Abstract
We consider the time discretization based on Lie-Trotter splitting, for the nonlinear Schrödinger equation, in the semi-classical limit, with initial data under the form of WKB states. We show that both the exact and the numerical solutions keep a WKB structure, on a time interval independent of the Planck constant. We prove error estimates, which show that the quadratic observables can be computed with a time step independent of the Planck constant. The functional framework is based on time-dependent analytic spaces, in order to overcome a previously encountered loss of regularity phenomenon.
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1 Introduction
This paper is devoted to the analysis of the numerical approximation of the solution to
in the semi-classical limit \(\varepsilon \rightarrow 0\). The nonlinearity is smooth and real-valued: \(\lambda \in {\mathbb R}\) and \({\sigma }\in {\mathbb N}\). The initial data that we consider are WKB states:
where \(\phi _0:{\mathbb R}^d\rightarrow {\mathbb R}\) is a real-valued phase, and \(a_0:{\mathbb R}^d\rightarrow {\mathbb C}\) is a possibly complex-valued amplitude. An important feature of such initial data is that in the context of the semi-classical limit for (1.1), they yield solutions which are in \(L^\infty ({\mathbb R}^d)\) uniformly in \(\varepsilon \), at least on some time interval [0, T] for some \(T>0\) independent of \(\varepsilon \). Also, note that even if \(\phi _0=0\) (no rapid oscillation initially), for \(\tau >0\) arbitrarily small and independent of \(\varepsilon \), \(u^\varepsilon (\tau )\) takes the form of a WKB state as in (1.2) with amplitude \(a^\varepsilon \) and phase \(\phi ^\varepsilon \not \equiv 0\) solving (2.2)–(2.3) below (see [8]).
We consider more precisely the time discretization for (1.1) based on Fourier time splitting. We denote by \(X^t_\varepsilon \) the map \(v^\varepsilon (0,\cdot )\mapsto v^\varepsilon (t,\cdot )\), where
We also denote by \(Y^t_\varepsilon \) the map \(w^\varepsilon (0,\cdot )\mapsto w^\varepsilon (t,\cdot )\), where
We consider the Lie-Trotter type splitting operator
The Lie-Trotter operator \(X^t_\varepsilon Y^t_\varepsilon \) could be handled in the same fashion. The advantage of splitting methods is that they involve sub-equations which are simpler to solve than the initial equation. In our case, (1.3) is solved explicitly by using the Fourier transform, defined by
since it becomes an ordinary differential equation
hence
Also, since \(\lambda \in {\mathbb R}\), in (1.4) the modulus of \(w^\varepsilon \) does not depend on time, hence
In the case \(\varepsilon =1\), several results exist to prove that the Lie-Trotter time splitting is of order one, and the Strang splitting of order two [6, 26]. The drawback of these proofs is that they rely on uniform Sobolev bounds for the exact solution, of the form \(u\in L^\infty ([0,T];H^s({\mathbb R}^d))\), for \(s\ge 2\). However, in the framework of (1.1), these norms are not uniformly bounded as \(\varepsilon \rightarrow 0\), in the sense that we rather have \(\Vert u^\varepsilon (t)\Vert _{H^s}\approx \varepsilon ^{-s}\), due to the oscillatory nature of \(u^\varepsilon \).
In the case of a linear potential (\(|u^\varepsilon |^{2{\sigma }}\) is replaced by a known function of x in (1.1)), error estimates are given in [3]; see also [14, 15]. In the nonlinear case, error estimates are established in [9], but for other nonlinearities than in (1.1)–(1.2). The proof there requires either to consider a weakly nonlinear regime, that is (1.1) is replaced by
with the same initial data (1.2), or to replace the nonlinearity in (1.1) with a smoothing nonlinearity of Poisson type. We recall in Sect. 2 why these assumptions are made in [9]. The goal of this paper is to prove error estimates which are similar to those established in [9], but for (1.1)–(1.2). Before stating our main result, we introduce a few notations. The Fourier transform is normalized as
A tempered distribution f is in \(H^s({\mathbb R}^d)\) if \(\xi \mapsto \langle \xi \rangle ^s \hat{f}(\xi )\;belongs\;to\;L^2(\mathbb R^d)\), where
Theorem 1.1
Suppose that \(d,{\sigma }\in {\mathbb N}\), \(d,{\sigma }\ge 1\), and \(\lambda \in {\mathbb R}\). Let \(\phi _0,a_0\) such that
for some \(\delta >0\), and \(u_0^\varepsilon \) given by (1.2). There exist \(T,\varepsilon _0,c_0>0\) and \((C_k)_{k\in {\mathbb N}}\) such that for all \(\varepsilon \in (0,\varepsilon _0]\), the following holds:
-
1.
(1.1)–(1.2) has a unique solution \(u^\varepsilon =S^t_\varepsilon u_0^\varepsilon \in C([0,T],H^\infty )\), where \(H^\infty =\cap _{s\in {\mathbb R}}H^s\). Moreover, there exist \(\phi ^\varepsilon \) and \(a^\varepsilon \) with, for all \(k\in {\mathbb N}\),
$$\begin{aligned} \mathop {\sup }\limits _{t \in [0,T]} \left( \Vert a^\varepsilon (t)\Vert _{H^k({\mathbb R}^d)} + \Vert \phi ^\varepsilon (t)\Vert _{H^k({\mathbb R}^d)}\right) \le {C_k}, \end{aligned}$$such that \(u^\varepsilon (t,x)=a^\varepsilon (t,x)e^{i\phi ^\varepsilon (t,x)/\varepsilon }\) for all \((t,x)\in [0,T]\times {\mathbb R}^d\).
-
2.
For all \({\Delta t}\in (0,c_0]\), for all \(n\in {\mathbb N}\) such that \(t_n=n{\Delta t}\in [0,T]\), there exist \(\phi _n^\varepsilon \) and \(a_n^\varepsilon \) with, for all \(k\in {\mathbb N}\),
$$\begin{aligned} \Vert a_n^\varepsilon \Vert _{H^k({\mathbb R}^d)}+\Vert \phi ^\varepsilon _n\Vert _{H^{k}({\mathbb R}^d)} \le C_k, \end{aligned}$$such that \({(Z_\varepsilon ^{{\Delta t}})}^{n} (a_0 e^{i\phi _0/\varepsilon }) = a_n^\varepsilon e^{i\phi _n/\varepsilon }\).
-
3.
For all \({\Delta t}\in (0,c_0]\), for all \(n\in {\mathbb N}\) such that \(n{\Delta t}\in [0,T]\), the following error estimate holds:
$$\begin{aligned} \left\| a_n^\varepsilon - a^\varepsilon (t_n)\right\| _{H^{k}}+ \left\| \phi _n^\varepsilon - \phi ^\varepsilon (t_n)\right\| _{H^{k}}\le C_k {\Delta t}. \end{aligned}$$
Example 1.2
The assumptions of Theorem 1.1 are satisfied as soon as \(\hat{\phi }_0\) and \({\hat{a}}_0\) are compactly supported, or in the case of Gaussian functions, typically.
Note that the first two points of the theorem imply that the functions a and \(\phi \) are not rapidly oscillatory: the oscillatory nature of both the exact and the numerical solutions is encoded in the exponential which relates the functions a and \(\phi \) to u.
We readily infer error estimates for the wave function and for quadratic observables,
Corollary 1.3
Under the assumptions of Theorem 1.1, there exist \(T>0\), \(\varepsilon _0>0\) and \(C,c_0\) independent of \(\varepsilon \in (0,\varepsilon _0]\) such that for all \({\Delta t}\in (0,c_0]\), for all \(n\in {\mathbb N}\) such that \(n{\Delta t}\in [0,T]\), and for all \(\varepsilon \in (0,\varepsilon _0]\),
This result is in agreement with the numerical experiments presented in [4]: to simulate the wave function \(u^\varepsilon \), the time step must satisfy \({\Delta t}=o(\varepsilon )\), while to observe the quadratic observables, \({\Delta t}=o(1)\) can be chosen independent of \(\varepsilon \).
It is very likely that the proof presented in this paper can be adapted to the case of Strang time-splitting schemes. The main modification to make would concern the local error estimate (presented in Sect. 6): here, we use a general abstract formula for the local error associated to Lie-Trotter schemes as established in [15]. Proving the analog of Theorem 1.1 would essentially require to adapt these computations.
This paper treats only the case of time discretization: the full numerical analysis would require to take spatial discretization as well. As pointed out in [4], the nature of the nonlinearity in (1.1), and more precisely the sign of \(\lambda \) (focusing or defocusing nonlinearity), alters the requirements to be made concerning the mesh size as a function of \(\varepsilon \). We do not address this delicate issue here.
As emphasized in [9], the fact that our analysis requires the analytical solution to have the form of a WKB state forces us to consider only finite time intervals: the good news is that the length of the time interval is independent of \(\varepsilon \), but on the other hand, the solution \(u^\varepsilon \) must be expected to keep the form of a monokinetic WKB state only for short time. This is related to the formation of singularities in compressible Euler equations, and we refer to [7, 9] for more precise discussions on that topic. However, the analytical framework presented in this paper might be useful for the long time analysis of numerical schemes, since the time dependent analytic norms that we consider were initially introduced by Ginibre and Velo to study the large time behavior of solutions to the Hartree equation (scattering theory). From this perspective, some other time discretization techniques could be tackled thanks to these tools, since several geometric numerical integrators have been proposed to overcome the lack of stability of time splitting over large time; see e.g. [5, 10, 16, 17, 21–23] and references therein.
2 Overview of the proof
We present the general strategy for the proof of Theorem 1.1 in the case of a more general nonlinearity,
with f real-valued. For initial data of the form (1.2), it has been noticed in [9] that the numerical discretization preserves such a structure, in the sense that the numerical solution satisfies the point 2. in Theorem 1.1. Indeed, the exact solution can be represented as \(u^\varepsilon = a^\varepsilon e^{i\phi ^\varepsilon /\varepsilon }\), where \(a^\varepsilon \) and \(\phi ^\varepsilon \) are given by
with initial data
The main feature of this representation is that even though they must be expected to depend on \(\varepsilon \), \(a^\varepsilon \) and \(\phi ^\varepsilon \) are bounded in Sobolev spaces uniformly in \(\varepsilon \in (0,1]\) on some time interval [0, T] for some T independent of \(\varepsilon \).
The idea of representing the solution \(u^\varepsilon \) under this form goes back to Grenier [20]. The main features of (2.2) is that the left hand side defines a symmetrizable hyperbolic system under the assumption \(f'>0\), and the right hand side is skew adjoint (hence plays no role at the level of energy estimates). Note that in the case of (1.1), this forces \(\lambda >0\) and \({\sigma }=1\) (cubic defocusing nonlinearity). For a nonlocal nonlinearity, \(f(|u|^2)=K*|u|^2\), the approach of Grenier can easily be adapted if \({\hat{K}}\) decays at least like \(|\xi |^{-2}\) for large \(|\xi |\) (see e.g. [9]). The approach of Grenier has also been generalized to more general nonlinearities: see [1, 13] for the defocusing case, and [28] for the focusing case. However, we do not use these results, as we now discuss.
Indeed, the splitting scheme for (2.1) amounts to some splitting technique on (2.2). Suppose that one solves the linear equation (1.3) with initial data \(v^\varepsilon (0)=a_0 e^{i\phi _0/\varepsilon }\). Then the solution \(v^\varepsilon \) can be written as \(v^\varepsilon (t)= a^\varepsilon (t) e^{i\phi (t)/\varepsilon }\), with \(a^\varepsilon \) and \(\phi \) given by
Similarly, the solution to (1.4) with initial data \(w^\varepsilon (0)=a_0 e^{i\phi _0/\varepsilon }\) can be written as \(w^\varepsilon (t)= a(t) e^{i\phi (t)/\varepsilon }\), with a and \(\phi \) given by
So computing the numerical solution amounts to solving successively (2.4) and (2.5), which turns out to be a splitting scheme on (2.2). We denote by \(\mathcal {X}^t_\varepsilon :(\phi _0,a_0)\mapsto (\phi (t,\cdot ),a^\varepsilon (t,\cdot ))\) the flow for (2.4) and by \(\mathcal {Y}^t_\varepsilon :(\phi _0,a_0)\mapsto (\phi (t,\cdot ),a(t,\cdot ))\) the flow for (2.5). The Lie-Trotter splitting operator we consider for (2.2) is then
Now in the case of a cubic defocusing nonlinearity (which enters the framework of [20]), we face a loss of regularity issue. Indeed, the reason why (2.2) is convenient lies first in the structure of the left hand side, which enjoys symmetry properties: the splitting leading to (2.4)–(2.5) ruins this property. Suppose for instance that at time \(t=0\), \(\phi _0\in H^s({\mathbb R}^d)\) and \(a_0\in H^k({\mathbb R}^d)\), for large s and k. In (2.4), the first equation propagates the \(H^s\) regularity on a small time interval, provided s is large. The second equation shows that \(a^\varepsilon \) cannot be more regular than \(H^{s-2}\), due to the last term of the left hand side. Now if we start with \(\phi _0\in H^s\) and \(a_0\in H^{s-2}\) in (2.5) (with \(f(|a|^2)=|a|^2\) for a cubic defocusing nonlinearity), we see that \(\phi \in H^{s-2}\) (provided \(s-2>d/2\)), and that no better regularity must be expected. So after one iteration of the operator \(Z_\varepsilon ^t\), \(\phi \) has lost two levels of regularity. When iterating Z with a small time step \({\Delta t}\), this loss becomes fatal. This is why in [9], it is assumed that either f is smoothing (to regain at least two levels of regularity) or that a factor \(\varepsilon \) is present in front of f, so that (2.5) is altered to
The main technical originality of this paper is based on the remark that if instead of working in Sobolev spaces, one works in time dependent analytic spaces, it is possible to control the loss of regularity. Such an idea goes back to [18], to solve (2.2). The fact that we consider decreasing time dependent weight to measure the analytic regularity is strongly inspired by the analysis of Ginibre and Velo in the context of long range scattering for Hartree equations [19], and is also reminiscent of the functional framework used by Chemin for the Navier–Stokes equation [12] and developed by Mouhot and Villani to prove Landau damping [27].
The main technical tools needed here are presented in Sect. 3. Thanks to these tools, we can prove that both the theoretical and the numerical solutions remain analytic in a suitable sense on some time interval [0, T] with \(T>0\) independent of \(\varepsilon \) (Sects. 4 and 5).
The next key estimate is the local error estimate, presented in Sect. 6. It is based on the general formula established in [15]. As noticed in [9], we must apply this formula to the system (2.4)–(2.5) and not only to (1.3)–(1.4).
With these propagating estimates and the local error estimate, the proof of Theorem 1.1 follows from the trick known as Lady Windermere’s fan (see e.g. [24]). Note however that because of the nonlinear context, where global bounds for the numerical solutions are not known a priori, the argument requires some extra care. We rely on the induction technique introduced in [25], which is sufficiently robust to be readily adapted to our case, as in [9].
3 Technical background
We recall here some of the technical tools introduced in [19]. We state the main properties established there concerning time dependent Gevrey spaces, and simplify as much as possible the framework, in view of the present context.
For \( 0<\nu \le 1\) and \(\rho >0\), we introduce the exponential weight
which is equivalent to \(\exp (\rho \left\langle \xi \right\rangle ^\nu )\). Define \(u_>\) and \(u_<\) by:
For \(k,\ell \in {\mathbb R}\) and \(0\le \ell _<<d/2\), the following families of norms are defined in [19]:
The first norm is well suited to estimate amplitudes, and the second is adapted to phases. As suggested by the above notations, the indices will be different for amplitudes and phases. This can be related to the fact that in the hydrodynamical setting with \(\lambda >0\), (2.2) with \(\varepsilon =0\) is a hyperbolic system in the unknown \((\nabla \phi ,a)\), and not in \((\phi ,a)\). Indeed, eventually there will be a shift of one index between the norm in \(\phi \) and the norm in a (see Lemma 3.3 and Proposition 4.1 below).
In the properties related to these norms which will be used in this paper, the value of \(\ell _<\) is irrelevant. Therefore, we set \(\ell _<=0\), and consider only one family of norms: for \(\ell \ge 0\), we set
Remark 3.1
The above definition is slightly different from the standard definition for Gevrey spaces, since low frequencies are smoothed out in the definition of the weight \(\mathtt{w}\): \(\max (1,|\xi |)\) (or \(\left\langle \xi \right\rangle \)) in \(\mathtt{w}\) is usually replaced with \(|\xi |\).
Note that the following estimate is a straightforward consequence of this definition: for any \(\alpha \in {\mathbb N}^d\), \(\ell \ge 0\),
Also, in view of the standard Sobolev embedding,
valid for \(s>d/2\), we have
with the same constant C independent of \(\rho \ge 0\).
The above notation may seem rather heavy: it is chosen so because the weight \(\rho \) will depend on time, as we now discuss. For a time-dependent \(\rho \), we have:
Even though \(\rho \) depends on time, we will consider below “continuous” \({\mathcal H}_\rho ^\ell \) valued functions. We mean functions that belong to
for some interval I.
To fix the technical framework once and for all, we recall another important result from [19]. Consider the system
for some time interval I, \(0<\mu \le d\), and where \(|\nabla |^\alpha \) denotes the Fourier multiplier defined by \(\mathcal F(|\nabla |^\alpha g)(\xi )=|\xi |^\alpha {\hat{g}}(\xi )\). Lemma 3.5 from [19], which uses (3.3) as well as rather involved estimates, implies that under the assumptions
any solution of (3.4) on I, such that \((\phi ,a)\in C(I,{\mathcal H}_\rho ^{\ell +1}\times {\mathcal H}_\rho ^{k})\cap L^2_\mathrm{loc}(I, {\mathcal H}_\rho ^{\ell +1+\nu /2}\times {\mathcal H}_\rho ^{k+\nu /2})\), satisfies
In the case of a cubic nonlinearity, we want to set \(\mu =d\) in order to get a local nonlinearity. Therefore, the above algebraic conditions
imply \(\nu \ge 1\), hence \(\nu =1\) and \(k=\ell \). In view of this remark, we suppose from now on \(\nu =1\), that is, we consider analytic functions (see [19]).
Since we consider only analytic functions, we borrow from [19] the only inequalities that we will really use, which appear in [19, Lemma 3.4]:
Lemma 3.2
Let \(m\ge 0\). Then,
-
1.
For \(k+s >m+d/2+2\), and \(k,s\ge m+1\),
$$\begin{aligned} \Vert \nabla \phi \cdot \nabla a\Vert _{{\mathcal H}_\rho ^m}\le C \Vert \phi \Vert _{{\mathcal H}_\rho ^{s}} \Vert a\Vert _{{\mathcal H}_\rho ^k}. \end{aligned}$$ -
2.
For \(k+s >m+2+d/2\), \(k\ge m\) and \(s\ge m+2\),
$$\begin{aligned} \Vert a\Delta \phi \Vert _{{\mathcal H}_\rho ^m}\le C \Vert \phi \Vert _{{\mathcal H}_\rho ^{s}} \Vert a\Vert _{{\mathcal H}_\rho ^k}. \end{aligned}$$ -
3.
For \(s>d/2\),
$$\begin{aligned} \Vert \psi _1 \psi _2\Vert _{{\mathcal H}_\rho ^m}\le C\left( \Vert \psi _1\Vert _{{\mathcal H}_\rho ^m}\Vert \psi _2\Vert _{{\mathcal H}_\rho ^s} + \Vert \psi _1\Vert _{{\mathcal H}_\rho ^s}\Vert \psi _2\Vert _{{\mathcal H}_\rho ^m}\right) . \end{aligned}$$(3.5)The various constants C are independent of \(\rho \).
We infer the important lemma:
Lemma 3.3
Set \(\nu =1\), and let \({\sigma }\in {\mathbb N}\), \(\lambda \in {\mathbb R}\), \(\ell >d/2\), and I be some time interval. Let \((\varphi ,b)\in C(I,{\mathcal H}_\rho ^{\ell +1}\times {\mathcal H}_\rho ^{\ell })\cap L^2_\mathrm{loc}(I, {\mathcal H}_\rho ^{\ell +3/2}\times {\mathcal H}_\rho ^{\ell +1/2})\). Then any solution \((\phi ,a^\varepsilon )\in C(I,{\mathcal H}_\rho ^{\ell +1}\times {\mathcal H}_\rho ^{\ell })\cap L^2_\mathrm{loc}(I, {\mathcal H}_\rho ^{\ell +3/2}\times {\mathcal H}_\rho ^{\ell +1/2})\) to
satisfies
where C is independent of \(\varepsilon \) and \(\rho \).
Proof
In view of (3.3) and (3.6), we have
Cauchy–Schwarz inequality yields
Inequality (3.5) with \(m=\ell +1/2\) and \(s=\ell \) yields
where we have used (3.1). The term involving b can be treated similarly. Indeed, using (3.5) on the one hand with \(m=\ell +1/2\) and \(s=\ell \) and on the other hand with \(m=\ell =s\), we can prove by induction on \(\sigma \) that
hence the first inequality for Lemma 3.3.
For the second inequality,
Remark that
so the Laplacian term is not present in energy estimates, which are therefore independent of \(\varepsilon \). Like before, Cauchy–Schwarz inequality yields
The last term is estimated thanks to the first point in Lemma 3.2, with
Similarly,
and the last term is estimated thanks to the second point in Lemma 3.2, with
The lemma follows easily. \(\square \)
4 A fundamental estimate
In the framework of Theorem 1.1, the initial datum \(u^\varepsilon _{\mid t=0}=a_0 e^{i\phi _0/\varepsilon }\) belongs to \(H^\infty \), so the existence of \(T^\varepsilon >0\) (depending a priori on \(\varepsilon \)), and of a unique solution \(u^\varepsilon \in C([0,T^\varepsilon ],H^\infty )\) to (1.1)–(1.2), stems from standard theory (see e.g. [11]). The fact that the existence time may be chosen independent of \(\varepsilon \), along with the rest of the first point of Theorem 1.1, stems from Proposition 4.1 below.
For a decreasing function \(\rho \), we introduce the norm defined by
Proposition 4.1
Let \(\lambda \in {\mathbb R}\), \(\ell >d/2+1\), \(M_0>0\) and \((\phi _0,a_0)\in {\mathcal H}_{M_0}^{\ell +1}\times {\mathcal H}_{M_0}^{\ell }\).
-
1.
There exists \(M\gg 1\) such that if \(T<M_0/M\) and \(\rho (t) =M_0-Mt\), (2.2)–(2.3) has a unique solution
$$\begin{aligned} (\phi ^\varepsilon ,a^\varepsilon )\in C([0,T],{\mathcal H}_\rho ^{\ell +1}\times {\mathcal H}_\rho ^{\ell })\cap L^2([0,T], {\mathcal H}_\rho ^{\ell +3/2}\times {\mathcal H}_\rho ^{\ell +1/2}), \end{aligned}$$with
$$\begin{aligned} |||\phi ^\varepsilon |||_{\ell +1,T}^2 \le 2\Vert \phi _0\Vert _{{\mathcal H}_{M_0}^{\ell +1}}^2+\Vert a_0\Vert _{{\mathcal H}_{M_0}^\ell }^{4\sigma },\quad |||a^\varepsilon |||_{\ell ,T}^2 \le 2\Vert a_0\Vert _{{\mathcal H}_{M_0}^{\ell }}^2. \end{aligned}$$(4.2) -
2.
If \(R>0\) and \((\phi _0,a_0), (\varphi _0,b_0)\in {\mathcal H}_{M_0}^{\ell +1}\times {\mathcal H}_{M_0}^{\ell }\), with
$$\begin{aligned} \Vert \phi _0\Vert _{{\mathcal H}_{M_0}^{\ell +1}}+\Vert a_0\Vert _{{\mathcal H}_{M_0}^\ell }\le R,\quad \Vert \varphi _0\Vert _{{\mathcal H}_{M_0}^{\ell +1}}+\Vert b_0\Vert _{{\mathcal H}_{M_0}^\ell }\le R , \end{aligned}$$there exists \(K=K(R)\) such that if M is chosen sufficiently large such that according to the first part of the proposition, (2.2)–(2.3) has solutions \((\phi ^\varepsilon ,a^\varepsilon )\) and \((\varphi ^\varepsilon ,b^\varepsilon )\) on [0, T] corresponding respectively to the initial data \((\phi _0,a_0)\) and \((\varphi _0,b_0)\) (with the same choice of \(\rho \) and the same assumption \(T<M_0/M\)), then
$$\begin{aligned} |||\phi ^\varepsilon -\varphi ^\varepsilon |||_{\ell +1,T}+|||a^\varepsilon - b^\varepsilon |||_{\ell ,T} \le K\left( \Vert \phi _0-\varphi _0\Vert _{{\mathcal H}_{M_0}^{\ell +1}} + \Vert a_0-b_0\Vert _{{\mathcal H}_{M_0}^{\ell }}\right) . \end{aligned}$$
Remark 4.2
The proof yields a rather implicit dependence of M upon \(M_0\) and \((\phi _0,a_0)\). As a consequence, it is not clear how to choose the best possible T, even for initial data whose Fourier transform is compactly supported. For our present concern, the important information is that we get some positive T independent of \(\varepsilon \).
Proof
To construct the solution, we use the standard scheme from hyperbolic symmetric systems (see e.g. [2]), that is, we consider the iterative scheme defined by
with \(f(|a|^2)=\lambda |a|^{2{\sigma }}\), initialized with \((\phi _0^\varepsilon ,a_0^\varepsilon )(t)=(\phi _0,a_0)\). For functions at the level of regularity of the norm (4.1) with \(\ell >d/2\), the above scheme is well defined: if \(|||\phi _j^\varepsilon |||_{\ell +1,T}+|||a_j^\varepsilon |||_{\ell ,T}\) is finite, then \(\phi _{j+1}^\varepsilon \) and \(a_{j+1}^\varepsilon \) are well-defined. Indeed, in the first equation, \(\phi _{j+1}^\varepsilon \) solves a linear transport equation with smooth coefficients, and the second equation is equivalent to the linear Schrödinger equation
through the relation \(v_{j+1}^\varepsilon = a_{j+1}^\varepsilon e^{i\phi _j^\varepsilon /\varepsilon }\). This is a linear Schrödinger equation with a smooth and bounded external time-dependent potential, for which the existence of an \(L^2\)-solution is granted.
The proof of the first part of the proposition goes in two steps: first, we prove that the sequence \((|||\phi _j^\varepsilon |||_{\ell +1,T}+|||a_j^\varepsilon |||_{\ell ,T})_{j\ge 0}\) is bounded for some \(T>0\) sufficiently small, but independent of \(\varepsilon \). Then we show that up to decreasing T, the series
is converging. Note that unlike in the case of hyperbolic symmetric systems in Sobolev spaces, the regularity is the same at the two steps of the proof (in Sobolev spaces, the standard proof involves first a bound in the large norm, then convergence in the small norm). This can be seen as an illustration of the fact that these time-dependent norms, through the decay of the weight \(\rho \), “neutralize” some nonlinear effects.
First step: the sequence is bounded By integration, Lemma 3.3 yields, for a decreasing \(\rho (t)\) and \(T>0\) to be chosen later,
Hölder and Cauchy–Schwarz inequalities yield
Recall that \(M_0>0\) is given. Take \(\phi _0\in {\mathcal H}_{M_0}^{\ell +1}\), \(a_0\in {\mathcal H}_{M_0}^{\ell }\) and set \(\rho (t)=M_0-Mt\). Under the condition
the previous inequalities imply
Let us now choose \(M=|\dot{\rho }(t)|\) sufficiently large such that (4.4) holds for \(j=0\) and such that
Note that in view of (4.1), for all \(T<M_0/M\) (so that \(\rho \) remains positive on [0, T]),
and similarly
so that our constraint on M only depends on \(\Vert \phi _0\Vert _{{\mathcal H}_{M_0}^{\ell +1}}\) and \(\Vert a_0\Vert _{{\mathcal H}_{M_0}^{\ell }}\). Then, for \(T<M_0/M\), the above inequalities yield, by induction, for all \(j\ge 1\),
Second step: the sequence converges For \(j\ge 1\), consider the difference of two successive iterates: in the case of the phase, we have
along with zero initial data. Inserting the term \(|\nabla \phi _j^\varepsilon |^2\), and denoting by \(\delta \phi _{j+1}^\varepsilon = \phi _{j+1}^\varepsilon - \phi _{j}^\varepsilon \), we can rewrite the above equation as
(3.3) yields, along with Cauchy–Schwarz inequality as in the first step of the proof of Proposition 4.1:
The first two terms are estimated thanks to the last point in Lemma 3.2, as in (3.7).
Since \(f(|z|^2)\) is a polynomial in \((z,{\bar{z}})\), the last point of Lemma 3.2 yields
where we have also denoted \(\delta a_j^\varepsilon = a_j^\varepsilon -a_{j-1}^\varepsilon \). We conclude:
where K stems from the first step. For M sufficiently large,
Similarly, \(\delta a_{j+1}^\varepsilon \) solves
The last term is skew-symmetric, and thus does not appear in energy estimates. Resuming the same estimates as in the proof of Lemma 3.3, we come up with:
hence
up to increasing M (hence decreasing T). For M possibly even larger, we infer that the series
converges geometrically. Uniqueness is a direct consequence of the estimates used in this second step. (4.2) is obtained by letting j go to infinity in the estimates at the end of the first step.
The Lipschitzean property of the flow follows from calculations similar to those of the second step of the proof. \(\square \)
5 Bounds on the numerical solution
Proposition 5.1
Let \(\lambda \in {\mathbb R}\), \({\sigma }\in {\mathbb N}\), and let \((\phi ^\varepsilon ,a^\varepsilon )\) be the solution of either of the systems (2.2), (2.4), or (2.5), with the notation \(f(|z|^2)=\lambda |z|^{2{\sigma }}\). Let \(s>d/2+1\), \(\mu >0\), and \(\ell \ge s\). Suppose that \((\phi ^\varepsilon ,a^\varepsilon )\) satisfies
where \(\rho (t)=M_0-Mt\) and \(0<T<M_0/M\), with
Then, up to increasing M (and decreasing T),
Note that the assumption carries over a regularity at level \(s>d/2+1\), while the conclusion addresses the regularity at level \(\ell \ge s\): the above proposition may be viewed as a tame estimate result.
Proof
First, remark that \(|||\phi ^\varepsilon (T)|||_{s+1}+|||a^\varepsilon (T)|||_{s}\) is a non-increasing function of M, provided that the constraint \(T<M_0/M\) remains fulfilled.
Second, note that it suffices to establish the result in the case of (2.2), since the other systems contain fewer terms, and we will estimate each term present in (2.2).
The idea of the result is then to view (3.3) as a parabolic estimate, with diffusive coefficient \(-\dot{\rho }=M\). Indeed, like in the proof of Lemma 3.3, we have
We then invoke Lemma 3.2 once more. Since the first two points in Lemma 3.2 involve the constraint \(k,s\ge m\), we can rely only on (3.5). We have
since \(s>d/2+1\). We have already used the estimate
so that Young inequality yields
Again, (3.5) yields
and
We end up with
Choosing \(2M \ge C\left( \mu +\mu ^{2{\sigma }-1}\right) \) thus yields the result. \(\square \)
We readily infer:
Corollary 5.2
Let \(\ell \ge s>d/2+1\), and \(\tau >0\). Suppose that the numerical solution
satisfies
where \(\rho (t)=M_0-Mt\). Then, up to increasing M (and possibly decreasing \(\tau \)),
Proof
Proposition 5.1 yields possibly three values for M, according to which flow is considered. Note that this value depends only on \(\mu \), and recall that if \({\tilde{M}}\ge M\), \({\tilde{\rho }}(t)= M_0-{\tilde{M}} t\), we have
Therefore, considering the maximum of the two values of M given by Proposition 5.1 (for the flows associated to (2.4) and (2.5)) yields the corollary. \(\square \)
6 Local error estimate
We resume the computations from [9], based on the general formula established in [15]. For a possibly nonlinear operator A, we denote by \({\mathcal E}_A\) the associated flow:
Theorem 6.1
(Theorem 1 from [15]) Suppose that \(F(u)=A(u)+B(u)\), and denote by
the exact flow and the Lie-Trotter flow, respectively. Let \({\mathcal L} (t,u) = {\mathcal Z}^t(u)-{\mathcal S}^t(u)\). We have the exact formula
where \([B,A](v)=B'(v)A(v)-A'(v)B(v)\).
In the case of the Lie-Trotter splitting (1.5) for Eq. (1.1), we would have
where we have omitted the dependence upon \(\varepsilon \) in the notations for the sake of brevity.
However, as pointed out in [9], using the above result directly in terms of the wave function \(u^\varepsilon \) does not seem convenient. In the context of WKB regime, we rather consider the operators A and B defined by
We note that with this approach, neither A nor B is a linear operator.
Lemma 6.2
Let A and B defined by (6.1). Their commutator is given by
As a consequence, if \(\ell >d/2+3\), \(\rho >0\), \(\Vert \phi \Vert _{{\mathcal H}^{\ell +1}_\rho }\le \mu \), \(\Vert a\Vert _{{\mathcal H}^{\ell }_\rho }\le \mu \), then there exists \(C=C(\mu )\) independent of \(\varepsilon \in [0,1]\) such that
Proof
Like in [9], we have
whereas unlike in [9], we consider a function f which is not necessarily linear, so that the linearized operator of B is given by
and thus
The explicit formula for [A, B] follows as in [9]. The estimates then follow directly from (3.5) and (3.1). \(\square \)
We have the explicit formula
and we readily infer
Finally, we compute that
solves the system
where \(\displaystyle \begin{pmatrix} \phi (t)\\ a(t) \end{pmatrix}={\mathcal E}_F\left( t, \begin{pmatrix} \phi _0\\ a_0 \end{pmatrix}\right) .\)
Lemma 6.3
Let \(\ell >d/2+1\), \(s\ge \ell \) and \((\varphi _0, b_0)\in {\mathcal H}_{M_0}^{\ell +1}\times {\mathcal H}_{M_0}^{\ell }\). Assume that \((\phi ,a)\in C([0,T],{\mathcal H}_\rho ^{s+1}\times {\mathcal H}_\rho ^{s})\cap L^2([0,T],{\mathcal H}_\rho ^{s+3/2}\times {\mathcal H}_\rho ^{s+1/2})\). Then for M sufficiently large and \(T<M_0/M\), the solution to (6.4) satisfies
Proof
The proof is quite similar to the one of Lemma 3.3 and Proposition 4.1. We take the \({\mathcal H}_\rho ^{\ell +1}\) scalar product of the first equation in (6.4) with \(\varphi \), and the \({\mathcal H}_\rho ^{\ell }\) scalar product of the second one with b. We get
Then, the use of (3.5) with \(m>d/2\) and integration in time yield, with estimates similar to those presented in the proof of Proposition 4.1,
We choose \(m=\ell >d/2+1\) in the estimate for \(\varphi \) and \(m=\ell -1>d/2\) in the estimate for b. Denoting
and
since the \({\mathcal H}_\rho ^k\) norms are increasing with k, Cauchy–Schwarz in time yields
Adding the last two inequalities, we deduce, for \(\mu \le C\),
from which the inequality of the lemma easily follows provided M is sufficiently large. \(\square \)
We infer the WKB local error estimate:
Theorem 6.4
(Local error estimate for WKB states) Let \(\ell >d/2+5\), \(M_0>0\), \(M\gg 1\), \(\rho (t)=M_0-Mt\) and \(\mu >0\). Let \((\phi _0,a_0)\in {\mathcal H}_{M_0}^{\ell +1}\times {\mathcal H}_{M_0}^{\ell }\) such that
There exist \(C,c_0>0\) (depending on \(\mu \)) independent of \(\varepsilon \in (0,1]\) such that
satisfies
The above result obviously involves a loss of regularity, between the initial assumptions and the conclusion. It is important to note that the local error estimate is used only once in the final Lady Windermere’s fan argument presented in the next section, so this loss is not a serious problem.
Proof
Let \(t\in [0,c_0]\), and fix \(\tau _1,\tau _2\) such that \(0\le \tau _2\le \tau _1\le t\). Introduce the following intermediary notations:
Then in view of Theorem 6.1, we have
Since \(\ell >d/2+1\), Proposition 4.1 for \(\lambda =0\) ensures that \((\phi _1,a_1^\varepsilon )\in {\mathcal H}_{\rho (\tau _1)}^{\ell +1}\times {\mathcal H}_{\rho (\tau _1)}^{\ell }\) is well defined provided \(\tau _1\le c_0< M_0/M\), with (according to (4.2) where we can remove the \(\Vert a_0\Vert _{{\mathcal H}_{M_0}^\ell }^{4\sigma }\) term because \(\lambda =0\))
(6.2) writes \((\phi _2^\varepsilon ,a_2^\varepsilon )=(\phi _1^\varepsilon -\lambda \tau _2|a_1^\varepsilon |^{2\sigma },a_1^\varepsilon )\) and thus (3.5) yields (in the calculations below, the constant C may depend on \(\mu \) and may change from line to line)
because \(\ell >d/2\). Similarly,
Next, since \(\ell -1>d/2+3\), Lemma 6.2 implies
In view of (6.3), we have
and therefore
since \(\ell -3>d/2\) and thanks to (3.5).
Finally, we prove that if \(\ell >d/2+5\), the \({\mathcal H}_{\rho (t)}^{\ell -3}\times {\mathcal H}_{\rho (t)}^{\ell -4}\) norm of
is uniformly bounded in \(t,\tau _1,\tau _2\) as long as \(0\le \tau _2\le \tau _1\le t\le T<M_0/M\). For this purpose, first note that since \(\ell -1>d/2+1\), it follows from (6.5) and Proposition 4.1 that we can choose \(M=M(\mu )\) sufficiently large such that if \(0<T-\tau _1<\rho (\tau _1)/M\),
with
(Note that \(\rho (\tau )=\rho (\tau _1)-M(\tau -\tau _1)\)). Then, thanks to (6.6) and Lemma 6.3, since \(\ell -4>d/2+1\) and \(s=\ell -1\ge \ell -4\), choosing possibly \(M=M(\mu )\) even larger,
The theorem follows. \(\square \)
Back to the wave functions, we obtain an estimate similar to the one presented in [15, Section 4.2.2]:
Corollary 6.5
Under the assumptions of Theorem 6.4, denoting
there exist \(C,c_0>0\) (depending on \(\mu \)) independent of \(\varepsilon \in (0,1]\) such that
Proof
With the same notations as in Theorem 6.4, the Sobolev embedding of \({\mathcal H}^{\ell -4}_{\rho (t)}\) into \(L^\infty \) (\(\ell >d/2+5\)) ensures
\(\square \)
This result will not be used in the sequel, but shows how a \(1/\varepsilon \) factor appears when going back to the wave function, in agreement with the observations in [4]. The above computation also shows how to infer the first point in Corollary 1.3 from Theorem 1.1.
7 Lady Windermere’s fan
Let \(M_0>0\), \(\ell >d/2+5\), and \(v_0=(\phi _0,a_0)\in {\mathcal H}_{M_0}^{\ell +1}\times {\mathcal H}_{M_0}^{\ell }\). For the sake of conciseness, we use the following notations: for \(t>0\), \(n\in {\mathbb N}\) and \(\Delta t>0\),
For \(\rho >0\) and \(v=(\phi ,a)\in {\mathcal H}_{\rho }^{\ell +1}\times {\mathcal H}_{\rho }^{\ell }\), we also denote
According to Proposition 4.1, if \(M>0\) is sufficiently large, \(T<M_0/M\) and \(\rho (t)=M_0-Mt\), (2.2)–(2.3) has a unique solution \(v^\varepsilon \in C([0,T],{\mathcal H}_\rho ^{\ell +1}\times {\mathcal H}_\rho ^{\ell })\), with
where \(R=2\Vert v_0\Vert _{M_0,\ell }\).
We recall the notation \(t_n=n\Delta t\), and we set \(\rho _n=\rho (t_n)\). We now prove by induction on n that there exists \(c_0>0\) such that if \(\Delta t\in (0,c_0]\), for every \(n\ge 0\) such that \(n\Delta t\le T\), we have
for some \(\delta ,\gamma >0\) that will be given later. (7.1)\(_n\)–(7.2)\(_{n}\)–(7.3)\(_n\) obviously hold for \(n=0\). Let \(n>0\) such that \(n\Delta t\le T\) and assume that (7.1)\(_j\)–(7.2)\(_j\)–(7.3)\(_j\) hold for all \(j\in \{0,\ldots ,n-1\}\). Then, for all \(j\in \{0,\ldots ,n-2\}\), (7.1)\(_{j+1}\) yields
On the other hand, for \(j\in \{0,\ldots ,n-2\}\), we also have
From (7.3)\(_j\), \(\Vert v_j^\varepsilon \Vert _{\rho _{j},\ell -4} \le R/2\), whereas \(\Vert v^\varepsilon (t_j)\Vert _{\rho _{j},\ell -4} \le R\) by choice of R. Thus, since \(\ell -4>d/2+1\), Proposition 4.1 and (7.2)\(_j\) imply (up to increasing M)
Therefore, if \(c_0>0\) is chosen sufficiently small such that \(K(R)\gamma c_0\le \delta \), we have
and (7.4), (7.5) and Proposition 4.1 ensure that for all \(j\in \{0,\ldots ,n-2\}\),
Moreover, the last estimate also holds for \(j=n-1\) if K is replaced by 1. According to (7.3)\(_j\) and Theorem 6.4, we deduce that for all \(j\in \{0,\ldots ,n-1\}\),
Piling up the last inequality for \(j\in \{0,\ldots ,n-1\}\), we conclude
which proves \(7.2_{n}\) with \(\gamma =\max (1,K(R+\delta ))C(R/2)T\). Then, (7.2)\(_n\) yields
Note that it does not prove (7.1)\(_n\) yet, because the choice of \(\delta =\gamma c_0\) may be incompatible with the previous constraint \(K(R)\gamma c_0\le \delta \). However, (7.3)\(_n\) follows from (7.7) and Corollary 5.2, once we have noticed that the proof of (7.7) also works if \(v_n^\varepsilon =\mathcal {Z}_\varepsilon ^{\Delta t}v_{n-1}^\varepsilon \) is replaced by \(\mathcal {Z}_\varepsilon ^{t}v_{n-1}^\varepsilon \) (and \(t_n\) by \(t_{n-1}+t\)), for any \(0\le t\le \Delta t\), so that
Then, (7.1)\(_n\) follows from (7.3)\(_n\), and any positive value for \(\delta \) is admissible.
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This work was supported by the French ANR projects SchEq (ANR-12-JS01-0005-01) and BECASIM (ANR-12-MONU-0007-04).
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Carles, R., Gallo, C. On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit II. Analytic regularity. Numer. Math. 136, 315–342 (2017). https://doi.org/10.1007/s00211-016-0841-y
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DOI: https://doi.org/10.1007/s00211-016-0841-y