sqrt{m^2-\omega _k^2}|x|}}{4\pi |x|}-\frac{e^{-m|x|}}{4\pi |x|}\Big )\nonumber \\= & {} \frac{\overline{C}_{k}}{2\pi ^2}\int \nolimits _0^{\infty } \Big (\frac{r^2}{r^2+m^2-\omega _k^2}-\frac{r^2}{r^2+m^2}\Big ) dr \nonumber \\= & {} \frac{\overline{C}_{k}}{2\pi ^2}\int \nolimits _0^{\infty } \Big (\frac{m^2}{r^2+m^2}-\frac{m^2-\omega _k^2}{r^2+m^2-\omega _k^2}\Big )dr\nonumber \\= & {} \frac{\overline{C}_{k}}{4\pi }(m-\sqrt{m^2-\omega _k^2}). \end{aligned}$$
(3.16)

Similarly,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0+}\lim _{x\rightarrow 0}K_m^{\epsilon }D_m^{-2}\varphi _{\omega _k}(x)= & {} \frac{\overline{C}_k}{2\pi ^2}\int \nolimits _0^{\infty } \frac{r^2 dr}{(r^2+m^2)(r^2+m^2-\omega _k^2)}\nonumber \\= & {} \frac{\overline{C}_{k}}{2\pi ^2\omega _k^2}\int \nolimits _0^{\infty } \Big (\frac{m^2}{r^2+m^2}-\frac{m^2-\omega _k^2}{r^2+m^2-\omega _k^2}\Big )dr\nonumber \\= & {} \frac{\overline{C}_{k}}{4\pi \omega _k^2}\big (m-\sqrt{m^2-\omega _k^2}\big ). \end{aligned}$$
(3.17)

Moreover,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0+}\lim \limits _{x\rightarrow 0}K_m^{\epsilon }\nabla _n D_m^{-2}\varphi _{\omega _k}(x)= & {} \frac{\overline{C}_{k}}{(2\pi )^3}\lim _{\epsilon \rightarrow 0+} \int \nolimits _{{{\mathbb {R}}}^3}\frac{\xi _n \xi ^2d^3\xi }{(\xi ^2+m^2-\omega _k^2)(\xi ^2+m^2)^{1+\epsilon }}=0,\nonumber \\ n= & {} 1,2,3. \end{aligned}$$
(3.18)

Substituting (3.16)–(3.18) into (3.15), we get

$$\begin{aligned}&\frac{1}{4\pi }\sum \limits _{k}C_{kj}e^{-i\omega _k t}\Bigg (m-\sqrt{m^2-\omega _k^2}+ \sigma _j\frac{m}{\omega _k}\bigg (m-\sqrt{m^2-\omega _k^2}\bigg )\Bigg )\nonumber \\&\quad =a_j (|\sum \limits _{k}C_{kj}e^{-i\omega _k t}|^2)\sum \limits _{k}C_{kj}e^{-i\omega _k t},\quad j=1, \ldots ,4. \end{aligned}$$
(3.19)

Lemma 3.4

Let \(C_{kj}\) be solutions to (3.19). Then for each fixed \(j=1, \ldots ,4\) only one of the coefficients \(C_{kj}\) is nonzero.

Proof

It suffices to consider the case \(j=1\) only. We should prove that may be no more than one nonzero \(c_k:=C_{k1}\). Assume, to the contrary, that \(c_{k_1},c_{k_2}, \ldots ,c_{k_n}\ne 0\) with \(k_1<k_2< \cdots <k_n\), where \(2\le n\). Then \(\omega _{k_1}<\omega _{k_2}< \cdots <\omega _{k_n}\) by (3.10). Denote \(\delta _{l,p}=\omega _{k_p}-\omega _{k_l}>0\), \(1\le l<p\le n\). Evidently, \(\delta :=\delta _{1,n}=\max \limits _{1\le l<p\le n} \delta _{l,p}\). Then

$$\begin{aligned} |\sum \limits _{k} c_{k}e^{-i\omega _k t}|^2=a+be^{i\delta t}+\overline{b}e^{-i\delta t} +\sum \limits _{(l,p)\not =(1,n)} (b_{l,p}e^{i\delta _{l,p} t}+\overline{b}_{l,p}e^{-i\delta _{l,p} t}) \end{aligned}$$

with some \(a>0\) and \(b\ne 0\). Hence, (3.3) implies

$$\begin{aligned} a_1 (|\sum \limits _{k} c_{k}e^{-i\omega _k t}|^2) =de^{i(N_1-1)\delta t}+\overline{d}e^{-i(N_1-1)\delta t}+R, \end{aligned}$$

where R consists of terms of the type \(Ce^{i\sigma t}\) with \(|\sigma |< (N_1-1)\delta \). Note that \(d\ne 0\) since \(a_1\) is a polynomial of degree \(N_1-1\ge 1\) due to (3.1) and (3.3). Now the right hand side of (3.19) contains the terms \(e^{-i[\omega _{k_1}t-(N_j-1)\delta ] t}\) and \(e^{-i[\omega _{k_n}t+(N_j-1)\delta ] t}\) with nonzero coefficients, which are absent on the left hand side. This contradiction proves the lemma. \(\square \)

The lemma and formulas (3.4) and (3.14) imply

$$\begin{aligned} \psi _j(x,t)= & {} \sum \limits _{k}\psi _{\omega _k,j}(x)\,e^{-i\omega _k t}= \sum \limits _{k}\varphi _{\omega _k,j}(x)e^{-i\omega _k t}+\Bigg (\sum \limits _{k}D_m^{-1}\omega _{k}\varphi _{\omega _k,j}(x)\,e^{-i\omega _{k} t}\Bigg )_j\\ {}= & {} \varphi _{\omega _{k_j},j}(x)\,e^{-i\omega _{k_j}t}+(D_m^{-1}\pi (x,t))_j,\quad j=1, \ldots ,4, \end{aligned}$$

where

$$\begin{aligned} \pi _j(x,t)=\omega _{k_j}\varphi _{\omega _{k_j},j}(x)\,e^{-i\omega _{k_j} t}. \end{aligned}$$

We can assume that \(k_j=j\). Then \(C_{k_jj}=C_{jj}\), \(\omega _{k_j}=\omega _j\), and \(\varphi _{\omega _j,j}(x)=\phi _{\omega _j}(x)\) from (3.7) with \(C_j=C_{jj}\). Then (3.5) follows. It remains to note that equation (3.19) in the case when \(C_{jk}=0\) for \(k\not =j\) is equation (3.8) for \(C_j=C_{jj}\). Proposition is completely proved. \(\square \)

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

Lemma 3.5

Let F satisfy (3.2)–(3.3) with \(M_j=-u_{1,j}>0\), where \(j\in \{1;2;3;4\}\). Then there exists an open subset \(I(M_j)\subset (-m,m)\) such that for any \(\omega _j\in I(M_j)\) the jth equation of (3.8) has nonzero solutions \(C_j=C_j(\omega _j)\). Moreover, \(I(M_j)=(-m,m)\) if \(M_j>m/(32\pi ^2)\).

We prove this lemma in Appendix D.

Now the solitary manifold \({\mathscr {S}}\) reads

$$\begin{aligned} {\mathscr {S}}=\left\{ \Phi _{\Omega }+D_m^{-1}\Psi _{\Omega }{\mathrm{:}}\ ~~\Omega =(\omega _1, \ldots ,\omega _4)\in {{\mathbb {R}}}^4\right\} , \end{aligned}$$
(3.20)

where

$$\begin{aligned} \Phi _{\Omega }(x)=(\phi _{\omega _1}(x), \ldots ,\phi _{\omega _4}(x)),\qquad \Psi _{\Omega }(x)=(\omega _1\phi _{\omega _1}(x), \ldots ,\omega _4\phi _{\omega _4}(x)). \end{aligned}$$

Our main result is the following theorem.

Theorem 3.6

Let (3.1) be satisfied, and let \(\psi (0):=\psi (x,0)=f(x)+\zeta _0\) with \(f\in H^{\frac{5}{2}+}\). Then the solution \(\psi (x,t)\) to (1.1) with initial function \(\psi (0)\) converges to solitary manifold \({\mathscr {S}}\) in the space \(L^{2}_{loc}({{\mathbb {R}}}^3)\):

$$\begin{aligned} \lim _{t\rightarrow \pm \infty } \mathrm{dist}_{L^{2}_{loc}({{\mathbb {R}}}^3)}(\psi (\cdot ,t),{\mathscr {S}})=0. \end{aligned}$$
(3.21)

It suffices to prove Theorem 3.6 for \(t\rightarrow +\infty \).

4 Dispersive component

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

Lemma 4.1

(cf. [22, Proposition 4.3]) Let \(\psi _f(x,t)\) be a solution to the free Dirac equation with initial function \(f\in H^2({{\mathbb {R}}}^3)\). Then \(\forall R>0\),

$$\begin{aligned} \Vert \psi _f(\cdot ,t)\Vert _{H^2(B_R)}\rightarrow 0,\qquad t\rightarrow \infty , \end{aligned}$$
(4.1)

where \(B_R\) is the ball of radius R.

Corollary 4.2

From (4.1) immediately follows that

$$\begin{aligned} \lambda (t)=\psi _{f}(0,t)\rightarrow 0,\quad t \rightarrow \infty . \end{aligned}$$
(4.2)

Now consider

$$\begin{aligned} \varphi _{g}^{+}(x,t)=\varphi _{g}(x,t)+iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)=\zeta _0(g(x)-\gamma (x,t)), \end{aligned}$$
(4.3)

where \(\varphi _{g}\) is the solution free Dirac equation with initial function \(\zeta _0g\), given by (2.13).

Lemma 4.3

\(\varphi ^{+}_{g}(x,t)=\zeta _0(g(x)-\gamma (x,t))\) decays in \(H^2_{loc}\) seminorms. That is, \(\forall R>0\)

$$\begin{aligned} \Vert \varphi ^{+}_{g}(\cdot ,t)\Vert _{H^2(B_R)}\rightarrow 0,\qquad t\rightarrow \infty . \end{aligned}$$
(4.4)

Proof

According to (2.12) the function \(h(x,t):=\gamma (x,t)-g(x)\) is the solutions to

$$\begin{aligned} \ddot{h}(x,t)=(\Delta -m^2)h(x,t), \qquad (h(x,t),\dot{h}(x,t))\vert _{_{t=0}}=(-g, 0). \end{aligned}$$
(4.5)

Then (4.4) follows by Lemma 3.3 of [27]. \(\square \)

In conclusion, let us show that

$$\begin{aligned} \varphi _{g}(\cdot ,t)\in C_{b}([0,\infty ), L^2). \end{aligned}$$
(4.6)

Indeed, the energy conservation for equation (4.5) implies that

$$\begin{aligned} (h(\cdot ,t), \dot{h}(\cdot ,t))\in C_{b}([0,\infty ), L^2({{\mathbb {R}}}^3)\oplus H^{-1}({{\mathbb {R}}}^3)). \end{aligned}$$

Hence,

$$\begin{aligned} (\gamma (\cdot ,t),{\dot{\gamma }}(\cdot ,t))=(h(\cdot ,t), \dot{h}(\cdot ,t))+(g(\cdot ),0)\in C_{b}([0,\infty ), L^2({{\mathbb {R}}}^3)\oplus H^{-1}({{\mathbb {R}}}^3)). \end{aligned}$$

Then (4.6) follows by (4.3).

5 Complex Fourier–Laplace transform

The conservation low (2.2) and a priory bound (2.3) imply that \(\psi (\cdot ,t)\in C_{b}([0,\infty ), L^2)\). Hence, (4.6) implies

$$\begin{aligned} \psi _S(\cdot ,t)=\psi (\cdot ,t)-\psi _f(\cdot ,t)-\varphi _{g}(\cdot ,t)\in C_{b}([0,\infty ), L^2). \end{aligned}$$
(5.1)

Let us analyze the Fourier–Laplace transform of \(\psi _S(x,t)\):

$$\begin{aligned} {\widetilde{\psi }}_S(x,\omega )={\mathscr {F}}_{t\rightarrow \omega }[\theta (t)\psi _S(x,t)] :=\int _ 0^\infty e^{i\omega t}\psi _S(x,t)\,dt,\quad \omega \in {{\mathbb {C}}}^{+},\quad x\in {{\mathbb {R}}}^3,\nonumber \\ \end{aligned}$$
(5.2)

where \({{\mathbb {C}}}^{+}:=\{z\in {{\mathbb {C}}}:\;{\mathrm{Im\,}}z>0\}\). Note that \({\widetilde{\psi }}_S(\cdot ,\omega )\) is an \(L^2\)-valued analytic function of \(\omega \in {{\mathbb {C}}}^+\) due to (5.1). Equation (2.16) implies that

$$\begin{aligned} -\omega {\widetilde{\psi }}_S(x,\omega )=D_m{\widetilde{\psi }}_S(x,\omega ) -D_m^{-1}{\widetilde{\zeta }}(\omega )\delta (x),\quad \omega \in {{\mathbb {C}}}^{+},\quad x\in {{\mathbb {R}}}^3, \end{aligned}$$
(5.3)

where \({\widetilde{\zeta }}(\omega )\) is the Fourier–Laplace transform of \(\zeta (t)\):

$$\begin{aligned} {\widetilde{\zeta }}(\omega )={\mathscr {F}}_{t\rightarrow \omega }[\theta (t)\zeta (t)]= \int _{0}^\infty e^{i\omega t}\zeta (t)\,dt. \end{aligned}$$

Applying the Fourier transform to (5.3), we get

$$\begin{aligned} \widehat{{\widetilde{\psi }}}_{S}(\xi ,\omega )= & {} \frac{(\alpha \cdot \xi +m\beta ){\widetilde{\zeta }}(\omega )}{(\alpha \cdot \xi +m\beta +\omega )(\xi ^2+m^2)} =\Big (\frac{1}{\xi ^2+m^2}-\frac{\omega }{(\alpha \cdot \xi +m\beta +\omega )(\xi ^2+m^2)}\Big ){\widetilde{\zeta }}(\omega )\nonumber \\= & {} \Big (\frac{1}{\xi ^2+m^2}+\frac{\omega ^2}{(\xi ^2+m^2-\omega ^2)(\xi ^2+m^2)} -\frac{\omega (\alpha \cdot \xi +m\beta )}{(\xi ^2+m^2-\omega ^2)(\xi ^2+m^2)}\Big ){\widetilde{\zeta }}(\omega )\nonumber \\= & {} \Big (\frac{1}{\xi ^2+m^2-\omega ^2} +\frac{\alpha \cdot \xi +m\beta }{\omega }\Big (\frac{1}{\xi ^2+m^2}-\frac{1}{\xi ^2+m^2-\omega ^2}\Big )\Big ) {\widetilde{\zeta }}(\omega ), \nonumber \\&\xi \in {{\mathbb {R}}}^3,\quad \omega \in {{\mathbb {C}}}^{+}. \end{aligned}$$
(5.4)

Denote

$$\begin{aligned} \varkappa (\omega )=\sqrt{\omega ^2-m^2}, \qquad {\mathrm{Im\,}}\varkappa (\omega )>0,\qquad \omega \in {{\mathbb {C}}}^{+}. \end{aligned}$$
(5.5)

The function \(\varkappa (\omega )\) is analytic on \({{\mathbb {C}}}^{+}\), and \({\widetilde{\psi }}_S(x,\omega )\) is given by

$$\begin{aligned} {\widetilde{\psi }}_S(x,\omega )= & {} V(x,\omega ){\widetilde{\zeta }}(\omega ), \quad {\mathrm{where}}\quad \nonumber \\ V(x,\omega )= & {} \frac{e^{i\varkappa (\omega )|x|}}{4\pi |x|}+\frac{1}{\omega }D_m\Big (\frac{e^{-m|x|}}{4\pi |x|}-\frac{e^{i\varkappa (\omega )|x|}}{4\pi |x|}\Big ), \quad \quad \omega \in {{\mathbb {C}}}^{+}. \end{aligned}$$
(5.6)

We then have, formally, for any \(\varepsilon >0\),

$$\begin{aligned} \psi _S(x,t)= & {} \frac{1}{2\pi }\int \nolimits _{{\mathrm{Im\,}}\omega =\varepsilon }e^{-i\omega t}V(x,\omega ){\widetilde{\zeta }}(\omega )\,d\omega \nonumber \\= & {} \frac{1}{2\pi }\int \nolimits _{{\mathbb {R}}}e^{-i\omega t}V(x,\omega +i0){\widetilde{\zeta }}(\omega +i0)\,d\omega ={\mathscr {F}}_{\omega \rightarrow t}^{-1} \big [V(x,\omega ){\widetilde{\zeta }}(\omega )\big ].\nonumber \\ \end{aligned}$$
(5.7)

We will justify this identities in the next section.

6 Traces on the real line

By (5.1) the Fourier transform \({\widetilde{\psi }}_S(\cdot ,\omega )={\mathscr {F}}_{t\rightarrow \omega }[\theta (t)\psi _S(\cdot ,t)]\) is a tempered \(L^2\)-valued distribution of \(\omega \in {{\mathbb {R}}}\). It is the boundary value of the analytic function (5.2) in the following sense:

$$\begin{aligned} {\widetilde{\psi }}_S(\cdot ,\omega ) =\lim \limits _{\varepsilon \rightarrow 0+}{\widetilde{\psi }}_S(\cdot ,\omega +i\varepsilon ),\qquad \omega \in {{\mathbb {R}}}, \end{aligned}$$
(6.1)

where the convergence holds in \({\mathscr {S}}'({{\mathbb {R}}},L^2)\). Indeed,

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

while \(\theta (t)\psi _S(\cdot ,t)e^{-\varepsilon t} \mathop {\longrightarrow }\limits _{\varepsilon \rightarrow 0+}\theta (t)\psi _S(\cdot ,t)\) in \({\mathscr {S}}'({{\mathbb {R}}},L^2)\). Therefore, (6.1) holds by the continuity of the Fourier transform \({\mathscr {F}}_{t\rightarrow \omega }\) in \({\mathscr {S}}'({{\mathbb {R}}})\).

Similarly to (6.1), the distribution \({\widetilde{\zeta }}(\omega )\), \(\omega \in {{\mathbb {R}}}\), is the boundary value of analytic in \({{\mathbb {C}}}^{+}\) function \({\widetilde{\zeta }}(\omega )\):

$$\begin{aligned} {\widetilde{\zeta }}(\omega )=\lim \limits _{\varepsilon \rightarrow 0+} {\widetilde{\zeta }}(\omega +i\varepsilon ), \quad \omega \in {{\mathbb {R}}}, \end{aligned}$$
(6.2)

since the function \(\theta (t)\zeta (t)\) is bounded. The convergence holds in the space of tempered distributions \({\mathscr {S}}'({{\mathbb {R}}})\).

Let us justify that the representation (5.6) for \({\widetilde{\psi }}_S(x,\omega )\) is also valid when \(\omega \in {{\mathbb {R}}}{\setminus }\{-m;m\}\). Namely,

Lemma 6.1

\(V(x,\omega )\) is a smooth function of \(\omega \in {{\mathbb {R}}}{\setminus }\{-m;m\}\) for any fixed \(x\in {{\mathbb {R}}}^3{\setminus } \{0\}\), and the identity

$$\begin{aligned} {\widetilde{\psi }}_S(x,\omega )=V(x,\omega ){\widetilde{\zeta }}(\omega ),\quad \omega \in {{\mathbb {R}}}{\setminus }\{-m;m\} \end{aligned}$$
(6.3)

holds in the sense of distributions.

Proof

This lemma follows from (6.1) and (6.2) by the smoothness of \(V(x,\omega )\) for \(\omega \not =\pm m\). \(\square \)

7 Absolutely continuous spectrum

Here we prove that the distribution \({\widetilde{\zeta }}(\omega )={\widetilde{\zeta }}(\omega +i0)\) is absolutely continuous for real \(|\omega |> m\).

Proposition 7.1

(cf. [21, Proposition 2.3]) \({\widetilde{\zeta }}(\omega )\in L^2_{loc}({{\mathbb {R}}}{\setminus } [-m,m])\otimes {{\mathbb {C}}}^4\).

Proof

We need to prove that

$$\begin{aligned} \int _{I}|{\widetilde{\zeta }}(\omega )|^2\,d\omega <\infty \end{aligned}$$
(7.1)

for any compact interval I such that \(I\cap [-m,m]=\emptyset \). The Parseval identity applied to

$$\begin{aligned} {\widetilde{\psi }}_S(x,\omega +i\epsilon )=\int _{0}^\infty \psi _S(x,t)e^{i\omega t-\epsilon t}\,dt, \quad \epsilon >0, \end{aligned}$$

gives

$$\begin{aligned} \int _{{{\mathbb {R}}}}\Vert {\widetilde{\psi }}_S(\cdot ,\omega +i\epsilon )\Vert _{L^2}^2\,d\omega =2\pi \int _{0}^\infty \Vert \psi _S(\cdot ,t)\Vert _{L^2}^2\, e^{-2\epsilon t}\,dt. \end{aligned}$$
(7.2)

The right-hand side of (7.2) does not exceed \(C_0/\epsilon \), with some \(C_0>0\), since \(\sup _{t\ge 0}\Vert \psi _S(\cdot ,t)\Vert _{L^2}<\infty \) by (5.1). Taking into account (5.6), we obtain

$$\begin{aligned} \int _{{{\mathbb {R}}}} |{\widetilde{\zeta }}(\omega +i\epsilon )|^2 \big \Vert V(\cdot ,\omega +i\epsilon )\frac{{\widetilde{\zeta }}(\omega +i\epsilon )}{|{\widetilde{\zeta }}(\omega +i\epsilon )|}\big \Vert _{L^2}^2\,d\omega \le \frac{C_0}{\epsilon }, \end{aligned}$$
(7.3)

since for any \(\varepsilon >0\) the set of zeros of analytic function \({\widetilde{\zeta }}(\omega +i\varepsilon )\) has measure zero.

Lemma 7.2

There exists \(C_I\) such that

$$\begin{aligned} \big \Vert V(\cdot ,\omega +i\epsilon )\frac{{\widetilde{\zeta }}(\omega +i\epsilon )}{|{\widetilde{\zeta }}(\omega +i\epsilon )|}\big \Vert _{L^2}^2\ge \frac{C_I}{\varepsilon }, \quad \omega \in I,\quad 0<\varepsilon \le |I|/2. \end{aligned}$$
(7.4)

Proof

For concreteness, we will consider the case \(I \subset (m,+\infty )\). Due to the middle line of (5.4), \({\widehat{V}}(\xi ,\omega )={\widehat{V}}_1(\xi )-{\widehat{V}}_2(\xi ,\omega )\), where

$$\begin{aligned} {\widehat{V}}_1(\xi )=\frac{1}{\xi ^2+m^2},\qquad {\widehat{V}}_2(\xi ,\omega )=\frac{\omega (\alpha \cdot \xi +m\beta -\omega )}{(\xi ^2+m^2-\omega ^2)(\xi ^2+m^2)}. \end{aligned}$$

One has

$$\begin{aligned} \big \Vert V_1(\cdot ) \frac{{\widetilde{\zeta }}(\omega +i\epsilon )}{|{\widetilde{\zeta }}(\omega +i\epsilon )|}\big \Vert _{L^2}^2= & {} \frac{1}{(2\pi )^3}\big \Vert {\widehat{V}}_1(\cdot ) \frac{{\widetilde{\zeta }}(\omega +i\epsilon )}{|{\widetilde{\zeta }}(\omega +i\epsilon )|}\big \Vert _{L^2}^2\\= & {} \frac{1}{4\pi }\int _0^{\infty }\frac{\rho ^2\,d\rho }{(\rho ^2+m^2)^2}=\mathrm{Const}. \end{aligned}$$

Hence it suffices to prove (7.4) for \(V_2\) only.

Denote by \(\Pi _{\pm }(\xi )\) orthogonal projections onto the eigenspaces of the operator \({\widehat{D}}_m(\xi )=\alpha \cdot \xi +\beta m\) corresponding to the eigenvalues \(\pm \sqrt{\xi ^2+m^2}\):

$$\begin{aligned} \Pi _{\pm }(\xi ):=\frac{1}{2}\Big (1\pm \frac{{\widehat{D}}_m(\xi )}{\sqrt{\xi ^2+m^2}}\Big ). \end{aligned}$$
(7.5)

Denote by \(e_{\pm }(\xi ,\omega )=\Pi _{\pm }(\xi )\frac{{\widetilde{\zeta }}(\omega )}{|{\widetilde{\zeta }}(\omega )|}\) the eigenvectors of the operator \(\alpha \cdot \xi +\beta m-\omega \). Then the function \( {\widehat{V}}_2(\xi ,\omega )\frac{{\widetilde{\zeta }}(\omega )}{|{\widetilde{\zeta }}(\omega )}\) for \(\omega \in {{\mathbb {C}}}^+\) can be expressed as

$$\begin{aligned} {\widehat{V}}_2(\xi ,\omega )\frac{{\widetilde{\zeta }}(\omega )}{|{\widetilde{\zeta }}(\omega )|}= & {} \omega \frac{(-\omega +\sqrt{\xi ^2+m^2})e_{+}(\xi ,\omega ) + (-\omega -\sqrt{\xi ^2+m^2})e_{-}(\xi ,\omega )}{(\xi ^2+m^2-\omega ^2)(\xi ^2+m^2)}\\= & {} \frac{\omega \, e_{+}(\xi ,\omega )}{(\sqrt{\xi ^2+m^2}+\omega )(\xi ^2+m^2)} -\frac{\omega \,e_{-}(\xi ,\omega )}{(\sqrt{\xi ^2+m^2}-\omega )(\xi ^2+m^2)}. \end{aligned}$$

Using the mutual orthogonality of \(e_+\) and \(e_-\) with respect to the \(L^2\)-product, we obtain for \(\quad \omega \in {{\mathbb {C}}}^+\)

$$\begin{aligned} \big \Vert V_2(\cdot ,\omega )\frac{{\widetilde{\zeta }}(\omega )}{|{\widetilde{\zeta }}(\omega )|}\big \Vert _{L^2}^2= & {} \frac{|\omega |^2}{(2\pi )^3}\int \Big (\frac{|e_{+}(\xi ,\omega )|^2}{|\sqrt{\xi ^2+m^2}+\omega |^2(\xi ^2+m^2)^2}\\&+\frac{|e_{-}(\xi ,\omega )|^2}{|\sqrt{\xi ^2+m^2}-\omega |^2(\xi ^2+m^2)^2}\Big )\,d\xi . \end{aligned}$$

Hence, for \(\omega \in I\subset (m,\infty )\) and \(\varepsilon >0\), we have

$$\begin{aligned} \big \Vert&V_2(\cdot ,\omega +i\varepsilon )\frac{{\widetilde{\zeta }}(\omega +i\varepsilon )}{|{\widetilde{\zeta }}(\omega +i\varepsilon )|}\big \Vert _{L^2}^2\nonumber \\&\quad \ge \frac{m^2}{(2\pi )^3}\int \nolimits _0^\infty \Big (\int \nolimits _{|\xi |=\rho }\frac{|e_{-}(\xi ,\omega +i\varepsilon )|^2}{((\sqrt{\xi ^2+m^2}-\omega )^2+\varepsilon ^2)(\xi ^2+m^2)^2}\,dS\Big )d\rho \nonumber \\&\quad \ge \frac{m^2}{(2\pi )^3}\int \nolimits _I\frac{Q(\omega +i\varepsilon ,r)\,dr}{((r-\omega )^2+\varepsilon ^2)r^3\sqrt{r^2-m^2}}, \end{aligned}$$
(7.6)

where

$$\begin{aligned} r=\sqrt{\rho ^2+m^2},\quad \mathrm{and}\quad Q(\omega +i\varepsilon ,r) :=\int \nolimits _{|\xi |=\sqrt{r^2-m^2}} |e_{-}(\xi ,\omega +i\varepsilon )|^2 dS,\quad r>m. \end{aligned}$$

Let us prove that \(q(I):=\inf \limits _{\varepsilon>0}~\inf \limits _{r,\omega '\in I}|Q(\omega '+i\varepsilon ,r)|>0\). By (7.5),

$$\begin{aligned} e_{-}(\xi ,\omega '+i\varepsilon )= & {} \frac{1}{2}\Big (1-\frac{\alpha \cdot \xi +m\beta }{\sqrt{\xi ^2+m^2}}\Big ) \frac{{\widetilde{\zeta }}(\omega '+i\varepsilon )}{|{\widetilde{\zeta }}(\omega '+i\varepsilon )|}\\= & {} \frac{1}{2r}(r-\alpha \cdot \xi -m\beta ) \frac{{\widetilde{\zeta }}(\omega '+i\varepsilon )}{|{\widetilde{\zeta }}(\omega '+i\varepsilon )|},\quad |\xi |=\sqrt{r^2-m^2}. \end{aligned}$$

The unit sphere \(S_1\) and the interval I are compact sets. Hence, it suffices to show that for any vector \({\mathrm{w}}\in S_1\) and any \(r\in I\) there exists \(\xi \in S_{\sqrt{r^2-m^2}}~\) such that

$$\begin{aligned} (r-\alpha \cdot \xi -m\beta ) {\mathrm{w}}\not =0. \end{aligned}$$

Indeed, suppose that \((r-\alpha \cdot \xi -m\beta ) {\mathrm{w}}=0\) for some \(\xi \in S_{\sqrt{r^2-m^2}}\). Then, \((\alpha \cdot \xi ) {\mathrm{w}}=(r-m\beta ){\mathrm{w}}\), and for \({\check{\xi }}=-\xi \) we have

$$\begin{aligned} (r-\alpha \cdot {\check{\xi }}-m\beta ) {\mathrm{w}}= & {} (r-m\beta ){\mathrm{w}}-(\alpha \cdot \check{\xi }) {\mathrm{w}}\\= & {} (r-m\beta ){\mathrm{w}}+(\alpha \cdot \xi ) {\mathrm{w}} =2(r-m\beta ){\mathrm{w}}\not =0 \end{aligned}$$

because of the nondegeneracy of the matrix \(r-m\beta \) for \(r>m\).

Now, (7.6) implies for any \(\varepsilon \in (0,|I|/2)\)

$$\begin{aligned} \big \Vert V_2(\cdot ,\omega +i\varepsilon )\frac{{\widetilde{\zeta }}(\omega +i\varepsilon )}{|{\widetilde{\zeta }}(\omega +i\varepsilon )|}\big \Vert _{L^2}^2\ge & {} \frac{m^2 q(I)}{(2\pi )^3}\int \nolimits _{I}\frac{dr}{((r-\omega )^2+\varepsilon ^2)r^3\sqrt{r^2-m^2}}\\\ge & {} C_I\int \nolimits _{I\cap [\omega -\varepsilon ,\omega +\varepsilon ]}\frac{dr}{2\varepsilon ^2}\ge \frac{C_I}{\varepsilon }. \end{aligned}$$

The last inequality is due to \(|I\cap [\omega -\varepsilon ,\omega +\varepsilon ]|\ge \varepsilon \), which follows from \(\omega \in I\) and \(\varepsilon <|I|/2\). \(\square \)

Substituting (7.4) into (7.3), we obtain

$$\begin{aligned} \int _{I}|{\widetilde{\zeta }}(\omega +i\varepsilon )|^2\,d\omega <C_0/C_I,\quad \varepsilon \in (0,|I|/2). \end{aligned}$$
(7.7)

We conclude that the set of functions \(g_{\varepsilon }(\omega )={\widetilde{\zeta }}(\omega +i\varepsilon )\), \(0<\varepsilon \le \varepsilon _I\) defined for \(\omega \in I\), is bounded in the Hilbert space \(L^2(I)\), and, by the Banach Theorem, is weakly compact. The convergence of the distributions (6.2) implies the weak convergence \(g_{\varepsilon }\mathop {-\!\!\!\!-\!\!\!\!\rightharpoonup }\limits _{\varepsilon \rightarrow 0+}g\) in the Hilbert space \(L^2(I)\). The limit function \(g(\omega )\) coincides with the distribution \({\widetilde{\zeta }}(\omega )\) restricted onto I. This proves the bound (7.1) and finishes the proof of the proposition. \(\square \)

8 Omega-limit compactness

Lemma 8.1

For any sequence \(s_{k}\rightarrow \infty \) there exists an infinite subsequence (which we also denote by \(s_{k}\)) such that

$$\begin{aligned} \zeta (t+s_{k})\rightarrow \eta (t), \quad k\rightarrow \infty ,\quad t\in {{\mathbb {R}}}, \end{aligned}$$
(8.1)

where \(\eta (t)\) is some function from \(C_b({{\mathbb {R}}})\otimes {{\mathbb {C}}}^4\). The convergence is uniform on \([-T,T]\) for any \(T>0\). Moreover, \(\eta (t)\) is the solution to

$$\begin{aligned}&-{\dot{\eta }}(t)+m\eta (t)-m\int \nolimits _0^{\infty }\frac{J_1(ms)}{s}\eta (t-s)ds\nonumber \\&\quad +\,im\beta \Big (\eta (t) -m\int \nolimits _0^\infty \Big (\int \nolimits _{ms}^{\infty } \frac{J_1(mu)}{u}du\Big )\eta (t-s)ds\Big ) =4\pi F(\eta (t)),~~ t\in {{\mathbb {R}}}.\nonumber \\ \end{aligned}$$
(8.2)

Proof

Theorem 2.1-iii), bound (2.8) and equation (2.24) imply that \(\zeta \in C^1_b({{\mathbb {R}}})\otimes {{\mathbb {C}}}^4\). Then (8.1) follows from the Arzelá-Ascoli theorem. Further, using the asymptotics of Bessel function [33, Formula 10.7.8], we obtain

$$\begin{aligned}&J_0(mt)\rightarrow 0,\quad t\int _{mt}^{\infty } \frac{J_1(u)du}{u}\\&\quad =t\int _{mt}^{\infty } \Big (\frac{\cos (u-3\pi /4)}{u^{3/2}}+{\mathscr {O}}(u^{-5/2})\Big )du\rightarrow 0,\quad t\rightarrow \infty . \end{aligned}$$

Moreover, for any \(t\in {{\mathbb {R}}}\)

$$\begin{aligned}&\int _0^{t+s_k}\frac{J_1(ms)}{s}\zeta (t+s_k-s)ds\rightarrow \int _{0}^{\infty }\frac{J_1(ms)}{s}\eta (t-s)ds,\quad k\rightarrow \infty , \\&\quad \int _0^{t+s_k}\Big (\int _{ms}^{\infty }\frac{J_1(mu)}{u}du\Big )\zeta (t+s_k-s)ds\\&\qquad \rightarrow \int _{0}^{\infty }\Big (\int _{ms}^{\infty }\frac{J_1(mu)}{u}\Big )\eta (t-s)ds,\quad j\rightarrow \infty \end{aligned}$$

by the Lebesgue dominated convergence theorem. Then equation (2.24) for \(\zeta (t)\) together with (2.15) and (4.2) implies (8.2). \(\square \)

Corollary 8.2

The distributions \({\widetilde{\eta }}_j(\omega )\), \(j=1, \ldots 4\), belongs to the space of quasimeasures which are defined as functions with bounded continuous Fourier transform.

Lemma 8.3

\(\mathop {\mathrm{supp}}{\widetilde{\eta }} \subset [-m,m]\).

Proof

Due to (8.1) and the continuity of the Fourier transform in \({\mathscr {S}}'({{\mathbb {R}}})\), we have

$$\begin{aligned} \chi (\omega ){\widetilde{\zeta }}(\omega )e^{-i\omega s_k}\mathop {{\mathop {\longrightarrow }\limits ^{{\mathscr {S}}'}}} \chi (\omega )\eta (\omega ),\quad k\rightarrow \infty . \end{aligned}$$

for any \(\chi \in C_0^{\infty }({{\mathbb {R}}})\) such that \(\mathop {\mathrm{supp}}\chi \cap [-m,m]=\emptyset \). The products \(\chi (\omega ){\widetilde{\zeta }}(\omega )\) are absolutely continuous measures since \({\widetilde{\zeta }}(\omega )\) is locally \(L^2\) for \(\omega \in {{\mathbb {R}}}{\setminus } [-m,m]\) by Proposition 7.1. Then \(\eta (\omega )=0\) for \(\omega \notin [-m,m]\) by the Riemann–Lebesgue Theorem. \(\square \)

9 Spectral inclusion and the Titchmarsh theorem

Here we will prove the following identity

$$\begin{aligned} \eta _j(t)=C_je^{-i\omega _j^+t},\quad t\in R,\quad \omega _j^+\in [-m,m], \quad j=1, \ldots ,4. \end{aligned}$$
(9.1)

We start with an investigation of \(\mathop {\mathrm{supp}}\widetilde{F_j(\eta _j)}\).

Lemma 9.1

The following spectral inclusion holds:

$$\begin{aligned} \mathop {\mathrm{supp}}\widetilde{F_j(\eta _j)}\subset \mathop {\mathrm{supp}}{\widetilde{\eta }}_j. \end{aligned}$$
(9.2)

Proof

Applying the Fourier transform to (8.2), we get by the theory of quasimeasures (see [20]) that

$$\begin{aligned} 4\pi \widetilde{F_j(\eta _j)}(\omega )= \big (i\omega +m-m{\widetilde{P}}(\omega )+im\sigma _j(1-m{\widetilde{Q}}(\omega ))\big ){\widetilde{\eta }}_j(\omega ),\quad j=1, \ldots ,4.\nonumber \\ \end{aligned}$$
(9.3)

where \(\sigma _j\) is defined in (3.9), \({\widetilde{P}}(\omega )\) and \({\widetilde{Q}}(\omega )\) are the Fourier transforms of the functions \(P(t)=\theta (t)\frac{J_1(mt)}{t}\) and \(Q(t)=\theta (t)\int \nolimits _{mt}^{\infty }\frac{J_1(mu)}{u}du\). Note that P(t) and Q(t) belong to \(L^1({{\mathbb {R}}})\). Therefore, the multiplication by \({\widetilde{P}}(\omega )\) and \({\widetilde{Q}}(\omega )\) is well-defined in the sense of quasimeasures (see Appendix B of [20]). Finally, (9.3) implies (9.2). \(\square \)

The second step is the following lemma

Lemma 9.2

For any omega-limit trajectory \(\eta _j(t)\) one has

$$\begin{aligned} |\eta _j(t)|={\mathrm{const}},\quad t\in {{\mathbb {R}}}. \end{aligned}$$
(9.4)

Proof

The assumption (3.2) implies that the function \(F_j(\eta _j(t))\), \(j=1, \ldots ,4\) admits the representation

$$\begin{aligned} F_j(\eta _j(t))=a_j(\eta _j(t))\eta _j(t), \end{aligned}$$
(9.5)

where, according to (3.3)

$$\begin{aligned} a_j(\eta _j)=\sum \limits _{n=1}^{N_j} 2 n u_{n,j}|\eta _j|^{2n-2}. \end{aligned}$$
(9.6)

The functions \(\eta _j(t)\) and \(a_j(\eta _j(t))\) are bounded continuous functions in \({{\mathbb {R}}}\) by Lemma 8.1. Hence, \(\eta _j(t)\) and \(a_j(\eta _j(t))\) are tempered distributions. Moreover, \(\mathop {\mathrm{supp}}{\widetilde{\eta }}_j\subset [-m,m]\) and \(\mathop {\mathrm{supp}}\widetilde{{\overline{\eta }}}_j\subset [-m,m]\) according to Lemma 8.3. Hence, \(\widetilde{a_j(\eta _j)}\) also has a bounded support. Denote \(\mathbf{F}_j=\mathop {\mathrm{supp}}\, \widetilde{F_j(\eta _j)}\), \(\mathbf{A}_j=\mathop {\mathrm{supp}}\,\widetilde{a_j(\eta _j)}\), \(\mathbf{Z}_j=\mathop {\mathrm{supp}}{\widetilde{\eta }}_j\). Then the spectral inclusion (9.2) gives

$$\begin{aligned} \mathbf{F}_j\subset \mathbf{Z}_j. \end{aligned}$$

On the other hand, applying the Titchmarsh convolution theorem [19, Theorem 4.3.3] to (9.5), we obtain

$$\begin{aligned} \inf \,\mathbf{F}_j=\inf \,\mathbf{A}_j+\inf \,\mathbf{Z}_j,\quad \sup \,\mathbf{F}_j=\sup \,\mathbf{A}_j+\sup \,\mathbf{Z}_j. \end{aligned}$$

Hence, \(\inf \,\mathbf{A}_j=\sup \,\mathbf{A}_j=0\), and then \(\mathbf{A}_j\subset \{0\}\). Thus, we conclude that \(\mathop {\mathrm{supp}}\,\widetilde{a_j(\eta _j)}=\mathbf{A}_j\subset \{0\}\), and therefore the distribution \(\widetilde{a_j(\eta _j)}(\omega )\) is a finite linear combination of \(\delta (\omega )\) and it’s derivatives. Then \(a_k(\eta _j(t))\) is a polynomial in t. By Lemma 8.1, \(a_j(\eta _j(t))\) is bounded then we conclude that \(a_j(\eta _j(t))={\mathrm{const}}\). Finally, (9.4) follows since \(a_j(\eta _j(t))\) is a polynomial in \(\eta _j(t)\), and its degree \(2N-2\ge 2\) by (3.1) and (9.6). \(\square \)

Now (9.4) means that \(\eta _j(t){\overline{\eta }}_j(t)\equiv C={\mathrm{const}}\), and then \({\widetilde{\eta }}_j*\widetilde{{\overline{\eta }}}_j=2\pi C\delta (\omega -\omega _j^+)\). Hence, if \(\eta _j\) is not identically zero, the Titchmarsh theorem implies that \(\mathbf{Z}_j=\omega _j\in [-m,m]\). Indeed,

$$\begin{aligned} 0=\sup \,\mathbf{Z}_j+\sup \,(-\mathbf{Z})_j=\sup \,\mathbf{Z}_j-\inf \,\mathbf{Z}_j, \end{aligned}$$

and hence \(\inf \,\mathbf{Z}_j=\sup \,\mathbf{Z}_j\). Therefore, \({\widetilde{\eta }}_j\) is a finite linear combination of \(\delta (\omega -\omega _j^+)\) and its derivatives. But the derivatives could not be present because of the boundedness of \(\eta _j(t)\). Thus \({\widetilde{\eta }}_j\sim \delta (\omega -\omega _j^+)\), which implies (9.1).

10 Convergence of singular component

Denote

$$\begin{aligned} \psi _S^-(x,t)=\psi _S(x,t)-iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)=\varphi _S(x,t)+iD_m^{-1}p_S(x,t), \end{aligned}$$
(10.1)

where \(\varphi _S(x,t)\) and \(p_S(x,t)\) are defined in (2.18) and (2.19). Here we prove that \(\psi _S^-(x,t)\) converges to some solitary wave.

Lemma 10.1

The convergence holds

$$\begin{aligned} \psi _S^-(\cdot ,t+s_{j})\rightarrow \phi _{\Omega ^+}(\cdot ,t)+iD_m^{-1}{\dot{\phi }}_{\Omega ^+}(\cdot ,t),\quad j\rightarrow \infty \end{aligned}$$
(10.2)

in the topology of \(C_b([-T,T],L^{2}_{loc}({{\mathbb {R}}}^3))\) for any \(T>0\). Here

$$\begin{aligned} \phi _{\Omega ^+,j}(x,t)=\phi _{\omega _j^+}(x)e^{-i\omega _j^+t}=C_j\frac{e^{-\sqrt{m^2-(\omega _j^+)^2}|x|}}{4\pi |x|}e^{-i\omega _j^+t},\quad j=1, \ldots ,4. \end{aligned}$$

Proof

Definition (2.18) of \(\varphi _S(x,t)\), Lemma 8.1 and identity (9.1) imply that for any \(x\not =0\)

$$\begin{aligned} \varphi _{S,j}(x,t+s_k)\rightarrow & {} \frac{C_je^{-i\omega _j^+(t-|x|)}}{4\pi |x|} -\frac{mC_j}{4\pi }\int _0^\infty \frac{\theta (s-|x|)J_1(m\sqrt{s^2-|x|^2})}{\sqrt{s^2-|x|^2}}e^{-i\omega _j^+(t-s)}ds\\= & {} \frac{C_je^{-i\omega _j^+t}}{4\pi } \Big (\frac{e^{i\omega _j^+|x|}}{|x|}-m{\widetilde{L}}(x,\omega _j^+)\Big )=C_je^{-i\omega _j^+ t} \frac{e^{-\sqrt{m^2-(\omega _j^+)^2}|x|}}{4\pi |x|}\\= & {} \phi _{\omega _j^+}(x)e^{-i\omega _j^+ t},\quad k\rightarrow \infty , \quad t\in {{\mathbb {R}}}\end{aligned}$$

by the Lebesgue dominated convergence theorem. Here \({\widetilde{L}}(x,\omega )=\frac{1}{m|x|}\big (e^{i|x|\omega }-e^{i|x|\sqrt{\omega ^2-m^2}}\big )\) is the Fourier transform of the function \(L(x,t)=\frac{\theta (t-|x|)J_1(m\sqrt{t^2-|x|^2})}{\sqrt{t^2-|x|^2}}\) (see Appendix in [27]). Hence, for any \(T>0\),

$$\begin{aligned} \varphi _{S}(\cdot ,t+s_k)\rightarrow \phi _{\Omega ^+}(\cdot ,t),\quad k\rightarrow \infty \end{aligned}$$
(10.3)

in \(C_b([-T,T],L^2_{loc}({{\mathbb {R}}}^3))\). It remains to prove that for any \(T>0\)

$$\begin{aligned} D_m^{-1}p_S(\cdot ,t+s_{k})\rightarrow D_m^{-1}{\dot{\phi }}_{\Omega ^+}(\cdot ,t)\quad k\rightarrow \infty \end{aligned}$$
(10.4)

in \(C_b([-T,T],H^1_{loc}({{\mathbb {R}}}^3))\). Lemma 8.1 and equation (2.24) imply that

$$\begin{aligned} {\dot{\zeta }}_j(t+s_{k})\rightarrow {\dot{\eta }}_j(t)=C_j(- i\omega _j^+ e^{-i\omega _j^+t}), \quad k\rightarrow \infty ,\quad t\in {{\mathbb {R}}}\end{aligned}$$
(10.5)

uniformly on \([-T,T]\) for any \(T>0\). Hence, using (2.21), we obtain similarly to (10.3) that for any \(T>0\),

$$\begin{aligned} p_{S}(\cdot ,t+s_{k})\rightarrow {\dot{\phi }}_{\Omega ^+}(\cdot ,t),\quad k\rightarrow \infty \end{aligned}$$
(10.6)

in \(C_b([-T,T],L^2_{loc}({{\mathbb {R}}}^3))\). Further, \(D_m^{-2} p_{S,j}(x ,t)=\displaystyle \int \nolimits _{{{\mathbb {R}}}^3}\displaystyle \frac{e^{-m|y|}}{4\pi |y|}\,p_{S,j}(x-y,t)dy\), and

$$\begin{aligned}&\int _{{{\mathbb {R}}}^3}\frac{e^{-m|y|}}{4\pi |y|}|p_{S,j}(x-y,t)|dy\\&\quad \le C\left( \int _{{{\mathbb {R}}}^3}\frac{e^{-m|y|}}{ |y||x-y|}dy+ \int _{{{\mathbb {R}}}^3}\frac{e^{-m|y|}}{ |y|}dy\right) \le C'\le \infty ,\quad x\in {{\mathbb {R}}}^3,\quad t\in {{\mathbb {R}}}\end{aligned}$$

by (2.21). Then for any \(x\in {{\mathbb {R}}}^3\),

$$\begin{aligned} \lim \limits _{ k\rightarrow \infty }D_m^{-2} p_{S}(\cdot ,t+s_{k})=D_m^{-2}\lim \limits _{ k\rightarrow \infty }p_{S}(\cdot ,t+s_{k}) =D_m^{-2}{\dot{\phi }}_{\Omega ^+}(x,y) \end{aligned}$$
(10.7)

by the Lebesgue dominated convergence theorem. Finally, from (10.6)–(10.7) it follows that for any \(T>0\),

$$\begin{aligned} D_m^{-1}p_{S}(\cdot ,t+s_{k})\rightarrow D_m^{-1}{\dot{\phi }}_{\Omega ^+}(\cdot ,t),\quad k\rightarrow \infty \end{aligned}$$

in \(C_b([-T,T],H^1_{loc}({{\mathbb {R}}}^3))\), which implies (10.4). \(\square \)

11 Proof of global attraction

Substituting (2.13) and (2.17) into (2.4), we obtain

$$\begin{aligned} \psi (x,t)= & {} \psi _f(x.t)+\varphi _{g}(x,t)+\psi _S(x,t)\\= & {} \psi _f(x.t)+\zeta _0\big (g(x)-\gamma (x,t)\big )-iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)\\&+\,\varphi _S(x,t)+iD_m^{-1}\zeta _0{\dot{\gamma }}(x,t)+iD_m^{-1}p_S(x,t)\\= & {} \psi _f(x.t)+\varphi _{g}^+(x,t)+\psi _S^-(x,t) \end{aligned}$$

by (4.3) and (10.1). Due to Lemmas 4.1 and 4.3 it suffices to prove that

$$\begin{aligned} \lim _{t\rightarrow \infty } \mathrm{dist}_{ L^{2}_{loc}({{\mathbb {R}}}^3)}(\psi _S^-(\cdot ,t),{\mathscr {S}})=0. \end{aligned}$$
(11.1)

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

$$\begin{aligned} \mathrm{dist}_{L^{2}_{loc}({{\mathbb {R}}}^3)}(\psi _S^-(\cdot ,s_k),{\mathscr {S}})\ge \delta ,\quad \forall k \end{aligned}$$
(11.2)

for some \(\delta >0\). According to Lemma  10.1, there exist a subsequence \(s_{k_n}\) of the sequence \(s_k\), \(\omega _j^+\in {{\mathbb {R}}}\) and functions \(\phi _{\omega _j^+}\) such that

$$\begin{aligned} \psi _S^-(\cdot ,t+s_{k_n})\rightarrow \phi _{\Omega ^+}(\cdot ,t)+iD_m^{-1}{\dot{\phi }}_{\Omega ^+}(\cdot ,t),\quad k_n\rightarrow \infty ,\quad t\in {{\mathbb {R}}}, \end{aligned}$$

in \(C_b([-T,T], L^{2}_{loc}({{\mathbb {R}}}^3))\) with any \(T>0\). This implies that

$$\begin{aligned} \psi _S^-(\cdot ,s_{k_n})\rightarrow \Phi _{\Omega _+}(\cdot )+D_m^{-1}\Psi _{\Omega ^+}(\cdot ),\quad k_n\rightarrow \infty , \end{aligned}$$
(11.3)

in \(L^{2}_{loc}({{\mathbb {R}}}^3).~\) Here

$$\begin{aligned} \Phi _{\Omega _+,j}(x)=\phi _{\Omega ^+,j}(x,0)=\phi _{\omega _j^+}(x),~~~~ \Psi _{\Omega ^+,j}(x)=i{\dot{\phi }}_{\Omega ^+,j}(x,0)=\omega _j^+\psi _{\omega _j^+}(x). \end{aligned}$$

The convergence (11.3) contradict (11.2) due to (3.20). This completes the proof of Theorem 3.6. \(\square \)