1 Introduction

In these pages we recall some recent results concerning the 1d-cubic non-linear Schrödinger equation (NLS) and other related systems. One of the main objectives is to explain in which sense

$$\begin{aligned} u_M(x,t)=c_M\sum _k e^{itk^2+ikx}, \end{aligned}$$
(1.1)

for some constant \(c_M\), is a “solution” of 1d-cubic NLS and to show the variety of phenomena it induces. Moreover, we will explain that it has a geometrical meaning due to its connection with the Binormal Flow (BF) of curves in three dimensions and the Schrödinger map (SM). Finally, we will explain how the so-called Talbot effect in Optics that is mathematically described by \(u_M\) is also present in the non-linear setting with data at the critical level of regularity. As it will be explained later on, in this geometrical interpretation the regular polygons of M sides play a crucial role and the choice of the constant \(c_M\) of (1.1) is particularly delicate. It is for this reason that we write \(c_M\) in (1.1) instead of using a generic complex constant c. Also the notion of solution is quite involved and we will try to give some hints in which sense the solution has to be understood.

Altogether we are speaking about a family of PDE problems. Consider first NLS which is a complex scalar equation with a cubic non-linear potential:

$$\begin{aligned} \left\{ \begin{array}{rl} \partial _t u&{}=i\left( \partial _x^2 u+(|u|^2-\mathcal {M}(t))\right) u,\qquad \mathcal {M}(t)\in \mathbb R,\\ u(x,0)&{}=u_0(x),\qquad x\in \mathbb R. \end{array}\right. \end{aligned}$$
(1.2)

Let us introduce next SM. Calling T(x, t) a unit vector in \(\mathbb R^3\) the Schrödinger Map onto the sphere is given by

$$\begin{aligned} \partial _tT=T\wedge \partial _x^2T. \end{aligned}$$
(1.3)

Finally, observe that the vector T(x, t) can be seen at any given time as the tangent vector of a 3d-curve \(\chi (x,t)\)

$$ \partial _x\chi = T, $$

with \(\chi \) a solution of

$$\begin{aligned} \partial _t\chi =\partial _x\chi \wedge \partial _x^2\chi . \end{aligned}$$
(1.4)

Da Rios [16] proposed (1.4) as a simplified model that describes the evolution of vortex filaments. Remember that Frenet equations read

$$\begin{aligned} T_x= & {} \qquad \kappa N\\ N_{x}= & {} -\kappa T\quad +\quad \tau B\\ B_{x}= & {} \qquad -\tau N \end{aligned}$$

with \(\kappa \) the curvature of the curve, \(\tau \) its torsion, N the normal vector, and B the binormal vector. Hence

$$ \partial _t\chi =\partial _x\chi \wedge \partial _x^2\chi =\kappa B. $$

That is the reason why sometimes the system of PDEs (1.4) is called the Binormal Flow.

The connection of the two systems BF and SM with (1.2) was established by Hasimoto in [26] through a straightforward computation. This computation is slightly simplified if instead of the Frenet frame one uses the parallel one. This is given by vectors \((T,e_1,e_2)\) that satisfy

$$\begin{aligned} T_x= & {} \alpha e_1+\beta e_2\nonumber \\ e_{1x}= & {} -\alpha T\\ e_{2x}= & {} -\beta T.\nonumber \end{aligned}$$
(1.5)

Defining \(u=\alpha + i \beta \) it is proved that u solves (1.2) for some given \(\mathcal {M}(t)\). From (1.3) and (1.4) we get

$$\begin{aligned} \partial _tT=\alpha _x e_2-\beta _x e_1 \end{aligned}$$
(1.6)

and

$$\begin{aligned} \partial _t\chi =\alpha e_2-\beta e_1. \end{aligned}$$

These two equations are gauge invariant, namely that given any real \(\varphi \) we can change \(e_1+ie_2\) into \(e^{i\varphi }(e_1+ie_2)\) and \(\alpha + i \beta \) into \(e^{i\varphi }(\alpha + i \beta )\) and (1.3) and (1.4) remain the same. Therefore, we have the freedom of choosing \(\mathcal {M}(t)\).

Observe that u, the solution of (1.2), gives the curvature and the torsion of \(\chi \). More concretely, taking \(e_1+ie_2= e^{i\int _0^x \tau (r)\,dr}(n+ib)\) we get that

$$ |u|^2=\alpha ^2+\beta ^2=\kappa ^2 $$

and after some calculation (see [26] and (4.2) below)

$$\begin{aligned} u(x,t)=\kappa e^{i\int _0^x \tau (r)\,dr}, \end{aligned}$$
(1.7)

with \(\varvec{\tau }\) denoting the torsion. (1.7) is usually called Hasimoto transformation.

A relevant simple example is

$$\begin{aligned} u_o=c_o\frac{1}{\sqrt{4\pi t}} e^{i x^2/4t},\qquad \mathcal {M}(t)=\frac{c_o^2}{4\pi t}, \end{aligned}$$
(1.8)

which is related to the self-similar solutions of SM and BF. Formally \(u_o(x,0)=c_o\delta \), and the corresponding \(\chi \) has a corner at \((x,t)=(0,0).\) Here \(\delta \) stands for the Dirac-\(\delta \) function. We will sometimes refer to this solution as either a fundamental brick or a coherent structure [35]. Our main interest is to consider rough initial data as polygonal lines and regular polygons. As we will see, the latter are related to (1.1) and therefore \(u_M\) could be understood as a superposition of infinitely many simple solutions \(u_o (x- j)\) centered at the integers j. As a consequence, the curve obtained from \(u_M\) can be seen as an interaction of these coherent structures (see https://www.youtube.com/watch?v=fpBcwuY57FU).

It is important to stress that to obtain \(\chi \) from T, besides integrating in the spatial variable the parallel system (1.5), one has to find the trajectory in time followed by one point of \(\chi _o\). This is not obvious even for (1.8), see [23]. It turns out that to compute that trajectory of, say, one corner of a regular polygon is rather delicate. The corresponding curve can be as complicated as the graph of the so-called Riemann’s non differentiable function:

$$\begin{aligned} \sum _k\frac{e^{itk^2}-1}{k^2}. \end{aligned}$$

See [5] for more details.

We will review some recent results regarding the IVP for (1.2) in Sections 2 and 3. In particular, we will show the existence of three new conservation laws (2.4), (3.7), and (3.9), valid at the critical level of regularity. As it is well known (1.2), and as a consequence also (1.3) and (1.4), are completely integrable systems with infinitely many conservation laws that start at a subcritical level of regularity, \(L^2\). For the others laws more regularity, measured in the Sobolev class, is needed. In Section 4 we will recall some work done on the transfer of energy for the Schrödinger map (1.3). Finally, in Section 5 we revisit the Talbot effect and modify some examples obtained in [3] to establish a connection with some recent work on Rogue Waves given in [20].

2 The Initial Value Problem

We start with the IVP associated to NLS equation (1.2):

$$ \left\{ \begin{array}{rl} \partial _t u&{}=i\left( \partial _x^2 u+(|u|^2-\mathcal {M}(t))\right) u,\qquad \mathcal {M}(t)\in \mathbb R,\\ u(x,0)&{}=u_0(x),\qquad x\in \mathbb R. \end{array}\right. $$

We are interested in initial data which are at the critical level of regularity. There are two symmetries that leave invariant the set of solutions that we want to consider. One is the scaling invariance: if u is a solution of (1.2), then

$$\begin{aligned} \lambda >0,\qquad u_\lambda (x,t)=\lambda u(\lambda x,\lambda ^2t), \end{aligned}$$
(2.1)

is also a solution of (1.2) with \(\lambda ^2\mathcal {M}(t)\) instead of \(\mathcal {M}(t)\). The second one is the so-called Galilean invariance: If

$$\begin{aligned} \nu \in \mathbb R,\qquad u^{\nu } (x,t)=e^{-it\nu ^2+i\nu x}u(x-2\nu t,t), \end{aligned}$$
(2.2)

then \(u^{\nu }\) is also a solution of (1.2) with the same \(\mathcal {M}(t).\) Hence, we want to work in a functional setting where the size of the initial data does not change under the scaling and Galilean transformations.

Let us review the classical results on NLS. The well-posedness of 1D cubic NLS on the full line and on the torus was firstly done in [8, 41] for data in \(L^2\). Observe that the space \(L^2(\mathbb R)\), although is invariant by Galilean symmetry (2.2), misses the scaling (2.1) by a power of 1/2 in the homogeneous Sobolev class \(\dot{H}^s\). In fact, the critical exponet for scaling is \(s=-1/2\) which is not invariant under the Galilean symmetry. The first result obtained beyond the \(L^2(\mathbb R)\) theory was given in [42] using some spaces of tempered distributions built on the well known Strichartz estimates. Later on in [21] well-posedness is studied in the Fourier–Lebesgue spaces that we denote by \( \mathcal {F}L^p \). These are spaces where the Fourier transform is bounded in \( L^p(\mathbb R) \) and therefore leave invariant (2.2). Moreover, \(\mathcal {F}L^{\infty } \) is also scaling invariant and therefore critical according to our definition. In [22] local well-posedness, also under periodic boundary conditions, was shown in \(\mathcal {F}L^p\) with \(2< p < + \infty \).

In the setting of Sobolev spaces of non-homogeneous type the progress has been remarkable. On the one hand, there is ill-posedness in \(H^{s}\) with \(s < 0\) in the sense that there is no uniformly continuous data-to-solution map and even some growth of the Sobolev norms has been proved, [11, 14, 29, 31, 32]. On the other hand, there is well-posedness in \(H^{s}\) for \(s > -1/2\) as has been shown in [27]. In this case a weaker notion of continuity for the data-to-solution map is used.

In this paper, we will consider solutions of (1.2) such that

$$\begin{aligned} \omega (\xi , t):=e^{it\xi ^2}\widehat{u}(\xi ,t)~\text { is }~2\pi \text {-periodic.} \end{aligned}$$
(2.3)

Here \(\widehat{u}\) denotes the Fourier transform of u,

$$ \int _{\mathbb R} e^{-ix\xi }u(x)\,dx. $$

To prove that this periodicity is preserved by the evolution is not completely obvious and it is a relevant property of (1.2). It can be proved writing the equation for \(\omega \), (\(\mathcal {M}(t)=0)\) in (1.2):

$$ \partial _t \omega (\eta , t)=\frac{i}{8\pi ^3}e^{-it\eta ^2}\int \int _{\xi _1+\xi _2+\xi _3-\eta =0} e^{it(\xi _1^2-\xi _2^2+\xi _3^2)} \omega (\xi _1)\bar{\omega }(\xi _2)\omega (\xi _3) \,d\xi _1d\xi _2d\xi _3. $$

Under the condition \(\xi _1-\xi _2+\xi _3-\eta =0\), we get

$$ \xi _1^2-\xi _2^2+\xi _3^2-\eta ^2=2(\xi _1-\xi _2)(\xi _1-\eta ). $$

The last quantity is invariant under translations so that the periodicity is formally preserved. Interestingly this calculation does not work for general dispersive systems as for example for modified KdV.

One of the three new conservation laws is precisely

$$\begin{aligned} \int _0^{2\pi }|\omega (\xi , t)|^2 \, d\xi = \text {constant}. \end{aligned}$$
(2.4)

This can be seen writing \(\omega (\xi ,t)=\sum _j A_j(t) e^{ij\xi }\) and looking for the ODE system that the Fourier coefficients \(A_j\) have to satisfy. Historically, our approach to this question has been different and this is what we explain next.

For solving (1.2) and following Kita in [30], we considered the ansatz

$$\begin{aligned} u(x,t)=\sum _j \tilde{A}_j(t) e^{it\partial _x^2}\delta (x-j) \end{aligned}$$
(2.5)

and therefore

$$ \widehat{u}(\xi ,t)=e^{-it\xi ^2}\sum _j \tilde{A}_j(t) e^{ij\xi }. $$

Define

$$\begin{aligned} V(y,\tau )=\sum _j \hat{V}_{\tau }(j)(\tau )e^{ ijy}, \end{aligned}$$
(2.6)

with \(\hat{V_\tau }(j)\) the j-Fourier coefficient of \(V(y,\tau )\). Then

$$\begin{aligned} u(x,t)= & {} \frac{1}{(4\pi it)^{1/2}}\sum _j \tilde{A}_j(t)e^{i\frac{(x-j)^2}{4t}}\\= & {} \frac{1}{(4\pi it)^{1/2}}e^{i\frac{|x|^2}{4t}}\sum _j \tilde{A}_j(t)e^{i\frac{j^2}{4t}-i\frac{x}{2t}j}\\:= & {} \frac{1}{(4\pi it)^{1/2}}e^{i\frac{|x|^2}{4t}}\overline{V}\left( \frac{x}{2 t},\frac{1}{t}\right) , \end{aligned}$$

with

$$\begin{aligned} \hat{V}_\tau (j)=\tilde{B}_j(\tau )e^{-i\frac{\tau }{4} j^2},\qquad \tilde{B}_j(\tau )=\overline{\tilde{A}_j}\left( \frac{1}{\tau }\right) . \end{aligned}$$
(2.7)

Finally, doing the change of variables

$$ y=\frac{x}{2 t},\qquad \tau =1/t, $$

we easily obtain that V solves

$$\begin{aligned} \partial _{\tau }V=i\left( \partial _y^2+\frac{1}{4\pi \tau }(|V|^2-m)\right) V;\qquad m(\tau )=\frac{4\pi }{\tau }\mathcal {M}\left( \frac{1}{\tau }\right) . \end{aligned}$$
(2.8)

We actually have that V is a pseudo-conformal transformation of u.

Remark 2.1

  1. 1.

    Observe that formally solutions of (2.8) remain periodic if they are periodic at a given time. That means that given the Fourier coefficients \({\hat{V}}_\tau (j)\) and using (2.7) to define \(\tilde{A}_j\) we conclude that the periodicity of \(\omega (\xi , t)=e^{it\xi ^2}\widehat{u}(\xi ,t)\) is also formally preserved.

  2. 2.

    The change of variable transforms \(t=0\) into \(\tau =\infty \). Hence, the initial value problem for u becomes a question about the scattering of the solutions of (2.8).

  3. 3.

    If \(V(1)=c_o\), and \(m=c_o^2\) then \(V(\xi ,t)=c_o\) for all t. The corresponding solution is the fundamental brick (1.8)

    $$ u(x,t)=u_o=\frac{c_o}{(4\pi it)^{1/2}} e^{i\frac{|x|^2}{4t}}, $$

    and \(u_0=c_o\delta \). This implies that unless we include the term \(\mathcal {M}(t)=c_o^2/(4\pi t)\) in (1.2) the IVP for the Dirac delta is ill–posed, something observed in [29]. It was proved in [1, 2] that even if this term is added and one looks for solutions of the type \(V=c_o+z\) with z small with respect to \(c_o\) the corresponding u of (1.2) cannot be defined for \(t=0\).

  4. 4.

    It immediately follows from the conservation of mass of (2.8) that

    $$\begin{aligned} m_0=\int _{0}^{2\pi } |V(\xi ,\tau )|^2 d\xi =\sum _{j}|\tilde{B}_j(\tau )|^2 \end{aligned}$$

    is formally constant for \(\tau >0\). And from (2.7) we also get that \(\sum _j|\tilde{A}_j(t)|^2=\sum _j|\tilde{B}_j(1/t)|^2=\int _0^{2\pi }|\omega (\xi ,t)|^2\,d\xi \) remains constant, which is (2.4).

3 Conservation Laws

In [3] a first result on the IVP (1.2) within the functional setting we have just described was obtained. The solution u is written as in (2.5) and for any \(a_j\) it is defined \(R_j\) by the identity

$$\begin{aligned} \,\tilde{A}_j(t) =a_j+R_j(t). \end{aligned}$$

Then, an infinite ODE system for \(R_j\) can be easily obtained. The corresponding solution is constructed through a fixed point argument in an appropriately chosen space which among other things implies that \(R_j(0)=0\). The condition on the data is that \(\sum _j |a_j|\) is finite (i.e. \(a_j\in l^1\)) but not necessarily small. This condition implies the

$$ \lim _{x\rightarrow \pm \infty }T(x,t)=A^{\pm },\qquad t\ge 0. $$

It is proved that \(A^{\pm }\) is independent of t and therefore

$$\begin{aligned} \widehat{T_x}(0,t)=\int _{-\infty }^{+\infty }\,T_x(x, t)\,dx=A^+-A^-. \end{aligned}$$
(3.1)

The result in [3] is local in time. A global result is obtained by assuming the extra condition

$$\begin{aligned} \sum _j j^2|a_j|^2<+\infty , \end{aligned}$$
(3.2)

whose evolution \(\sum _j j^2|\tilde{A}_j(t)|^2\) is easy to determine as we explain next.

First of all, it is much more convenient to work with the solution V of (2.8) which was defined in (2.6). Moreover, we define

$$\begin{aligned} A_j(t)= \tilde{A}_j(t) e^{i\phi _j(t)}, \qquad \phi _j(t)=e^{i\frac{|a_j|^2}{4\pi }\log t}, \end{aligned}$$
(3.3)

and correspondingly (cf. (2.7))

$$ B_j(\tau )= \bar{A}_j(1/\tau ). $$

Let us introduce for each k the non-resonant set

$$ NR_k=\{(j_1,j_2,j_3),~k-j_1+j_2-j_3=0,~ k^2-j_1^2+j_2^2-j_3^2\ne 0\}. $$

Similarly the resonant set is given by the solutions of

$$ k-j_1+j_2-j_3=0,\quad k^2-j_1^2+j_2^2-j_3^2= 0. $$

Define

$$ w_{k,j_1,j_2}:=k^2-j_1^2+j_2^2-j_3^2. $$

It is immediate to see that if \(k-j_1+j_2-j_3=0\) then

$$\begin{aligned} w_{k,j_1,j_2}=2(k-j_1)(j_1-j_2). \end{aligned}$$
(3.4)

Taking \(m=2\sum _j|a_j|^2\). After some computation one gets that the ODE system that \( B_j\) has to verify is

$$\begin{aligned} i\partial _\tau B_k(\tau )= & {} \frac{1}{8\pi \tau }\sum _{NR_k}\!e^{-i\tau (k^2-j_1^2+j_2^2-j_3^2)-\frac{i}{4\pi }(|a_k|^2-|a_{j_1}|^2+|a_{j_2}|^2-|a_{j_3}|^2)\log t} B_{j_1}(\tau )\overline{ B_{j_2}(\tau )} B_{j_3}(\tau )\nonumber \\{} & {} -\frac{1}{8\pi \tau }(| B_k|^2 -|a_k|^2) B_k. \end{aligned}$$
(3.5)

Hereafter this is the system we are going to solve. Recall that we have got it choosing \(m=2\sum _j|a_j|^2<+\infty \). For a regular polygon, an example we will consider below, m is not finite any more and therefore we are doing a renormalization as the Wick renormalization done in [9] and in [13].

For any real function c(k) we want to understand the behavior of

$$\begin{aligned} \frac{d}{d\tau } \sum _k c(k)|B_k(\tau )|^2 \end{aligned}$$
(3.6)

as a function of \(\tau \). A symmetrization argument makes to appear \(c(k)-c(j_1)+c(j_2)-c(j_3)\) that vanishes in the resonant set because of (3.4). Then, we have that

$$\begin{aligned}{} & {} \frac{d}{d\tau } \sum _k c(k)|B_k(\tau )|^2\\{} & {} =\frac{1}{2 i\tau }\sum _{k-j_1+j_2-j_3=0}(c(k)-c(j_1)+c(j_2)-c(j_3))e^{-i\tau w_{k,j_1,j_2}}B_{j_1}(\tau )\overline{B_{j_2}(\tau )}B_{j_3}(\tau )\overline{B_k(\tau )}\\{} & {} =\frac{1}{2i\tau }\sum _{k;NR_k}(c(k)-c(j_1)+c(j_2)-c(j_3))e^{-i\tau w_{k,j_1,j_2}}B_{j_1}(\tau )\overline{B_{j_2}(\tau )}B_{j_3}(\tau )\overline{B_k(\tau )}. \end{aligned}$$

Relevant examples are \(c(j)=1\) that gives (2.4), the \(L^2\) conservation law already mentioned, and \(c(j)=j\), that yields a second conservation law:

$$\begin{aligned} \sum _j j|B_j(\tau )|^2=\sum _j j|A_j(1/\tau )|^2~\text { is constant.} \end{aligned}$$
(3.7)

The final example is \(c(j)=j^2\), cf. (3.2). In this case (3.6) can be written in terms of V as

$$ \int _0^{2\pi } |\partial _y V (y,\tau )|^2\,dy. $$

Then, defining

$$ E(\tau )=\int |\partial _y V (y,\tau )|^2-\frac{1}{16\pi \tau }(|V(y,\tau )|^2-m)^2\,dy, $$

a direct computation gives that solutions of (2.8) satisfy

$$ \frac{d}{d\tau }E(\tau )=\frac{1}{16\pi \tau ^2}\int (|V(y,\tau )|^2-m)^2\,dy. $$

The next step about (1.2) with the ansatz (2.5)-(3.3) was given in [5], where the Picard iteration is done measuring more carefully the first iterate. Particular attention is given to the example

$$\begin{aligned} a_j=1\quad \text { for }~|j|\le N\quad \text {and zero otherwise}, \end{aligned}$$
(3.8)

see [5].

Finally in [10], Bourgain’s approach [8] is followed. This amounts to use the Sobolev spaces of the coefficients \(B_j(\tau )\). The results in that paper can be summarized as follows, for initial datum in \(l^p\), \(p \in (1,+\infty )\):

  1. 1.

    Local well-posedness with a smallness assumption in \(l^p\) for the initial datum: for any \(T > 0\), there exists \(\varepsilon (T) > 0\) such that if the \(l^p\) norm of the initial datum \(\{a_{j}\}\) is smaller then \(\varepsilon (T)\), then there exists a unique solution of (1.2)-(2.5)-(3.3) in [0, T] in an appropriate sense.

  2. 2.

    Local well-posedness with a smallness assumption in \(l^{\infty }\) for the initial datum: if the \(l^{\infty }\) norm of \(\{a_j\}\) is small enough then there exists a time \(T(\Vert \alpha \Vert _{l^{\infty }},\Vert \alpha \Vert _{l^p})\) such that a unique solution of (1.2)-(2.5)-(3.3) exists in [0, T] in an appropriate sense.

  3. 3.

    For \(p = 2\), global in time well-posedness with a smallness assumption in \(l^{\infty }\) for the initial datum. As it can be expected this result follows from (2) and the \(l^2\) conservation law. The smallness condition comes from the linear term that is treated as a perturbation. We don’t know if this smallness condition can be removed.

For establishing the third conservation law we have to observe that \(w_{k,j_1,j_2}\) given in (3.4) is invariant under translations. This implies that solutions of (3.5) such that \(B_{j+M}=B_j\) at a given time formally preserved this property for all time. Therefore,

$$\begin{aligned} \sum _{j=1}^M|a_j|^2=\sum _{j=1}^M|A_j(t)|^2=\sum _{j=1}^M|B_j(1/t)|^2=\text {constant}. \end{aligned}$$
(3.9)

This conservation law is much stronger than (2.4) because it just assumes an \(l^\infty \) condition on \(a_j\).

As a consequence, in Proposition 1 of [10] an explicit solution of (3.5) is obtained with

$$\begin{aligned} c_M=a_j\qquad \text {for all } j \end{aligned}$$
(3.10)

as in (1.1). For this solution \(|B_j(\tau )|=c_M\) and \(B_j(\tau )\) is independent of j.

Recall that the ansatz (2.5) has to be modified as in (3.3) to construct a solution of (1.2). The logarithmic correction given by (3.3) implies that even though there is a limit of the amplitudes \(|A_j(t)|=|B_j(1/t)|\) when t approaches zero, the limit of the phases does not exist, and therefore the IVP (1.2) is ill posed. As it was proved in [3] this loss is irrelevant when (1.2) is understood in connection to BF and SM. For example, for BF the solutions can be perfectly defined at \(t=0\) as a polygonal line. Moreover, the behavior close to a corner is determined by a self-similar solution (1.8). This self-similar solution, and the precise theorem given about them in [23], gives the necessary information at \(t=0\) so that the flow can be continued for \(t<0\). A crucial ingredient in this process is the precise relation established in [23] between \(c_o\) and the angle \(\theta _o\) of the corresponding corner, namely

$$\begin{aligned} \sin \theta _o=e^{-\frac{\pi c_o^2}{2}}. \end{aligned}$$

For a regular polygon with M sides the angle is \(\theta _M=2\pi /M\). Chossing \(c_M\) in such a way that

$$ \sin \left( \frac{2\pi }{M}\right) =e^{-\frac{\pi c_M^2}{2}}, $$

and using it in (3.10), we obtain a solution, once the Wick renormalization is done, for the case of a regular polygon at the level of NLS. This choice is the one conjectured in [17] based on the numerical simulations done in [28].

4 Transfer of Energy

Solutions of (1.2) have associated a natural energy/mass density which is \(|u(x,t)|^2\,dx\). For solutions of (1.3), this is the same density associated to \(|T_x|^2\) thanks to Hasimoto transformation (1.7). In Section 3 we have constructed solutions of (1.2) whose energy density is also well described in terms of \(|\widehat{u}|^2\) as \(|\widehat{u}|^2=|\omega |^2\) with \(\omega \) given in (2.3). Also remember that \(\omega \) is related to V through (2.7) and (2.8). Even though there is no reason a priori to think that \(\big |\widehat{T_x}(\xi ,t)\big |^2 d\xi \) is related to a density energy, in Theorem 1.1 of [4] it is proved that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{2\pi n}^{2\pi (n+1)}\left| \widehat{T_x}(\xi ,t)\right| ^2 d\xi =\int _0^{2\pi } |V(\xi ,t)|^2 d\xi . \end{aligned}$$
(4.1)

This equality suggests that at least in the limit of large frequencies \(\big |\widehat{T_x}(\xi ,t)\big |^2 d\xi \) is related to an energy density. Recall that it was proved in Sections 2 and 3 that

$$ \int _0^{2\pi } |V(\xi ,t)|^2 d\xi = \text {constant} $$

and therefore, that there is no flux of energy for \(\widehat{u}\) at least for \(0<t<1\). The situation for \(T_x\) is different. It was proved in [6] that there is some cascade of energy. More concretely we have the following result that was motivated by some numerical experiments done in [18].

Theorem 4.1

Assume

$$ \left\{ \begin{array}{rl} a_{-1}&{}=a_{+1}\ne 0,\\ a_j&{}=0\quad \text {otherwise.} \end{array}\right. $$

Then there exists \(c>0\) such that

$$ \sup _\xi \left| \widehat{T_x} (\xi ,t)\right| ^2 \ge \sup _{\xi \in B\left( \pm \frac{1}{t},\sqrt{t}\right) }\left| \widehat{T_x} (\xi ,t)\right| ^2\ge c|\log t|,\qquad t>0. $$

This type of energy cascade is an alternative to the ones in [15, 25], where results about the growth of Sobolev norms are proved. In these papers the authors consider Sobolev spaces \(H^s\) with \(s>1\), so that the growth is a consequence of a transfer of energy from low Fourier modes to high Fourier modes. From (4.1) we conclude that in our setting \(\widehat{T_x}\) does not tend to zero in the large frequency limit, so that \(T_x\) does not belong to \(L^2\). Also observe that the initial data given in Theorem 4.1 satisfies (3.2) that together with (3.1) give that \(\widehat{T_x}\) remains bounded for small frequencies for \(0<t<1\). It is also proved in [6] that \(\widehat{T_x}(\cdot ,1)\) is bounded. Nevertheless, the \(L^\infty \) norm grows logarithmically when t tends to zero for frequencies \(\xi \) such that \(|\xi |\sim 1/ t\).

Recall that if \(u=\alpha + i \beta \) is the solution of (1.2) then T can be obtained from (1.5), and the equation for \(T_t\) is (1.6). After a simple calculation one gets that

$$\begin{aligned} \partial _x( e_1\cdot e_{2 t})= -\frac{1}{2}\partial _x(\alpha ^2+\beta ^2). \end{aligned}$$
(4.2)

This fact, together with the property that \((T, e_1,e_2)\) is an orthonormal frame for all (x, t) gives

$$\begin{aligned} T_t= & {} -\beta _x e_1+ \alpha _x e_2\\ e_{1_t}= & {} \beta _x T +\frac{1}{2}((\alpha ^2+\beta ^2)-\mathcal {M}(t))e_2\\ e_{2_t}= & {} -\alpha _x T-\frac{1}{2}((\alpha ^2+\beta ^2)-\mathcal {M}(t))e_1, \end{aligned}$$

for some real function \(\mathcal {M}(t)\).

Notice that this is just a linear system of equations which is hamiltonian and that satisfies the three conservation laws (2.4), (3.7), and (3.9) given in the previous sections for u of type (2.5). Nevertheless, Theorem 4.1 applies and therefore this system also exhibits a cascade of energy.

5 Talbot Effect and Rogue Waves

In this section we want to revisit the examples on the Talbot effect showed in [3]. The Talbot effect was originally observed in linear Optics and has received plenty of attention since the pioneering work of M. Berry and collaborators (see for example [7]). In [17] the fractal properties of this effect is considered, taking in (1.4) regular polygons as initial data. These examples suggest a possible connection with the turbulent dynamics observed in non-circular jets (see for instance [24]). At a more regular level, the Talbot effect for the linear and nonlinear Schrödinger equations with initial data periodic and given by functions with bounded variation was studied in [12, 19, 33, 37, 40]. The fractal behavior was already seen numerically in the setting of the architecture of aortic valve fibers in [36, 38]. These works use (1.4) but with boundary conditions which are not periodic.

The Talbot effect is very well described by (1.1). As it will be shown below in (5.2) and (5.3), the values of (1.1) at times which are rational multiples of the period can be written in a closed form: if the rational is p/q then Dirac deltas appear at all the rationals of \(\mathbb Z/q\) in space, and the amplitudes are given by a corresponding Gauss sum. Going either backward or forward in time a phenomenon of constructive/destructive interference appears. We think that there is a similitude of this phenomenon with the one exhibited in [20] related to the so-called Rogue Waves.

The example we propose is very similar to (3.8). Recall that the construction of the solution we have sketched in the previous sections is perturbative and therefore, it always implies some smallness condition. This condition is measured in terms of \(\sum |a_j|\) that can be small without the corresponding solution u being small. For example, from (2.5) it is immediate that at least for small times the \(L^\infty \) norm of u is not small. Something similar can be said for the \(L^1_{\text {loc}}\) norm. At this respect it is relevant to notice the definition of u in terms of V given in (2.8). Observe that

$$ |u(x,t)|=\frac{1}{\sqrt{4\pi t}}\, |V(x/2t,1/t)|, $$

and therefore \(L^1_{\text {loc}}\) grows with t.

We have the following result.

Theorem 5.1

(Appearance of Rogue Waves) Let \(0<\eta <\frac{1}{4}\) and let \(p\in \mathbb N\) large. There exists \(u_0\) with \(\widehat{u_0}\) a \(2\pi \)-periodic function, supported modulo \(2\pi \) in \(\left[ -\eta \frac{2\pi }{ p},\eta \frac{2\pi }{ p}\right] \), such that the solution u(x, t) of (1.2) obtained from \(a_k=\widehat{u_0}(k)\) satisfies the following properties:

  • If \(t_{\tilde{p},\tilde{q}}=\frac{1}{2\pi }\frac{\tilde{p}}{\tilde{q}}\) with \(\tilde{p}\approx \tilde{q}\approx 1\), coprime, \(\tilde{p}<\tilde{q}\), and \(\tilde{q}\) odd, then at times \(t_{\tilde{p},\tilde{q}}\) on the interval \([-\frac{1}{2\tilde{q}},\frac{1}{2\tilde{q}}]\) the solution u(x, t) is localized and has a large amplitude;

  • If \(t_{p,q}=\frac{1}{2\pi }\frac{p}{q}\) with \(q\approx p\), coprime, and q odd, then at times \(t_{p,q}\) on the interval \([-\frac{1}{2\tilde{q}},\frac{1}{2\tilde{q}}]\) the solution u(x, t) has a small amplitude.

We start with a computation for the linear Schrödinger equation on the line, concerning the Talbot effect related to (1.1).

Proposition 5.2

(Talbot effect for linear evolutions) Let \(0<\eta <\frac{1}{4}\), \(p\in \mathbb N\) and \(u_0\) such that \(\widehat{u_0}\) is a \(2\pi \)-periodic function, supported modulo \(2\pi \) in \([-\eta \frac{2\pi }{ p},\eta \frac{2\pi }{ p}]\). For all \(t_{p,q}=\frac{1}{2\pi }\frac{p}{q}\) with q odd and for all \(x\in \mathbb R\) we define

$$ \xi _{x} :=\frac{\pi q}{p} \,d\left( x,\frac{1}{q}\mathbb Z\right) \in \left[ 0,\frac{\pi }{p}\right) . $$

Then, there exists \(\theta _{x,p,q}\in \mathbb R\) such that

$$\begin{aligned} e^{it_{p,q}\varDelta }u_0(x)=\frac{1}{\sqrt{q}} \,\widehat{u_0}(\xi _x)\, e^{-it_{p,q}\,\xi _x^2+ix\,\xi _x+i\theta _{x,p,q}}. \end{aligned}$$
(5.1)

In particular \(|e^{it_{p,q}\varDelta }u_0|\) is \(\frac{1}{q}\)-periodic and if \(d(x,\frac{1}{q}\mathbb Z)>\frac{2\eta }{q}\) then \(e^{it_{p,q}\varDelta }u_0(x)\) vanishes.

Proof

We start by recalling the Poisson summation formula \(\sum _{k\in \mathbb Z}f(k)=\sum _{k\in \mathbb Z}\hat{f}(2\pi k)\) for the Dirac comb:

$$ \left( \sum _{k\in \mathbb Z}\delta _k\right) (x)=\sum _{k\in \mathbb Z}\delta (x-k)=\sum _{k\in \mathbb Z}e^{i2\pi kx}, $$

as

$$ \widehat{\delta (x-\cdot )}(2\pi k)=\int _{-\infty }^\infty e^{-i2\pi ky}\delta (x-y)\,dy=e^{-i2\pi kx}. $$

The computation of the free evolution with Dirac comb data is

$$\begin{aligned} e^{it\varDelta }\left( \sum _{k\in \mathbb Z}\delta _k\right) (x)=\sum _{k\in \mathbb Z}e^{-it(2\pi k)^2+i2\pi kx}. \end{aligned}$$
(5.2)

For \(t=\frac{1}{2\pi }\frac{p}{q}\) we have (choosing \(M=2\pi \) in formulas (37) combined with (42) from [17])

$$\begin{aligned} e^{it\varDelta }\left( \sum _{k\in \mathbb Z}\delta _k\right) (x)=\frac{1}{q}\sum _{l\in \mathbb Z}\sum _{m=0}^{q-1}G(-p,m,q)\delta \left( x-l-\frac{m}{q}\right) , \end{aligned}$$
(5.3)

which describes the linear Talbot effect in the periodic setting. Here \(G(-p,m,q)\) stands for the Gauss sum

$$ G(-p,m,q)=\sum _{l=0}^{q-1}e^{2\pi i\frac{-pl^2+ml}{q}}. $$

Now we compute the free evolution of data \(u_0\) with \(\widehat{u_0}\) a \(2\pi \)-periodic function, i.e. \(\widehat{u_0}(\xi )=\sum _{k\in \mathbb Z}\alpha _ke^{-ik\xi }\) and \(u_0=\sum _{k\in \mathbb Z}\alpha _k\delta _k\):

$$\begin{aligned} e^{it\varDelta }u_0(x)= & {} \frac{1}{2\pi }\int _{-\infty }^\infty e^{ix\xi }e^{-it\xi ^2}\widehat{u_0}(\xi )\,d\xi \\= & {} \frac{1}{2\pi }\sum _{k\in \mathbb Z}\int _{2\pi k}^{2\pi (k+1)} e^{ix\xi -it\xi ^2}\widehat{u_0}(\xi )\,d\xi \\= & {} \frac{1}{2\pi }\int _0^{2\pi }\widehat{u_0}(\xi )\sum _{k\in \mathbb Z}e^{ix(2\pi k+\xi )-it(2\pi k+\xi )^2}\,d\xi \\= & {} \frac{1}{2\pi }\int _0^{2\pi }\widehat{u_0}(\xi )e^{-it\xi ^2+ix\xi }\sum _{k\in \mathbb Z}e^{-it\,(2\pi k)^2+i2\pi k (x- 2t\xi )}\,d\xi . \end{aligned}$$

Therefore, for \(t_{p,q}=\frac{1}{2\pi }\frac{p}{q}\) we get using (5.2)–(5.3):

$$ e^{it_{p,q}\varDelta }u_0(x)=\!\frac{1}{q}\int _0^{2\pi }\widehat{u_0}(\xi )e^{-it_{p,q}\xi ^2+ix\xi }\sum _{l\in \mathbb Z}\sum _{m=0}^{q-1}G(\!-p,m,q)\delta \left( x\!-2t_{p,q}\xi \!-l\!-\frac{m}{q}\right) \,d\xi . $$

For q odd \(G(-p,m,q)=\sqrt{q}e^{i\theta _{m,p,q}}\) for some \(\theta _{m,p,q}\in \mathbb R\) so we get for \(t_{p,q}=\frac{1}{2\pi }\frac{p}{q}\)

$$ e^{it_{p,q}\varDelta }u_0(x)=\frac{1}{\sqrt{q}}\int _0^{2\pi }\widehat{u_0}(\xi )e^{-it_{p,q}\xi ^2+ix\xi }\sum _{l\in \mathbb Z}\sum _{m=0}^{q-1}e^{i\theta _{m,p,q}}\delta \left( x-2t_{p,q}\xi -l-\frac{m}{q}\right) \,d\xi . $$

For a given \(x\in \mathbb R\) there exists a unique \(l_x\in \mathbb Z\) and a unique \(0\le m_x<q\) such that

$$ x-l_x-\frac{m_x}{q} \in \left[ 0,\frac{1}{q}\right) ,\quad \xi _{x} =\frac{\pi q}{p}\left( x-l_x-\frac{m_x}{q}\right) =\frac{\pi q}{p}\,d\left( x,\frac{1}{q}\mathbb Z\right) \in \left[ 0,\frac{\pi }{p}\right) . $$

We note that for \(0\le \xi <\eta \frac{2\pi }{p}\) we have \(0\le 2t\xi <\frac{1}{2q}\). As \(\widehat{u}_0\) is supported modulo \(2\pi \) only in a neighborhood of zero of radius less than \(\frac{\eta }{2\pi }{ p}\) then we get the expression (5.1). \(\square \)

Proof of Theorem 5.1

We shall construct sequences \(\{\alpha _k\}\) such that \(\sum _{k\in \mathbb Z} \alpha _k \delta _k\) concentrates in the Fourier variable near the integers. To this purpose we consider, for \(s>\frac{1}{2}\), a positive bounded function \(\psi \in H^{s}\) with support in \([-1,1]\) and maximum at \(\psi (0)=1\). We define the \(2\pi \)-periodic function satisfying

$$ f(\xi ):=p^{\beta }\psi \left( \frac{p}{2\pi \eta } \xi \right) ,\quad \forall \xi \in [-\pi ,\pi ], $$

with \(\beta <\frac{1}{2}-\frac{3}{2}s\), introduce its Fourier coefficients:

$$ f(\xi ):=\sum _{k\in \mathbb Z} \alpha _k e^{ik\xi }, $$

and the distribution

$$ u_0:=\sum _{k\in \mathbb Z} \alpha _k \delta _k. $$

In particular, on \([-\pi ,\pi ]\), we have \(\widehat{u_0}=f\) and the restriction of \(\widehat{u_0}\) to \([-\pi ,\pi ]\) has support included in a neighborhood of zero of radius less than \(\eta \frac{2\pi }{p}\). We then get from (5.1):

$$\begin{aligned} e^{it_{p,q}\varDelta }u_0(x)=\frac{1}{\sqrt{q}} \,\widehat{u_0}\left( \frac{\pi q}{p} \,d\left( x,\frac{1}{q}\mathbb Z\right) \right) \, e^{-it_{p,q}\,\xi _x^2+ix\,\xi _x+i\theta _{m_x}}, \end{aligned}$$

that

$$\begin{aligned} \left| e^{it_{p,q}\varDelta }u_0(0)\right|= & {} \frac{1}{\sqrt{q}} \,|f(0)|=\frac{1}{\sqrt{q}} p^{\beta }\psi (0)=\frac{p^{\beta }}{\sqrt{q}},\end{aligned}$$
(5.4)
$$\begin{aligned} \left\| e^{it_{p,q}\varDelta }u_0\right\| _{L^\infty }\le & {} \frac{1}{\sqrt{q}} \,\Vert f\Vert _{L^\infty }=\frac{p^{\beta }}{\sqrt{q}}, \end{aligned}$$
(5.5)

and

$$\begin{aligned} e^{it_{p,q}\varDelta }u_0(x)=0\quad \text{ if } ~d\left( x,\frac{1}{q}\mathbb Z\right) >\frac{2\eta }{q}. \end{aligned}$$
(5.6)

We note that

$$ \Vert \alpha _k\Vert _{l^{2,r}}^2=\sum _k|k|^{2r}|\alpha _k|^2=\Vert f\Vert _{\dot{H}^r}^2=\frac{p^{2(\beta +r-\frac{1}{2})}}{(2\pi \eta )^{2(r-\frac{1}{2})}}\Vert \psi \Vert _{\dot{H}^r}. $$

Since \(\beta <\frac{1}{2}-s\) and p is large, it follows that \(\Vert \alpha _k\Vert _{l^{2,s}}\) is small enough so that we can use the results in [3] to construct a solution up to time \(t=\frac{1}{2\pi }\) for (1.2) of type

$$ u(x,t)=\sum _{k\in \mathbb Z}e^{i(|\alpha _k|^2-2\sum _j|\alpha _j|^2)\log t}(\alpha _k+R_k(t))e^{it\varDelta }\delta _k(x). $$

Hence

$$\begin{aligned}{} & {} \left| u(t,x)-e^{it_{}\varDelta }\left( \sum _{k\in \mathbb Z}\alpha _k\delta _k\right) (x)\right| \\{} & {} \le \sum _{k\in \mathbb Z}\left( 1-e^{i(|\alpha _k|^2-2\sum _j|\alpha _j|^2)\log t_{}}\right) \alpha _k e^{it_{}\varDelta }\delta _k(x)\!+\!\sum _{k\in \mathbb Z}e^{i(|\alpha _k|^2-2\sum _j|\alpha _j|^2)\log t_{}}R_k(t_{})e^{it_{}\varDelta }\delta _k(x)\\{} & {} \le \frac{C}{\sqrt{t_{}}}\Vert \alpha _k\Vert _{l^{2}}^2\Vert \alpha _k\Vert _{l^{2,s}} +\frac{Ct^\gamma }{\sqrt{t}}\Vert \alpha _k\Vert _{l^{2,s}}^3=C(\eta )\frac{p^{3(\beta +s-\frac{1}{2})}}{\sqrt{t}}(p^{-s}+t^\gamma ), \end{aligned}$$

for \(\gamma <1\).

Therefore, in view of (5.4), (5.5), and (5.6) we have for times \(t_{p,q}\) and \(t_{\tilde{p},\tilde{q}}\) both of size \(\frac{1}{2\pi }\), but with rational representation of type \(q\approx p\) which is fixed to be large, and \(\tilde{p}\approx \tilde{q}\approx 1\) with \(\tilde{p}<\tilde{q}\), that:

  • at time \(t_{p,q}\) the modulus \(|u(t_{p,q},x)|\) is a \(\frac{1}{q}\)-periodic function of maximal amplitude \(\frac{1}{p^{\frac{1}{2}-\beta }}\) plus a remainder term of size \(\frac{1}{p^{3(-\beta -s+\frac{1}{2})}}\), that is negligible provided that \(\beta <\frac{1}{2}-\frac{3}{2}s\). So modulo negligible terms \(|u(t_{p,q},x)|\) has plenty of \(\frac{1}{p}\)-period waves of small amplitude \(\frac{1}{p^{\frac{1}{2}-\beta }}\),

  • at time \(t_{\tilde{p},\tilde{q}}\) the modulus \(|u(t_{\tilde{p},\tilde{q}},x)|\) is a \(\frac{1}{\tilde{q}}\)-periodic function of maximal amplitude \(\frac{1}{\tilde{p}^{\frac{1}{2}-\beta }}\) plus a remainder term of size \(\frac{1}{\tilde{p}^{3(-\beta -s+\frac{1}{2})}}\), that is again negligible provided that \(\beta <\frac{1}{2}-\frac{3}{2}s\). So modulo negligible terms \(|u(t_{\tilde{p},\tilde{q}},x)|\) has in the interval \(I:=[-\frac{1}{2\tilde{q}},\frac{1}{2\tilde{q}}]\) a wave of amplitude \(\frac{1}{\tilde{p}^{\frac{1}{2}-\beta }}\), and is upper-bounded by a smaller value on \(I\setminus [-\frac{2\eta }{\tilde{q}},\frac{2\eta }{\tilde{q}}]\).

Therefore, observing what happens in the interval I we have at time \(t_{p,q}\) small waves while at time \(t_{\tilde{p},\tilde{q}}\) a localized large-amplitude (with respect to \(\eta \)) structure emerges. \(\square \)

Remark 5.3

  1. 1.

    In the above argument we need \(\eta \) to be small. As a consequence, the \(L^\infty \) norm and therefore the \(L^1_{\text {loc}}\) norm of the solution is small. This can be avoided by considering \(u_{\lambda }=\frac{1}{\lambda }u(x/\lambda , t/\lambda ^2\)), where u is any of the solutions constructed above. If \(\lambda >1\) the \(L^\infty \) norm grows, while for \(\lambda <1\) the \(L^1_{\text {loc}}\) norm around the corresponding bump grows.

  2. 2.

    The size of the error can be made smaller following the ideas developed in [5]. This is due to the type of data (3.8) we are using. In this case the size of the first Picard iterate is indeed much smaller than the \(l^1\) norm we are using in the above argument.