On the hyperbolic nonlinear Schrödinger equations

Abstract

Here, we consider here Hyperbolic Nonlinear Schrödinger Equations (HNLS) that occur as asymptotic models in the modulational regime when the Hessian of the dispersion relation is not positive (or negative) definite. We review classical examples, well-known results, and main open questions.

1 Introduction

The cubic nonlinear Schrödinger equation (NLS)

$$ iu_{t}+\Delta u\pm \vert u \vert ^{2}u=0, \qquad u=u(x,t),\quad (x,t)\in \mathbb{R}^{n} \times \mathbb{R}, $$

(1.1)

is one of the most popular nonlinear dispersive equations. It is a paradigm of equations appearing as asymptotic models for various systems with dispersion in the so-called modulational regime. Here, one aims to approximate wave packets, fast oscillating waves whose amplitude varies slowly. One obtains equations or systems of nonlinear Schrödinger type, where the real part of the unknown is an approximation of the slowly varying wave amplitude.

The first derivation of a nonlinear Schrödinger equation as the equation of the envelope of wave trains with slowly varying amplitude was performed in the pioneering paper [2]. The formal derivation of such an equation in the context of infinite-depth water waves was obtained by Zakharov [54] (see below for more details).

At this point, it is worth noticing that the Laplace operator in (1.1) is by no way the only possible second-order operator appearing in this kind of equation. Actually, in the derivation, one obtains the operator:

$$ \begin{aligned} L=\sum_{i,j}\omega _{ij} \frac{\partial ^{2}}{\partial x_{i}\partial x_{j}}, \end{aligned} $$

(1.2)

where $(\omega _{ij})$ is the Hessian of the dispersion relation ω of the original system. A Laplace operator occurs only when this Hessian is positive (or negative) definite. If this is not the case, one obtains a “hyperbolic” (or “nonelliptic” in the terminology of [23]) nonlinear Schrödinger equation.

More specifically, the evolution of a weakly nonlinear wave packet in a strongly dispersive cubic medium is governed by a universal NLS equation, which is relevant in all areas of continuum physics (water waves, plasma waves, atmospheric waves, elastic waves, etc.). Denoting $\omega =\omega (k_{1},k_{2},k_{3})$ the dispersion relation between the frequency ω and the wave vector $k=(k_{1},k_{2},k_{3})$, this NLS equation can be expressed as

$$ \begin{aligned} \frac{\partial A}{\partial t}+ \vec{V_{g}}\cdot \nabla A-\frac{i}{2} \nabla ^{t} \biggl( \frac{\partial ^{2}\omega}{\partial k_{i}\partial k_{j}} \biggr) \nabla A-i\beta \vert A \vert ^{2}A=0, \end{aligned} $$

(1.3)

where A is the complex amplitude of the perturbation, $\vec{V_{g}}=\nabla \omega$ is the group velocity, and β is the nonlinear coupling constant.

We now describe the classical physical situations leading to such an equation.

The equation derived by Zakharov in the modulational regime for water waves with infinite depth reads (see also [15], equation (2.18)) in the notation of Yuen [53]

$$ i \biggl(A_{t}+\frac{\omega _{0}}{2k_{0}}A_{x} \biggr)- \frac{\omega _{0}}{8k_{0}^{2}}A_{xx}+\frac{\omega _{0}}{4k_{0}^{2}}A_{yy}- \frac{1}{2}k_{0}^{2} \vert A \vert ^{2}A=0. $$

(1.4)

The relationship of the free surface η to A is given by:

$$ \begin{aligned} \eta (x,y,t)=A(x,y,t)\exp \bigl[i(k_{0} x-\omega _{0} t) \bigr]+c. \end{aligned} $$

Note that a similar equation holds for the case of the (large) finite depth, but for moderate depths, one obtains the classical cubic NLS. We refer to [35] for further developments and a rigorous derivation. In the same regime, one can find more general systems (Benney-Roskes, Davey-Stewartson) involving a nonelliptic NLS.

Next, as noticed in [6], the theory of self-focusing wave packets in plasma or nonlinear optics is commonly based on the following nonlinear Schrödinger equation in the notations of [6]:

$$ \begin{aligned} \frac{2ik_{c}}{\omega '} \biggl( \frac{\partial E}{\partial t}+\omega ' \frac{\partial E}{\partial z} \biggr)+\nabla ^{2}_{\perp }E+s \frac{\partial ^{2}E}{\partial z^{2}}+2k_{c}^{2}n \bigl( \vert E \vert ^{2} \bigr)E=0, \end{aligned} $$

(1.5)

where $s=\frac{\omega '' k_{c}}{\omega '}$, $k_{c}$ is the carrier wave number, and $\omega '=\partial \omega /\partial k|_{k_{c}}$, $\omega ''=\partial ^{2}\omega /\partial k^{2}|_{k_{c}}$ representing the group velocity and the dispersion coefficient along the wave-propagation axis in the z direction,respectively, and $\nabla _{\perp}^{2}=\partial _{x}^{2}+\partial _{y}^{2}$ is the perpendicular component of the dispersive operator describing the diffraction of the wave in the transverse plane $(x,y)$.

Depending on the dispersive properties of the medium, the coefficient s can be either positive in the case of a so-called anomalous medium or negative in the case of a normal medium [3, 5–7]. We restrict ourselves here to this last case, which leads to a nonelliptic dispersive operator. We also limit ourselves to a cubic medium corresponding to $n(|E|^{2})=|E|^{2}$.

In the study of optical self-focusing of short light pulses in nonlinear media, the relevant nonlinear Schrödinger equation writes in the case of normal dispersion [3, 5]:

$$ i\partial A/\partial z+\Delta _{\perp }A-\partial ^{2} A/\partial t^{2}+2 \vert A \vert ^{2}A=0, \qquad \Delta _{\perp}=\partial ^{2}/ \partial x^{2}+ \partial ^{2}/ \partial y^{2}, $$

(1.6)

where the first term models the propagation of the pulse along the z-axis, the second term describes the effect of the transverse diffraction, and the third and fourth terms account for the pulse temporal dispersion and the Kerr nonlinearity. Note that in this context, z plays the role of a time variable.

We will use here the more standard form:

$$ \begin{aligned} i\psi _{t}+\Delta _{\perp }\psi \pm \psi _{zz}+\beta \vert \psi \vert ^{2} \psi =0 \quad \text{in } \mathbb{R}^{d+1}, d=1,2, \end{aligned} $$

(1.7)

where $\Delta _{\perp}=\frac{\partial ^{2}}{\partial _{xx}}+ \frac{\partial ^{2}}{\partial _{yy}}$ (resp. $\frac{\partial ^{2}}{\partial _{xx}}$).

As emphasized in Zakharov and Kuznetsov [55], in the case of three spatial variables, there are four canonical NLS equations, namely

$$\begin{aligned} &i\psi _{t}+\Delta \psi + \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(1.8)

$$\begin{aligned} &i\psi _{t}+\Delta _{\perp}\psi -\psi _{zz} + \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(1.9)

$$\begin{aligned} &i\psi _{t}+\Delta \psi - \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(1.10)

$$\begin{aligned} &i\psi _{t}+\Delta _{\perp }\psi -\psi _{zz} - \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(1.11)

where Δ is the three-dimensional Laplacian, and $\Delta _{\perp}$ the Laplacian in $(x,y)$. (1.8) and (1.10) are the focusing and defocusing cubic NLS equations, respectively. (1.9) and (1.11) are there modifications when adding a dispersive term with a different sign.

The corresponding local Cauchy problem for (1.8) and (1.10) is well known since the pioneering work by Ginibre and Velo [25]. It is based on a Picard iteration scheme on the Duhamel formulation of the equation using Strichartz estimates satisfied by the underlying linear group. As noticed in [23], (1.9) and (1.11) have the same Strichartz estimates, and thus, one obtains the same result for the local Cauchy problem of the four equations. Actually, equations (1.8)–(1.11) are $H^{1}$-subcritical and $L^{2}$-supercritical, and the Cauchy problem is locally well-posed in $H^{1}(\mathbb{R}^{3})$.

Remark 1.1

In the two-dimensional case, there are only three different cubic NLS, namely

$$\begin{aligned} & i\psi _{t}+\psi _{xx}-\psi _{yy}+ \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(1.12)

$$\begin{aligned} & i\psi _{t}+\psi _{xx}+\psi _{yy}+ \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(1.13)

$$\begin{aligned} &i\psi _{t}+\psi _{xx}+\psi _{yy}- \vert \psi \vert ^{2}\psi =0. \end{aligned}$$

(1.14)

Equations (1.12)–(1.14) are $H^{1}$-subcritical and $L^{2}$-critical so that the Cauchy problem is locally well-posed in $H^{1}(\mathbb{R}^{2})$ and $L^{2}(\mathbb{R}^{2})$, globally well-posed for small $L^{2}$ initial data.

Despite this, the local Cauchy problems for those equations are the same; the global behavior of solutions is, of course, expected to be very different. For instance, (1.8) (resp. (1.13)) are focusing $L^{2}$-supercritical (resp. $L^{2}$-critical) cubic NLS equation that admits finite-time blow-up solutions while (1.10) and (1.14) are cubic defocusing energy subcritical NLS with global solutions that scatter.

On the other hand, (1.9), (1.11), and (2.3) are hyperbolic NLS for which the global well-posedness/versus finite-time blow-up of solutions is still unknown.

Note, however, that when (1.9) is posed in $\mathbb{R}^{2}_{x,y}\times \mathbb{T}_{z}$, the finite blow-up is possible using the two-dimensional blowing solutions of the $L^{2}(\mathbb{R}^{2})$-critical equation (1.13), see, for instance, [47].

Remark 1.2

One can find in [19] physically relevant variants of the hyperbolic NLS equation derived from the Maxwell equation and in [33] the derivation of an hyperbolic equation for the propagation of electromagnetic waves.

Up to now, we have considered only cubic nonlinearities. We now mention a physically relevant quadratic case. Since the nonlinear terms have to be smooth in the physical models (they occur from some Taylor expansion of a smooth term), they cannot be of the type $|u|u$, and the models are quadratic systems with polynomial nonlinearities. They occur in the modeling of the propagation of light in the so-called quadratic ($\chi ^{2}$) media; see [3, 9] for illuminating physical surveys and [12] for a mathematical study together with a summary of the physical context.

In the notations of [12], one obtains the system

$$ \textstyle\begin{cases} i\frac{\partial u}{\partial t}+ \Delta _{\perp}u + \gamma _{1}\frac{\partial ^{2} u}{\partial z^{2}} +\bar{u} v=0, & x \in \mathbb{R}^{d}, z \in \mathbb{R}, t>0, \\ 2i\frac{\partial v}{\partial t}+ \Delta _{\perp}v + \gamma _{2}\frac{\partial ^{2} v}{\partial z^{2}} -\beta v+ \frac{1}{2}u^{2}=0, & x\in \mathbb{R}^{d}, z \in \mathbb{R}, t>0, \\ u(0,x,z)=u_{0}(x,z),\qquad v(0,x,z)=v_{0}(x,z), & x \in \mathbb{R}^{d}, z \in \mathbb{R}, \end{cases} $$

(1.15)

where $u,v= u(x,z,t), v(x,z,t)$, $x\in \mathbb{R}$, $z\in \mathbb{R}^{2}$, $d=1,2$, $t>0$ and $\Delta _{\perp}=\partial ^{2}/\partial x^{2}+ \partial ^{2}/ \partial y^{2}$ (resp. $\Delta _{\perp}=\partial ^{2}/ \partial x^{2}$). Here, $\gamma _{1}$, $\gamma _{2}$, and β are real constants.

One obtains at least one nonelliptic NLS when $\gamma _{1}$ and/or $\gamma _{2}$ is negative. Whatever the sign of $\gamma _{1}$, $\gamma _{2}$ is. The situation is quite different from the cubic case. In fact, it is established in [12] that the system is globally well-posed in $L^{2}(\mathbb{R}^{d+1})$ and in $H^{1}(\mathbb{R}^{d+1}), d=1,2$, using that the problem is $L^{2}$-subcritical.

Open problem 1

Global behavior of the solutions to the above quadratic system in function of the parameters $\gamma _{1}$, $\gamma _{2}$.

Remark 1.3

There are nonlocal versions of hyperbolic NLS, for instance, the “hyperbolic-elliptic” Davey-Stewartson systems in the terminology of [21], one of them, the defocusing DS-II system being integrable. See below for some details and to [34], Chap. 4 for an up-to-date survey.

A more general class is that of Zakharov-Schulman systems introduced in [56]; see also [30] to model the interactions of small amplitude, high-frequency waves with acoustic-type waves and studied mathematically in [22, 31]. They read in $\mathbb{R}^{d}\times \mathbb{R}, d=2,3$,

$$ \begin{aligned} &i\partial _{t}\psi +L_{1} \psi +\psi \Phi =0, \\ &L_{2}\Phi =L_{3} \vert \psi \vert ^{2}, \end{aligned} $$

(1.16)

where

$$ \begin{aligned} L_{n}=\sum_{j,\ k=1}^{d} C_{jk}^{n} \frac{\partial ^{2}}{\partial x_{j}\partial x_{k}}, \quad n=1,2,3 \end{aligned} $$

are second-order differential operators with constant coefficients, and the matrices $(C_{j,k}^{n} )_{1\leq j\leq k\leq d}$ are real and symmetric (but not necessarily positive or negative).

When $L_{2}$ is the Laplace operator, one can express $\Phi =\Delta ^{-1}L_{3}|\psi |^{2}$, and if $L_{1}$ is nonelliptic, one obtains a (nonlocal) nonelliptic NLS.

2 Review of previous results and conjectures

We briefly describe here some of the known properties of nonelliptic nonlinear Schrödinger equation, referring to [34] Sect. 4.6 for further details.

2.1 The Cauchy problem

As mentioned above, it was noticed in [23], that the Strichartz estimates for linear Schrödinger operators

$$ \begin{aligned} Lu=iu_{t}+\sum _{i,j}\omega _{ij} \frac{\partial ^{2}}{\partial x_{i}\partial x_{j}}u \end{aligned} $$

are the same as those of the usual Schrödinger operator $iu_{t}+\Delta u$, as soon as the Hessian $\Sigma _{i,j}\omega _{ij}$ is not degenerate, as well as the local smoothing property [13].

Thus, one gets the classical local well-posedness results known for the standard NLS equation [25, 47].

For equations (1.4), (1.7), and (1.15), we already recalled that one obtains the local well-posedness of the Cauchy problem in $H^{1}$. An approach to global existence for sufficiently smooth, not necessary small, solutions was taken in [49] for (1.7) in 2D, but the proof suffers from some gaps.

One expects here scattering of arbitrary large solutions. Such a result was obtained for radial initial data by Dodson [17] for the $L^{2}$-critical defocusing NLS in two and three dimensions.

Conjecture 1

The Cauchy problem for equation (1.12) is globally well-posed in $L^{2}(\mathbb{R}^{2})$, and the solutions scatter.

Concerning the $L^{2}$ setting, we observe that (1.4) and (1.7) in space dimension two are $L^{2}$-critical so that one gets the local well-posedness in $L^{2}$ and the global well-posednes for small initial data in $L^{2}$. The global well-posedness for arbitrary initial data in $L^{2}$ is expected but not yet proven.

The global well-posedness have been obtained for the integrable defocussing Davey-Stewartson system, (DS-II) based on the integrabiloity. More precisely, Sung [48] proved the global well-posedness for arbitrary data in the Schwartz class $\mathcal {S}(\mathbb{R}^{2})$. Moreover, he proved that the sup-norm of the solution decays as $1/t$, that is the decay rate for the linear problem. This result was extended by P. Perry [42] for arbitrary data in the weighted Sobolev space $H^{1,1}(\mathbb{R}^{2})$. Furthermore, he proved the scattering of those global solutions. The best-known result concerning the Cauchy problem was obtained in [39], where the decay condition of the initial data was excluded, proving global well-posedness in $L^{2}(\mathbb{R}^{2})$. The integrable defocusing DS-II equation can be written as

$$ \begin{aligned} i\psi _{t}+\psi _{xx}-\psi _{yy}+2\Delta ^{-1}(\partial _{yy}- \partial _{xx}) \vert \psi \vert ^{2}\psi =0. \end{aligned} $$

(2.1)

The “general”, nonintegrable DS-II systems write

$$ \begin{aligned} i \partial _{t}\psi + \partial _{xx}\psi -\partial _{yy}\psi +2\rho \Delta ^{-1}{} \bigl[ \bigl(\partial _{yy}+(1-2\beta )\partial _{xx} \bigr) \vert \psi \vert ^{2} \bigr] \psi = 0, \end{aligned} $$

(2.2)

which involves the zero-order nonlocal operator

$$ \begin{aligned} \Delta ^{-1}{} \bigl[ \bigl(\partial _{yy}+(1-2\beta )\partial _{xx} \bigr) \bigr] . \end{aligned} $$

The parameter $\beta \in \mathbb{R}$ in (2.2) determines the contribution of the mean field Φ to the nonlinearity in the nonlinear Schrödinger equation.

The completely integrable DS-II systems are now obtained when $\beta =1$ (focusing when $\rho =-1$, defocusing when $\rho =1$).

Conjecture 2

The global results in [39, 42, 48] hold for the nonintegrable defocusing DS-II system (2.2), that is, $\rho =1$, $\beta \neq 1$. We refer to [34] for numerical simulations.

In particular, for the two-dimensional HNLS:

$$ \begin{aligned} i\psi _{t}+\psi _{xx}-\psi _{yy}+ \vert \psi \vert ^{2}\psi =0, \end{aligned} $$

(2.3)

one expects global well-posedness in $L^{2}(\mathbb{R}^{2})$ and scattering.

We refer to [1] for a numerical analysis of a possible asymptotic regime of the 2D HNLS equation. See also [57] for previous work.

Coming back to the possibility of a finite-time blow-up for $L^{2}$ solutions to (1.15) when $d=1$ (two-dimensional case), we mention that using a refined Strichartz estimate, Rogers and Varga [43] proved that if the solution blows up, then there exists a mass-concentration phenomenon, more precisely

Theorem 2.1

([43])

Suppose that ψ is an $L^{2}$ solution of (1.15) with $d=1$ that blows up at $T_{\mathrm{max}}<\infty$. Then, where R denotes a rectangle in x-space

$$ \begin{aligned} \limsup_{T\to T_{\mathrm{max}}}\sup _{ \vert R \vert \leq T_{\mathrm{max}}-t} \biggl( \int _{R} \bigl\vert \psi (x,t) \bigr\vert ^{2}\,dx \biggr)^{1/2}>\epsilon , \end{aligned} $$

where ϵ is a constant depending only on β and $\|\psi _{0}\|_{L^{2}(\mathbb{R}^{2})}$.

More recently, a profile decomposition for a bounded sequence in $L^{2}(\mathbb{R}^{2})$ is given in [18]. A key argument is again a refined Strichartz estimate. The profiles are given in terms of the group $e^{it\partial _{x}\partial _{y}}$ as an application of the profile decomposition, one constructs a minimal mass solution ψ of (1.15) such that there exists a time $T^{\star}$ such that

$$ \begin{aligned} \int _{-T^{\star}}^{T^{\star}} \int _{\mathbb{R}^{2}} \vert \psi \vert ^{4}\,dx\, dy\, dt =+ \infty . \end{aligned} $$

In other words, this is a solution of least mass for which the small mass global argument fails.

We now focus on the three spatial dimensional case. As mentioned above, equations (1.7), (1.9), and (1.10) are $L^{2}$-supercritical, and no local $L^{2}$ theory is possible. On the other hand, they are energy subcritical, yielding the local well-posedness of the Cauchy problem in $H^{1}(\mathbb{R}^{3})$ [25].

No rigorous results on a possible blow-up in finite time or on the global existence of “large” solutions are known in the three-dimensional case. The situation might be different for equations (1.9) and (1.11) since (1.11) is in some sense “closer” to a defocusing equation.

Numerical simulations in [3, 4] for equation (1.6) (that is (1.9)) suggest the dispersion of “small” solutions and a pulse splitting for larger initial data. No obvious finite-time blow-up appears.

Open problem 2

Long-time behavior of the local $H^{1}(\mathbb{R}^{3})$ solutions to the cubic nonelliptic cubic Schrödinger equations in three spatial dimensions.

Remark 2.2

A WKB analysis is justified in [10] for (2.4) and related equations. Since, contrary to the classical NLS equation, the leading order system in this analysis is not hyperbolic, one needs to work in analytic classes.

Remark 2.3

(Periodic setting)

We focus in this paper on the Cauchy problem posed on the space $\mathbb{R}^{n}$. The case of the two-dimensional periodic problem in $\mathbb{T}^{2}$ has been considered in [26, 52] (see also [37] for a more general context). Wang [52] and Godet-Tzvetkov [26] have established Strichartz estimates with derivative losses. We give below the result as stated in [26].

Theorem 2.4

([26])

Let $P=-\partial _{x}^{2}+\partial _{y}^{2}$. Let $(p,q)$ be such that

$$ \frac{1}{p}+\frac{1}{q}{=\frac{1}{2}},\quad p>2. $$

There exists a constant $C>0$ such that for any initial data $u_{0}\in H^{\frac{1}{p}}(\mathbb{T}^{2})$,

$$ \begin{aligned} \bigl\Vert e^{-itP}u_{0} \bigr\Vert _{L^{p}_{t\in [0,1]}L_{xy}^{q}(\mathbb{T}^{2})}\leq C \Vert u_{0} \Vert _{H^{\frac{1}{p}}(\mathbb{T}^{2})}. \end{aligned} $$

Moreover, this estimate is optimal in the sense that the inequality

$$ \begin{aligned} \bigl\Vert e^{-itP}u_{0} \bigr\Vert _{L^{p}_{t\in [0,1]}L_{xy}^{q}(\mathbb{T}^{2})}\leq C \Vert u_{0} \Vert _{H^{s}(\mathbb{T}^{2})} \end{aligned} $$

is false when $s<\frac{1}{p}$.

On the other hand, Wang [52] proved that the Cauchy problem is locally well-posed in $H^{s}(\mathbb{T}^{2}), s>1/2$ while it is ill-posed in $H^{s}(\mathbb{T}^{2}), s<1/2$.

The semi-periodic problem, posed in $\mathbb{R}\times \mathbb{T}$, has been studied by Tzvetkov in [50], who proved the Strichartz estimate

$$ \begin{aligned} \bigl\Vert e^{-itP}u_{0} \bigr\Vert _{L^{4}([0,1]\times \mathbb{R}\times \mathbb{T})} \leq C \Vert u_{0} \Vert _{H^{s}(\mathbb{R}\times \mathbb{T})}, \end{aligned} $$

for any $s>0$, which is the standard $L^{4}_{x,t}$ Strichartz estimate of the equation posed in $\mathbb{R}^{2}$ with an ϵ loss.

We are not aware of results on the periodic or semi-periodic Cauchy problem in three dimensions.

Open problem 3

Study of the Cauchy problem for the 3D HNLS in the periodic or semi-periodic setting.

An important issue is that of the transverse stability properties of the Schrödinger line soliton with respect to the nonelliptic NLS, see below.

We will consider here the different equations

$$\begin{aligned}& i\psi _{t}+\psi _{xx}-\psi _{yy}+ \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(2.4)

$$\begin{aligned}& i\psi _{t}+\psi _{xx}-\Delta _{\perp }\psi + \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(2.5)

$$\begin{aligned}& i\psi _{t}-\psi _{xx}+\Delta _{\perp }\psi + \vert \psi \vert ^{2}\psi =0, \end{aligned}$$

(2.6)

where $\Delta _{\perp}=\partial ^{2}_{yy}+\partial ^{2}_{zz}$.

For functions ψ depending only on y and z, the two last equations (2.5) and (2.6) are the defocusing (resp. focusing) two-dimensional cubic NLS. Adding the $\psi _{xx}$ term adds a focusing (resp. defocusing) contribution, which may change the nature of the equation.

In the two first equations (2.4) and (2.5), there exists the NLS bright (line) soliton depending only on x

$$ Q(x)e^{it}, Q(x)=\frac{\sqrt {2}}{\cosh (x)}, $$

(2.7)

while the last one (2.6) possesses a two-dimensional solution corresponding to the ground state solution of the two-dimensional NLS:¹

$$ i\psi _{t}+\Delta _{\perp}\psi + \vert \psi \vert ^{2}\psi =0. $$

(2.8)

2.2 Solitary waves

The question is here of the existence of (fully) localized solutions of the form $e^{it\omega}\Phi (x)$ for equations of the type

$$ iu_{t}+\sum_{i,j} \omega _{ij} \frac{\partial ^{2} u}{\partial x_{i}\partial x_{j}}+f \bigl( \vert u \vert ^{2} \bigr)u=0 \quad \text{in } \mathbb{R}^{n} \times \mathbb{R}, $$

(2.9)

where f is real-valued continuous function such that

$$ \begin{aligned} \bigl\vert f(s) \bigr\vert \leq c \bigl(1+ \vert s \vert ^{p} \bigr),\quad p\leq \frac{2}{n-2}\ (1\leq p< \infty \text{ if }n=2). \end{aligned} $$

It is established in [24] via suitable Pohojaev identities that for any $\omega \in \mathbb{R}$, no nontrivial solitary waves exist when the nonsingular matrix $\omega _{ij}$ is not positive or negative definite.

As noticed in [23], this result does not exclude the existence of nontrivial nonfully localized traveling waves, different from line solitons. Let us consider for instance the cubic nonelliptic NLS

$$ iu_{t}+u_{xx}-u_{yy}+ \vert u \vert ^{2}u=0. $$

(2.10)

Let $f\in H^{1}(\mathbb{R}^{2})$ be the unique positive radial solution $(f(x,y)=R((x^{2}+y^{2})^{1/2}))$ of

$$ \begin{aligned} -f+\Delta f+f^{3}=0. \end{aligned} $$

It is well known that $f\in C^{\infty}(\mathbb{R}^{2})$ and therefore that $R(r)=T(r^{2})$. Moreover, $T(\sigma )$ is exponentially decaying as $\sigma \to + \infty$. One can easily verify that $\phi (x,y)=T(x^{2}-y^{2})$ is a solution to

$$ \begin{aligned} -\phi +\phi _{xx}-\phi _{yy}+ \phi ^{3}=0, \end{aligned} $$

and therefore that $e^{it}\phi (x,y)$ is a nontrivial traveling wave for (2.10).

Constructions on nonlocalized, infinite $H^{1}$ norm solutions are provided in [14, 29, 32, 36, 51]; see [34, Sect. 4.6] for details. We first mention the existence, proven in [14], of plane wave solutions of the form $\psi (t,x,{\mathbf{y}})=f(t,x-c\cdot {\mathbf{y}})$, where ${\mathbf{y}}=y$ or $(y,z)$ and $c\in \mathbb{R}$ or $\mathbb{R}^{2}$ are fixed and non-zero. Thus, f should satisfy the following equation

$$ if_{t}+ \bigl(1- \vert c \vert ^{2} \bigr)f_{zz}+ \vert f \vert ^{2}f=0. $$

(2.11)

As noticed in [14], the size of $|c|$ determines the nature of (2.11). For instance, when $|c|<1$, (2.11) is the focusing cubic 1D NLS while it is the defocusing, when $|c|>1$. When $|c|=1$, one gets the explicit solution

$$ \begin{aligned} f(t,z)=f(0,z)e^{i{|f(0,z)|}^{2}t},\quad t\in \mathbb{R}. \end{aligned} $$

Note that in the three cases, the plane wave solution is globally defined.

In [36], Lu establishes the existence of two families of standing waves of (2.10). Solutions in the first family are periodic in y and localized in x, while solutions in the second family are periodic in both x and y. The tools used in the analysis are the center manifold reduction and Nash-Moser iteration. The second family of solutions was numerically observed in [51].

In [32], introducing hyperbolic coordinates allows for the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions.

Remark 2.5

The physical relevance of those non-$L^{2}$ solutions is not clearly established. Nevertheless, as pointed out in [14], it might be mathematically interesting to study the Cauchy problem in functional spaces comprising those solutions.

Remark 2.6

The linear equation

$$ i\frac{\partial \phi}{\partial t}+\frac{1}{2} \biggl( \frac{\partial ^{2}\phi}{\partial x^{2}}+ \frac{\partial ^{2} \phi}{\partial y^{2}} \biggr) -\frac{1}{2} \frac{\partial ^{2}\phi}{\partial z^{2}}=0 $$

(2.12)

was considered in [11], where explicit wave packet solutions were exhibited. They write

$$ I(x,y,z,t)= I_{0}J_{0}^{2}(r/r_{0})Ai^{2} \biggl(\epsilon \frac{z}{z_{0}}- \frac {t^{2}}{4t_{0}^{2}} \biggr), $$

(2.13)

where $J_{0}$ is the Bessel function, Ai is the Airy function, $r=\sqrt{x^{2}+y^{2}}$, $I_{0}$ is the peak intensity, $\epsilon =\pm 1$ determines the direction of the Airy function envelope, and $r_{0}$ and $z_{0}$ determine the radial and transverse widths of the wave packet, respectively.

2.2.1 Transverse stability/instability

A natural question is that of the possible stability of the one-dimensional (resp. two-dimensional) solitary waves. Many numerical investigations of the transverse instability of the KdV line soliton have been performed (see [16, 40] and the references there in) for the 2D HNLS equation. A study of the spectral stability is provided in [41], where it is proven that the line soliton is spectrally unstable under transverse perturbations of arbitrary small periods, i.e., short waves. Linear or nonlinear (ins)-stability results (à la Rousset-Tzvetkov, [44–46]) seem to be unknown.

We are not aware of such studies in the 3D case in the context of equations (2.5) and (2.6). Note that the issues are different here. The issue for (2.5) is that of the transverse stability of the cubic NLS line soliton with respect to transverse two-dimensional perturbations while for (2.6) the question is the transverse stability of the ground state solution of the 2D focusing NLS with respect to one-dimensional transverse perturbations.

Open problem 4

Analysis of transverse stability/instability issues for the 3D HNLS equations.

2.3 Solitary waves in quadratic media

For (1.15), a solitary wave solution of the form $u(t,x,z)\equiv u(x,z)e^{i\omega t}$ and $v(t,x,z)\equiv v(x,z)e^{i\omega t }$ should satisfy the system

$$ \begin{gathered} -\omega u+\Delta _{\perp }u+ \gamma _{1} \frac{\partial ^{2} u}{\partial z^{2}} +\bar{u}v=0, \\ -(4\omega +\beta )v+\Delta _{\perp }v+ \gamma _{2} \frac{\partial ^{2} v}{\partial z^{2}} +\frac{1}{2}u^{2}=0. \end{gathered} $$

(2.14)

With similar Pohojaev-type arguments, it is established in [12] that (2.14) has no nontrivial solution for any ω and β when $(\gamma _{1},\gamma _{2})=(-1,-1)$.

Open problem 5

What happens when $(\gamma _{1},\gamma _{2})=(1,-1)$ or $(-1,1)$?

2.4 Conclusion

This paper is focussed on known results and open questions for “nonelliptic” second-order cubic or quadratic Schrödinger equations. There are physically relevant “nonelliptic” nonlinear Schrödinger equations obtained by considering higher-order dispersive effects (see [20, 28]) in the context of surface water waves and [8] nonlinear optics. Their linear dispersive part is of the third order, and they offer new challenging issues, see [27, 38].