Abstract
We prove global attraction to stationary orbits for 3D Dirac equation with concentrated nonlinearity. We show that each finite energy solution converges as \(t\rightarrow \pm \infty \) to the set of four-frequency “nonlinear eigenfunctions”. The global attraction is caused by nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation.
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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
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 \({{\mathbb {R}}}^3\),
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
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:
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
The system (1.1) is \({\mathbf {U}}(1)\)-invariant; that is,
Our main results are as follows. First, for initial data of type
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
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:
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
-
(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) -
(ii)
in [30] for 1D Dirac equations with more regular nonlinearity \(D_m^{-1}\delta (x)F(\psi (0,t))\);
-
(iii)
in [21, 22] for nD Klein–Gordon and Dirac equations with nonlinearity of type \(\rho (x)F(\langle \psi ,\rho \rangle )\);
-
(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
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
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
which generally is not a linear space. Note that the first equation of (1.1) can be written in the other form
(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
-
(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 \(\zeta (t)\in C^1[0,\infty )\).
-
(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) -
(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) -
(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
where \(\psi _f(x,t)\) and \(\varphi _{g}\) are the unique solutions to the free Dirac equation with initial functions f and \(\zeta _0g\):
and \(\psi _S(x,t)\) is the solution to
Evidently,
Hence,
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
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
satisfies
Hence,
where
is the solution to
Here \(J_1\) is the Bessel function of the first order and \(\theta \) is the Heaviside function. Finally, (2.9) and (2.10) imply
where \(\varphi _{g}^{+}(x,t):=\zeta _0(g(x)-\gamma (x,t))\).
Lemma 2.2
For any \(t>0\) there exists
We prove this lemma in Sect. A. Note that the function \(\mu (t)\) is continuous for \(t>0\), and there exists
Moreover,
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
is given by
where \(\gamma \) is defined in (2.11), and
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
In the case \(m=0\) this is well-known formula [14, Section 175]. Hence,
\(\square \)
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
Proposition 2.5
For any \(\zeta (t)\in C^1[0,\infty )\otimes {{\mathbb {C}}}^4\) there exists
Substituting these limits into the second equation of (2.5) and taking into accunt (2.14), we obtain the equation for \(\zeta (t)\):
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
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
-
(i)
the identity holds
$$\begin{aligned} {{\widetilde{U}}}(\zeta )= U(\zeta ),\quad |\zeta |\le \Lambda (\psi (0)), \end{aligned}$$(2.26) -
(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) -
(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
-
(i)
there exists a unique solution \((\psi _S(x,t),\zeta (t))\) to (2.5) such that
$$\begin{aligned} \psi _{reg}^-(\cdot ,t):= \psi _S(\cdot ,t)+\varphi _{g}(\cdot ,t)-\zeta (t)g(\cdot )\in C([0,\tau ], H^{\frac{3}{2}-}), \quad \zeta \in C^2[0,\tau ]\otimes {{\mathbb {C}}}^4; \end{aligned}$$ -
(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:
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
-
(i)
for sufficiently small \(\tau >0\) the Cauchy problem (2.29) has a unique solution \(\zeta \in C^1[0,\tau ]\otimes {{\mathbb {C}}}^4\);
-
(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
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 \(t\in [0,\tau ]\)
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
satisfies
Indeed, \(\psi _{reg}^{-}(x,t)\) is a solution to
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
Then, for (2.31) we need to show that
Applying the Fourier transform, we obtain
Hence, integration by parts gives
where \(\ddot{\zeta }\in C[0,\tau ]\otimes {{\mathbb {C}}}^4\) by (2.7), (2.14) and (2.29). Therefore,
Similarly,
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
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
Proof
Equations (2.32) and (2.34) imply for any \(t\in (0,\tau ]\)
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 \(\varepsilon \ge 0\)
Hence, uniformly in \(t\in [0,\tau ]\), we have
Therefore,
in the sense of distributions. Then (2.35) follows. \(\square \)
Corollary 2.10
The following identity holds
Proof
First note that
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,
Hence, (2.35) implies the a priory bound
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
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
where
Definition 3.1
-
(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.
-
(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
where \(\Omega =(\omega _1, \ldots ,\omega _4)\) with \(|\omega _j|<m\),
and \(C_j=C_j(\omega _j)\in {{\mathbb {R}}}\) are solutions to
with
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):
Consider the function
where
Hence,
Further, (3.11) implies that
by the first equation of (1.1). Hence,
by (3.11). Therefore,
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
Moreover,
Substituting this into (3.12), we obtain by (3.13)
where we denote
Now we are able to find coefficients \(C_{kj}\). The second equation of (1.1) together with (3.4) and (3.14) imply
Note, that
Similarly,
Moreover,
Substituting (3.16)–(3.18) into (3.15), we get
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
with some \(a>0\) and \(b\ne 0\). Hence, (3.3) implies
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
where
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
where
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 \(L^{2}_{loc}({{\mathbb {R}}}^3)\):
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 \(\forall R>0\),
where \(B_R\) is the ball of radius R.
Corollary 4.2
From (4.1) immediately follows that
Now consider
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, \(\forall R>0\)
Proof
According to (2.12) the function \(h(x,t):=\gamma (x,t)-g(x)\) is the solutions to
Then (4.4) follows by Lemma 3.3 of [27]. \(\square \)
In conclusion, let us show that
Indeed, the energy conservation for equation (4.5) implies that
Hence,
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
Let us analyze the Fourier–Laplace transform of \(\psi _S(x,t)\):
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
where \({\widetilde{\zeta }}(\omega )\) is the Fourier–Laplace transform of \(\zeta (t)\):
Applying the Fourier transform to (5.3), we get
Denote
The function \(\varkappa (\omega )\) is analytic on \({{\mathbb {C}}}^{+}\), and \({\widetilde{\psi }}_S(x,\omega )\) is given by
We then have, formally, for any \(\varepsilon >0\),
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:
where the convergence holds in \({\mathscr {S}}'({{\mathbb {R}}},L^2)\). Indeed,
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 \({\widetilde{\zeta }}(\omega )\):
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
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
for any compact interval I such that \(I\cap [-m,m]=\emptyset \). The Parseval identity applied to
gives
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
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
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
One has
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 \(\pm \sqrt{\xi ^2+m^2}\):
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
Using the mutual orthogonality of \(e_+\) and \(e_-\) with respect to the \(L^2\)-product, we obtain for \(\quad \omega \in {{\mathbb {C}}}^+\)
Hence, for \(\omega \in I\subset (m,\infty )\) and \(\varepsilon >0\), we have
where
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),
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
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
because of the nondegeneracy of the matrix \(r-m\beta \) for \(r>m\).
Now, (7.6) implies for any \(\varepsilon \in (0,|I|/2)\)
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
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
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
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
Moreover, for any \(t\in {{\mathbb {R}}}\)
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
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
We start with an investigation of \(\mathop {\mathrm{supp}}\widetilde{F_j(\eta _j)}\).
Lemma 9.1
The following spectral inclusion holds:
Proof
Applying the Fourier transform to (8.2), we get by the theory of quasimeasures (see [20]) that
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
Proof
The assumption (3.2) implies that the function \(F_j(\eta _j(t))\), \(j=1, \ldots ,4\) admits the representation
where, according to (3.3)
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
On the other hand, applying the Titchmarsh convolution theorem [19, Theorem 4.3.3] to (9.5), we obtain
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,
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
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
in the topology of \(C_b([-T,T],L^{2}_{loc}({{\mathbb {R}}}^3))\) for any \(T>0\). Here
Proof
Definition (2.18) of \(\varphi _S(x,t)\), Lemma 8.1 and identity (9.1) imply that for any \(x\not =0\)
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 \(T>0\),
in \(C_b([-T,T],L^2_{loc}({{\mathbb {R}}}^3))\). It remains to prove that for any \(T>0\)
in \(C_b([-T,T],H^1_{loc}({{\mathbb {R}}}^3))\). Lemma 8.1 and equation (2.24) imply that
uniformly on \([-T,T]\) for any \(T>0\). Hence, using (2.21), we obtain similarly to (10.3) that for any \(T>0\),
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
by (2.21). Then for any \(x\in {{\mathbb {R}}}^3\),
by the Lebesgue dominated convergence theorem. Finally, from (10.6)–(10.7) it follows that for any \(T>0\),
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
by (4.3) and (10.1). Due to Lemmas 4.1 and 4.3 it suffices to prove that
Assume by contradiction that there exists a sequence \(s_k\rightarrow \infty \) such that
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
in \(C_b([-T,T], L^{2}_{loc}({{\mathbb {R}}}^3))\) with any \(T>0\). This implies that
in \(L^{2}_{loc}({{\mathbb {R}}}^3).~\) Here
The convergence (11.3) contradict (11.2) due to (3.20). This completes the proof of Theorem 3.6. \(\square \)
References
Adami, R., Dell’Antonio, G., Figari, R., Teta, A.: The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity. Ann. Inst. Henri Poincare 20, 477–500 (2003)
Adami, R., Noja, D., Ortoleva, C.: Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three. J. Math. Phys. 54(1), 013501 (2013)
Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. American Mathematical Society, Providence (2005)
Albeverio, S., Høegh-Krohn, R.: Point interactions as limits of short range interactions. J. Oper. Theory 6, 313–339 (1981)
Albeverio, S., Figari, R.: Quantum fields and point interactions. Rend. Mat. Appl. (7) 39, 161–180 (2018)
Berezin, F.A., Faddeev, L.D.: A remark on Schrödinger’s equation with a point interaction. Sov. Math. Dokl. 2, 372–375 (1961)
Buslaev, V., Komech, A., Kopylova, E., Stuart, D.: On asymptotic stability of solitary waves in nonlinear Schrödinger equation. Commun. Partial Differ. Equ. 33(4), 669–705 (2008)
Buslaev, V., Perelman, G.: On the stability of solitary waves for nonlinear Schrödinger equations. In: Nonlinear Evolution Equations, vol. 164 of Amer. Math. Soc. Transl. Ser. 2, 75–98. Amer. Math. Soc., Providence, RI (1995)
Buslaev, V., Sulem, C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20, 419–475 (2003)
Cacciapuoti, C., Finko, D., Noja, D., Teta, A.: The point-like limit for a NLS equation with concentrated nonlinearity in dimension three. J. Funct. Anal. 273, 1762–1809 (2017)
Carlone, R., Correggi, M., Tentarelli, L.: An introduction to the two-dimensional Schrödinger equation with nonlinear point interactions. Nanosyst. Phys. Chem. Math. 9(2), 187–195 (2018)
Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54, 1110–1145 (2001)
Dirac, P.A.M.: Classical theory of radiating electrons. Proc. R. Soc. A 167, 148–169 (1938)
Erdelyi, A., et al.: Tables of Integral Transforms, vol. 1. McGraw-Hill Book Company, New York (1954)
Fukaya, N., Georgiev, V., Ikeda, M.: On stability and instability of standing waves for 2D-nonlinear Schrödinger equation with point interaction. arXiv:2109.04680v1
Gittel, H.-P., Kijowski, J., Zeidler, E.: The relativistic dynamics of the combined particle-field system in renormalized classical electrodynamics. Commun. Math. Phys. 198, 711–736 (1998)
Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford University Press, Oxford (1985)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic Press Inc., San Diego (2000)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. Springer Study Edition, vol. 1. Springer, Berlin (1990)
Komech, A.I., Komech, A.A.: Global attractor for a nonlinear oscillator coupled to the Klein–Gordon field. Arch. Ration. Mech. Anal. 185, 105–142 (2007)
Komech, A.I., Komech, A.A.: Global attraction to solitary waves for Klein–Gordon equation with mean field interaction. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(3), 855–868 (2009)
Komech, A.I., Komech, A.A.: Global attraction to solitary waves for nonlinear Dirac equation with mean field interaction. SIAM J. Math. Anal. 42(6), 2944–2964 (2010)
Komech, A., Kopylova, E., Stuart, D.: On asymptotic stability of solitons in a nonlinear Schrödinger equation. Commun. Pure Appl. Anal. 11(3), 1063–1079 (2012)
Kopylova, E.: On asymptotic stability of solitary waves in discrete Schrödinger equation coupled to nonlinear oscillator. Nonlinear Anal. Ser. A Theory Methods Appl. 71(7–8), 3031–3046 (2009)
Kopylova, E.: On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to nonlinear oscillator. Appl. Anal. 89(9), 1467–1493 (2010)
Kopylova, E.: On global well-posedness for Klein–Gordon equation with concentrated nonlinearity. J. Math. Anal. Appl. 443(2), 1142–1157 (2016)
Kopylova, E.: Global attraction to solitary waves for Klein–Gordon equation with concentrated nonlinearity. Nonlinearity 30(11), 4191–4207 (2017)
Kopylova, E.: On global attraction to stationary state for wave equation with concentrated nonlinearity. J. Dyn. Differ. Equ. 30(1), 107–116 (2018)
Kopylova, E., Komech, A.: On global attractor of 3D Klein–Gordon equation with several concentrated nonlinearities. Dyn. PDEs 16(2), 106–124 (2019)
Kopylova, E., Komech, A.: Global attractor for 1D Dirac field coupled to nonlinear oscillator. Commun. Math. Phys. 375, 573–603 (2020)
Masaki, S., Murphy, J., Segata, J.-I.: Asymptotic stability of solitary waves for the 1d NLS with an attractive delta potential. arXiv:2008.11645
Noja, D., Posilicano, A.: Wave equations with concentrated nonlinearities. J. Phys. A 38(22), 5011–5022 (2005)
Olver, F.W.J., et al.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Smirnov, V.I.: A Course of Higher Mathematics. Pergamon Press, II, Oxford (1964)
Soffer, A., Weinstein, M.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990)
Soffer, A., Weinstein, M.: Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Differ. Equ. 98, 376–390 (1992)
Yafaev, D.R.: On a zero-range interaction of a quantum particle with the vacuum. J. Phys. A 25, 963–978 (1992)
Yafaev, D.R.: A point interaction for the discrete Schrödinger operator and generalized modeling polynomial. J. Math. Phys. 58(6), 063511 (2017)
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Appendices
Proof of Lemma 2.2
Note that
Hence it suffices to prove that
Applying the Fourier transform \({\widehat{f}}(\xi )={\mathscr {F}}_{x\rightarrow \xi } f(x)\), we get
Then for (A.1) it suffices to justify the following permutation of limits:
We will do it for each term in (A.3) separately.
Step i) Applying the Lebesgue dominated convergence theorem, we obtain
Step ii) Let us prove that
Indeed, for \(\rho :=|x|<t\),
since
On the other hand,
by [14, Formula 1.3.(7)]. Here \(\mathbf{K}_\nu \) is the modified Bessel function, and \(\Gamma \) is the gamma function. One can justify the last limit, splitting the integral into a sum of integrals over the intervals [0, 1] and \([1,\infty )\), and integrating by parts in the second one. Further,
since
by [33, Formulas 5.7.1 and 10.39.2].
Step iii) It remains to check that
One has
Evidently,
Hence, (A.7) will follow from
One has
where
The last estimate implies
Moreover,
that easily follows by means of integration by parts. Finally, (A.12)–(A.13) imply (A.9).
Proof of Proposition 2.4
For any \(t>0\),
Hence, it suffices to prove that
The Fourier transform of \(\varphi _S(x,t)-\zeta (t)g(x)\) for any \(t>0\) reads
Therefore, because of the continuity of the Fourier transform in \({\mathscr {S}}'\), it remains to prove that
Due to splitting (A.2), equality (B.2) will follow from the following three equalities
Step i) For any \(s\in {{\mathbb {R}}}\) and \(\varepsilon \ge 0\) the function \(\widehat{T}_\varepsilon (\xi ,s):=\frac{\cos (s\sqrt{\xi ^2+m^2})}{\xi ^2(\xi ^2+m^2)^{1+\varepsilon }}\in L^1({{\mathbb {R}}})\). Hence \(T_\varepsilon (x,s)={\mathscr {F}}^{-1}_{\xi \rightarrow x} {\widehat{T}}_\varepsilon (\xi ,s)\) is continuous in x. Moreover it is uniformly continuous in s. Therefore,
by uniformly continuity of \(T_\varepsilon (0,s)\) in \(\varepsilon \).
Step ii) Let us prove (B.4). Due to (A.8)
where \(\rho =|x|\), and
Applying the Lebesgue dominated convergence theorem, we obtain
where \(S_{\varepsilon }(0,s)=\lim \limits _{\rho \rightarrow 0}S_{\varepsilon }(\rho ,s)=\int \nolimits _{0}^{2m} \frac{\cos (s\sqrt{r^2+m^2})-\cos (sr)}{(r^2+m^2)^{\varepsilon }}dr\), \(\varepsilon \ge 0\). Further, (A.10) implies
where
by (A.11) and the Lebesgue theorem. Finally,
which easily follows by means of integration by parts.
Step iii) Let us prove (B.5). Due to (A.5) we need to prove that
Taking into account (A.4), we obtain
since \(\int \nolimits _0^\infty \frac{\sin u}{u}=\frac{\pi }{2}\). Hence (B.7) is equivalent to
By [14, Formula 2.3.(28)],
where
and \(_1F_2(a,b,c; x)\) is the generalized hypergeometric function:
where the series converges for all finite values of \(x\in {{\mathbb {C}}}\) and defines an entire function (see [33, \(\S \) 16.2 (ii)]). It is easy to see that
Hence,
Further,
Hence,
where we denote
In the case \(k=0\) , taking into account (B.11), we obtain for any small \(\nu >0\)
where \(\eta (\nu )\in [0,\nu ]\). Because of the arbitrariness of \(\nu >0\), we get
Together with (B.9), (B.10) and (B.12), this gives
Now for (B.8) it remains to prove that
One has
Hence, (B.9) implies that
Therefore (B.13) follows.
Proof of Proposition 2.5
Note that \(D_m^{-1}=D_m D_m^{-2}=-i\alpha \cdot \nabla D_m^{-2}+m\beta D_m^{-2}\). Then the Proposition 2.5 will follow from the next three lemmas:
Lemma C.1
The following equality holds,
Lemma C.2
The following limit holds,
Lemma C.3
The following limit holds,
1.1 Proof of Lemma C.1
Note that \(p_S(x,t)\) is a solution to (2.20) with \({\dot{\zeta }}(t)\) instead of \(\zeta (t)\). Hence,
and for (C.1) it suffices to prove that
We split the integrand in (C.5) as
and justify the permutation of the limits (C.5) for integrals of each terms in the RHS of (C.6) separately.
The proof of equality
is similar to the proof of equality (B.3). By the same arguments,
since
It remains to prove that
Applying the Lebesgue theorem, we obtain
by [14, Formula 2.3.(6)]. Here \(\mathrm{I}_{\varepsilon }(z)\) is the modified Bessel function, and \(\mathbf{L}_{-\varepsilon }(z)\) is the modified Struve function, satisfying
by Formulas (10.30.1) and (11.2.2) of [33]. On the other hand,
By the Lebesgue theorem
Further, applying [18, Formula 3.742 (1)], we obtain
Moreover,
by [14, Formula 2.2.(15)]. Therefore,
Now (C.11), (C.12), (C.13) and [14, Formula 2.2.(26)] imply
which coincides with the right hand side of (C.9). Hence, (C.8) follows.
1.2 Proof of Lemma C.2
Integrating by parts in (C.4), we obtain
Let us calculate the inverse Fourier transform of \(\frac{\cos (s\sqrt{\xi ^2+m^2})}{\xi ^2+m^2}\) and of \(\frac{\sin (t\sqrt{\xi ^2+m^2})}{(\xi ^2+m^2)\sqrt{\xi ^2+m^2}}\). In the sense of distributions, we obtain
Hence, (C.14) -(C.16) imply for \(t>0\) and \(|x|\le t\)
One has
Further, changing the order of integration gives
Substituting this into (C.17), we obtain
since \(\displaystyle \int \nolimits _0^{\infty }\frac{J_1(u)du}{u}=1\) by [18, Formula 6.561(17)].
1.3 Proof of Lemma C.3
Note that
Taking into account (C.7), we obtain
Hence, for (C.4) it remains to prove that
Formula 1.3(7) of [14], and formulas 10.27.4 and 10.25.2 of [33] imply
where we denote
One has
Hence
Further,
Denoting \(\alpha _k:=2k+2\varepsilon \). In the case \(\tau :=\frac{s}{|x|}\ge 1\), we obtain
One has
where \(C_1(k)= (\alpha _k+2)(3^{\alpha _k}+1)\), \(~~C_2(k)=\Bigg ((\frac{3}{2})^{\alpha _k-2}+\frac{1}{3}\Bigg )(\alpha _k+1)\alpha _k\). Hence,
Therefore,
It remains to prove that
In the case \(s<|x|\), we obtain similarly to (C.20)
Hence, for small |x|,
which implies (C.21).
1.3.1 The case \(\varepsilon =0\)
Note, that the limit (C.18) does not exist for \(\varepsilon =0\) i.e. without the smoothing operator \(K_m^\varepsilon \). Namely, in this case
by [18, Formula 3.741(1)]. Hence,
Evidently, the first limit in the RHS is zero. Further, L’Hopital’s rule implies
Hence, the second limit in the RHS is zero, and the third limit does not exist.
Existence of nonzero solitary waves
Here we prove the lemma 3.5. It suffices to consider the case \(j=1\) only, since the equation (3.8) for \(j=2\) is the same, and for \(j=3,4\) is similar. We rewrite the equation (3.8) with \(j=1\) as
where
Necessarily, equation (D.1) has nonzero solutions \(C_1=C_1(\omega _1)\) for \(\omega _1\in (-m,m)\), satisfying the condition
Obviously, (D.2) holds for any \(M>0\) and \(\omega _1\) sufficiently close to 0 or to \(-m\). Moreover, (D.2) holds for any \(M>0\) and for \(\omega _1\in (-m,m)\) such that
which is equivalent to
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Kopylova, E. Global attractor for 3D Dirac equation with nonlinear point interaction. Nonlinear Differ. Equ. Appl. 29, 27 (2022). https://doi.org/10.1007/s00030-022-00758-3
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DOI: https://doi.org/10.1007/s00030-022-00758-3