Two dimensional nonlinear Schrödinger equation is integrable

Abhijeet Joshi¹, Vivek M. Vyas¹ and Prasanta K. Panigrahi^2,3

¹ Department of Applied Physics, Indian Institute of Information Technology Vadodara, Sector 28, Gandhinagar 382028, India
² Centre for Quantum Science and Technology, Siksha ‘O’ Anusandhan University, Bhubaneswar, Odisha 751030, India
³ Department of Physical Sciences, Indian Institute of Science Education & Research (IISER) Kolkata, Mohanpur, Nadia 741 246, India

Abstract. The classical dynamics of two dimensional Schrödinger model is studied in this work. Owing to the presence of $\mathfrak{sl}(2,\mathbb{R})$ algebra it is found that the model exhibits a set of infinitely many independently conserved charges, which establishes its Liouville integrability.

1. Introduction

Nonlinear Schrödinger equation is an important partial differential equation in physics, which manifests in various physical contexts like superconductivity, superfluidity, nonlinear optics, plasma physics and so on [1–3]. In one spatial dimension this celebrated equation was found to be integrable and exactly soluble half a century back [4, 5]. Various mathematical formalisms have been developed to shed light on the aspects of integrability of this model like inverse scattering method [5, 6], bi-Hamiltonian structures [7] and Hirota bilinear method [8, 9]. Crucially these theoretical developments provide different routes to understand the existence of infinite number of conserved charges admitted by the one dimensional nonlinear Schrödinger equation, establishing its integrability in the classical Liouville sense [10]. The quantum version of this model has also been studied in great detail, and was found to be integrable as also exactly soluble [11, 12].

While considerable attention has been devoted to the study of one dimensional nonlinear Schrödinger equation, both in the classical and quantum realms, the same system in two spatial dimensions has yet not been studied so intensely [13, 14]. The two dimensional system is well known to display a class of solutions known as Townes soliton, which have been experimentally observed in optical and cold atomic systems [15–17]. It is also known to display self-focussing singularity, decided by the Vlasov-Petrishchev-Talanov criteria [13, 14, 18]. This equation is found to play an important role in cold atom systems, wherein it is known to capture the mean field dynamics [19–21]. It was also found to exhibit a class of solutions called the breathing modes, which were experimentally observed in cold atoms experiments [20]. It is certainly of significant relevance to the macroscopic description of planar superconductors and superfluids [22, 23]. While these works are important in their own respective rights, they however have not thrown a decisive light on the question whether the system at hand is integrable or not.

Here we show that the two dimensional nonlinear Schrödinger equation is integrable in the Liouville sense, wherein it admits an infinite set of conserved charges which are in involution with each other. This is achieved by employing the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$ which is known to exist in the system [20, 25]. The present derivation is rather straightforward and does not rely on known mathematical frameworks like Lax pairs, Hirota method or bi-Hamiltonian structures.

2. Two dimensional nonlinear Schrödinger equation

The celebrated nonlinear Schrödinger partial differential equation in two spatial dimensions is given by:

i \partial_t \psi = -\frac{1}{2m}\nabla^2\psi + g|\psi|^2\psi (1)

for a complex field $\psi(\vec{x},t)$ . This equation is an initial value problem which dictates the time evolution of field $\psi(\vec{x},t)$ at any time given the initial value $\psi(\vec{x},t_{i})$ at any earlier time $t_i$ . In this discussion we shall consider that the field configurations are such that $\psi(\vec x,t) \rightarrow 0$ as $|\vec{x}| \rightarrow \infty$ so that Dirichlet boundary condition is respected.

The Lagrangian density giving rise to this equation is [24]:

L = \frac{i}{2}(\psi^*\dot\psi - \psi\dot\psi^*) - \frac{1}{2m}|\nabla\psi|^2 - \frac{g}{2}|\psi|^4

While the Hamiltonian functional corresponding to this system is:

H[\psi,\psi^*] = \int d^2x (\frac{1}{2m}|\nabla\psi|^2 + \frac{g}{2}|\psi|^4) (2)

The canonical Poisson brackets in this case read [24]:

{\psi(x,t), \psi^*(x',t)} = -i \delta^2(x - x'), {\psi(x,t), \psi(x',t)} = 0

It can be checked that the Hamiltonian equation of motion $\dot\psi = \{\psi, H\}$ indeed reproduces the equation (1).

Owing to the global $U(1)$ phase invariance in this system, the Noether theorem dictates the occurrence of locally conserved density and current respectively as:

\rho = |\psi|^2, \vec j = \frac{1}{2im}(\psi^*\nabla\psi - \psi\nabla\psi^*)

In light of the canonical brackets one immediately sees that the brackets between density and current read:

Poisson brackets of density and current: {\rho,\rho} = 0, {\rho, j_k} = \rho \partial_k \delta^2, {j_k, j_l} = j_l \partial_k \delta^2 - j_k \partial_l \delta^2 (3)

Let us set $m=1$ without losing generality, and define the quantities:

D = \int d^2x \vec x\cdot\vec j, K = \frac12\int d^2x \vec x^2 \rho (4)

$D$ is known in the literature as the generator of scale transformation or dilation [25]. Its infinitesimal action on the field $\psi(\vec x, t)$ is given by:

\delta\psi = \epsilon{\psi, D} = -\epsilon(1 + \vec x\cdot\nabla)\psi

where $\epsilon$ is an infinitesimal number. The corresponding finite transformation reads:

\psi'(\vec x,t) = \lambda\psi(\vec x',t) (5)

where $\vec x' = \lambda\vec x$ for some real $\lambda$ . It is worth mentioning that under the action of this finite scale transformation, the charge $N = \int d^{2}x'\,\rho(\vec x', t)$ remains unchanged, so that $\{D, N\} = 0$ .

$K$ in the literature is known to be the generator of special conformal transformations [25]. In this discussion, we shall not engage into it further, and shall only content by noting the important fact that:

Under the effect of finite scale transformation (5) one sees that the Hamiltonian $H$ and $K$ behave self-similarly:

H[\psi'] = \lambda^2 H[\psi], K[\psi'] = K[\psi]/\lambda^2

This can be traced to the fact that $\{D,H\}=2H, \{D,K\}=-2K$ . Thus one finds the existence of a Lie algebra in this problem defined by the relations:

With the identification $T_{+} = H$ , $T_{-} = K$ and $T_{3} = \frac{D}{2}$ as the generators, one sees that the algebra is $\mathfrak{sl}(2,\mathbb{R})\cong\mathfrak{so}(2,1)\cong\mathfrak{su}(1,1)$ obeying $\{T_{3}, T_{\pm}\} = \pm T_{\pm}, \; \{T_{-}, T_{+}\} = 2T_{3}$ [20, 25–28].

Given a generator $G$ that generates a flow with flow parameter $\tau$ , any functional $A(\tau)$ evolves under the action of $G$ via the Lie derivative $dA/d\tau=\{A,G\}$ . One can exponentiate the Lie derivative $dA/d\tau=\{A,G\}$ to get a finite transformation as [24]:

A(h) = e^{h\{\cdot,G\}}A(0) = \sum_l \frac{h^l}{l!} (l nested brackets with G) (7)

Considering the case when $G=H+K$ , and $A(\tau) = H(\tau)$ with the identification $H(\tau = 0) = H$ we get a finite transformation of $H$ under the action of $G$ as:

H(\tau) = (H+K)/2 + (H-K)/2 \cos 2\tau - D/2 \sin 2\tau

Very interestingly for $\tau = \frac{\pi}{2}$ this expression gives: $H(\frac{\pi}{2}) = K$ . This means that under the effect of canonical transformation generated by $G = H + K$ , Hamiltonian $H$ transforms to $K$ :

Let us now construct a transformed field $\Phi(\vec x, t)$ using the finite action on $\psi(\vec x,t)$ :

\Phi(\vec x,t) = e^{-\frac{\pi}{2}\{\cdot,G\}}\psi(\vec x,t) (9)

If one thinks of $\psi(\vec x,t)$ as the initial field configuration given to us, then the above relation essentially allows us to think of $G$ as an effective Hamiltonian evolving $\psi(\vec x,t)$ to $\Phi(\vec x, t)$ as time flows from $0$ to $-\frac{\pi}{2}$ . Interestingly since $G = H + K$ , it implies that the corresponding infinitesimal time evolution is governed by nonlinear Schrödinger equation with a harmonic potential term.

Let us now investigate the temporal change of the density $I(\vec x, t) = \Phi^{\ast}(\vec x, t)\Phi(\vec x, t) = e^{-\frac{\pi}{2}\{\cdot,G\}}\rho(\vec x, t)$ :

\partial_t I = {I, H} = ... = e^{-\frac{\pi}{2}\{\cdot,G\}}{\rho, K} = 0 (10)

in light of (3).

Thus one sees that the density $I(\vec x)$ is actually conserved under time evolution. Furthermore owing to the brackets (3) it follows that:

{I(\vec x), I(\vec y)} = e^{-\frac{\pi}{2}\{\cdot,G\}}{\rho(\vec x), \rho(\vec y)} = 0 (11)

Thus the density $I(\vec x)$ essentially defines a set of infinitely many conserved quantities which are in involution with each other.

Let us consider the scenario wherein the system is defined on a plane of length and breadth $L$ such that $\psi(0,y,t) = \psi(L,y,t)$ and $\psi(x,0,t) = \psi(x,L,t)$ . In such a case it is possible to define the Fourier components of $I$ as:

Q_{mn} = \int \frac{d^2k}{(2\pi)^2} I(\vec x) e^{i\vec k\cdot\vec x} (12)

wherein $\vec k = \frac{2\pi}{L}(m\hat{i} + n\hat{j})$ , with $m,n$ being integers. The infinite conservation laws are then encoded into the algebra of charges $Q_{mn}$ which follow from (10) and (11):

{Q_{kl}, Q_{mn}} = 0, {H, Q_{kl}} = 0 (13)

for integers $m,n,k,l$ .

This shows that the nonlinear Schrödinger system in two dimensions is Liouville integrable.

3. Conclusion

It has long been known that the nonlinear Schrödinger equation plays an important role governing different physical phenomena like superconductivity, superfluidity, nonlinear optical pulse propagation and so on. Half a century back the celebrated one dimensional nonlinear Schrödinger equation was discovered to be integrable, admitting infinitely many conserved charges, which were explicitly constructed using methods like inverse scattering transform. In the present work, we show that the two dimensional nonlinear Schrödinger equation is also Liouville integrable by explicitly constructing infinitely many conserved charges exploiting the presence of $\mathfrak{sl}(2,\mathbb{R})$ algebra in the system.

Formally it should be possible to extend the mathematical technique employed here to the quantum version of the system to show its integrability as well. Work along these lines is in progress and shall be reported in due course.

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