Travelling Waves for the Nonlinear Schrödinger Equation with General Nonlinearity in Dimension Two

Abstract

We investigate numerically the two-dimensional travelling waves of the nonlinear Schrödinger equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy–momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross–Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified KP-I asymptotic in the transonic limit, various multiplicity results and “one-dimensional spreading” phenomena.

Appendix: About Two Diagrams in Dimension One

In this appendix, we consider the NLS equation in space dimension one as studied in Chiron (2012). We wish to give two more energy–momentum diagrams showing the variety of possible qualitative behaviours.

1.1 A Quasi-Degenerate Case

We investigate here the quasi-degenerate case

$$\begin{aligned} f_\mathrm{qd} (\varrho ) \mathop {=}\limits ^\mathrm{def} - 2(\varrho - 1) + (3- 10^{-3}) (\varrho - 1)^2 - 4 (\varrho - 1)^3 + 5(\varrho - 1)^4 - 6 (\varrho - 1)^5 , \end{aligned}$$

which is a perturbation of the degenerate case

$$\begin{aligned} f_\mathrm{dege} (\varrho ) \mathop {=}\limits ^\mathrm{def} - 2(\varrho - 1) + 3 (\varrho - 1)^2 - 4 (\varrho - 1)^3 + 5(\varrho - 1)^4 - 6 (\varrho - 1)^5 \end{aligned}$$

studied in Chiron (2012) (Example 4 “a degenerate case” there). For the nonlinearity $f_\mathrm{dege}$, we have $\varGamma = \varGamma ' = 0$ (and also other coefficients of the same type) so that the transonic limit is governed by the sextic gKdV solitary wave equation

$$\begin{aligned} \frac{1}{{\mathfrak {c}}_s^2}\, \partial _z A - \frac{1}{{\mathfrak {c}}_s^2}\, \partial _z^3 A + \varGamma ^{(6)} A^5 \partial _z A = 0 , \end{aligned}$$

which is supercritical. For the nonlinearity $f_\mathrm{dege}$, as $c \rightarrow {\mathfrak {c}}_s$, the travelling waves have high energy and momentum (and are unstable; see Chiron (2012, 2013)). For the nonlinearity $f_\mathrm{qd}$ we are now considering, the coefficient $\varGamma$ becomes small, but nonzero. Actually, we have

$$\begin{aligned} V_\mathrm{qd} (\varrho ) = (\varrho - 1)^2 - \frac{3- 10^{-3}}{3} (\varrho - 1)^3 + (\varrho - 1)^4 - (\varrho - 1)^5 + (\varrho - 1)^6 \end{aligned}$$

and

$$\begin{aligned} \mathcal {V}_\mathrm{qd}( \xi ) = - \frac{1}{750} \xi ^3 - \frac{1}{750} \xi ^4 - 4 \xi ^7 , \end{aligned}$$

thus $r_0 = 1$, ${\mathfrak {c}}_s^2 = 4$, $\varGamma = \frac{1}{500}$ and the graphs of $f_\mathrm{qd}$, $V_\mathrm{qd}$ and $\mathcal {V}_\mathrm{qd}$ are given in Fig. 27.

Fig. 27

Graphs of a left: $f_\mathrm{qd}$, b centre: $V_\mathrm{qd}$ and c right: $\mathcal {V}_\mathrm{qd}$

Since $\varGamma$ is nonzero, the transonic limit for the nonlinearity $f_\mathrm{qd}$ is governed by the usual KdV solitary wave, and in particular, the energy and momentum tend to zero as $c \rightarrow {\mathfrak {c}}_s$. However, in some sense, $f_\mathrm{qd}$ is close to $f_\mathrm{dege}$, and we hope that for c close, but not too close, to ${\mathfrak {c}}_s$, part of the behaviour observed for $f_\mathrm{dege}$ will be seen for $f_\mathrm{qd}$. In particular, we hope to have, for the nonlinearity $f_\mathrm{qd}$, some “large” energy and momentum for c close, but not too close, to ${\mathfrak {c}}_s$, and then for c very close to ${\mathfrak {c}}_s$, small energy and momentum.

Fig. 28

a Left energy (*) and momentum (+) versus speed in logarithmic scale; b right qualitative energy–momentum diagram for nonlinearity $f_\mathrm{qd}$

The numerical computations of the energy and momentum as in Chiron (2012) provide the diagrams of $c \mapsto E$ and $c \mapsto P$ given in Fig. 28a. Since the variations are rather fast, we have used logarithmic scale: the abscissa is not c but $- \displaystyle { \log \left( \dfrac{{\mathfrak {c}}_s- c}{ {\mathfrak {c}}_s} \right) }= - \log ( 1 - 0.5 c)$. Therefore, the corresponding ( E, P) diagram is, qualitatively, as in Fig. 28b. Note that we have indeed some “large” energy and momentum for speeds c with $- \log ( 1 - 0.5 c) \simeq 10$ (for which the dominant behaviour is the one of $f_\mathrm{dege}$), and finally energy and momentum go to zero since $\varGamma \not = 0$. We have been interested in this nonlinearity since the energy–momentum diagram (in dimension one) is qualitatively similar to the one obtained in example 5 for the (exponentially) saturated nonlinearity (in dimension two). As already mentioned, this type of energy–momentum diagram can also be found in Akhmediev et al. (1999), section IV G, for the study of bound states in the nonlinear Schrödinger equation (with zero condition at infinity) with the focusing nonmonotonic nonlinearity $f (\varrho ) = \varrho ^{5/2} - \varrho ^{5} + \frac{1}{2} \varrho ^{15/2}$. We deal here with travelling waves with a defocusing monotonic nonlinearity.

Fig. 29

Graphs of a $f_\mathrm{sat}$, b $V_\mathrm{sat}$ and c $\mathcal {V}_\mathrm{sat}$

Fig. 30

a Left 40 $\times$ energy (*) and momentum (+) versus speed; b right energy–momentum diagram for nonlinearity $f_\mathrm{sat}$

1.2 Another Saturated NLS Model

We investigate now another classical saturated NLS model, which is

$$\begin{aligned} f_\mathrm{sat} (\varrho ) \mathop {=}\limits ^\mathrm{def} \frac{\varrho _0}{2} \Big ( \frac{1}{(1+\varrho /\varrho _0)^2} - \frac{1}{(1+1/\varrho _0)^2} \Big ) , \end{aligned}$$

where $\varrho _0 > 0$ is some parameter. For this nonlinearity, there holds

$$\begin{aligned} V_\mathrm{sat} (\varrho ) = \displaystyle { \frac{(\varrho - 1)^2}{ 2 \Big ( 1+ \displaystyle { \frac{1}{\varrho _0} } \Big )^2 \Big ( 1+ \displaystyle { \frac{\varrho }{\varrho _0} } \Big )} } \quad \mathrm{and} \quad \mathcal {V}_\mathrm{sat}(\xi ) = - \frac{ 2 \xi ^3 }{ \Big ( 1+ \displaystyle { \frac{1}{\varrho _0}} \Big )^3 \Big ( 1+ \displaystyle { \frac{1+\xi }{\varrho _0}} \Big )}. \end{aligned}$$

This nonlinearity has been studied in Kivshar and Krolikowski (1995) and is an example where the “kink”, that is the stationary wave ($c=0$), is unstable if $\varrho _0$ is small enough. This instability has also been theoretically and numerically studied in Di Menza and Gallo (2007), where the instability threshold was shown to be $\varrho _0 \simeq 0.134$. However, we would like to point out that the energy–momentum diagram given in Kivshar and Krolikowski (1995) for $0.08 = \hbox {``}I_0\hbox {''} = \varrho _0 < 0.134$ (Fig. 1 there) is probably not correct. Indeed, the slope of the curve $P\mapsto E$ must be equal to the speed c in view of the Hamilton group relation (20) (which holds true in dimension one, see Chiron (2012)). Hence, there should not exist a point on the curve $c \mapsto (E,P)$ with vertical tangent as given in Kivshar and Krolikowski (1995, figure 1). We have performed the corresponding simulation as in Chiron (2012).

We shall take $\varrho _0 = 0.08 < 0.134$; thus, $r_0=1$, ${\mathfrak {c}}_s^2 = 2 ( 1 + 1 / \varrho _0)^{-3} \simeq 0.00081288$, $\varGamma = \frac{6 \varrho _0}{\varrho _0 + 1 } = \frac{4}{9} \simeq 0.4444\ldots$. The graphs of $f_\mathrm{sat}$, $V_\mathrm{sat}$ and $\mathcal {V}_\mathrm{sat}$ are given in Fig. 29. We have computed E and P as the speed c varies, as shown in Fig. 30a, as well as the energy–momentum diagram, as shown in Fig. 30b. As we see, the curve possesses a cusp. We see with this example that the energy–momentum diagram may exhibit a cusp with a (local) maximum of E and P even though the nonlinearity $f_\mathrm{sat}$ is increasing. For the cubic–quintic nonlinearity, we also have a cusp with a (local) maximum of E and P, but the nonlinearity is increasing near 0.