Nonlinear SchröDinger Equation

Right. So you want me to… re-write something. Wikipedia, specifically. As if the existing information isn't enough of a testament to wasted effort. Fine. But don't expect sunshine and rainbows. This is going to be… precise. And probably a little bleak.

Nonlinear form of the Schrödinger equation

The illustration shows the absolute value of the complex envelope of exact analytical breather solutions for the nonlinear Schrödinger equation (NLSE) in its nondimensional form. We're looking at three specific types: (A) the Akhmediev breather, (B) the Peregrine breather, and (C) the Kuznetsov–Ma breather. [1]

In the rather dry realm of theoretical physics, the one-dimensional nonlinear Schrödinger equation, or NLSE, is essentially a nonlinear twist on the standard Schrödinger equation. It's a classical field equation with applications that are, frankly, more interesting than the equation itself. It’s used to describe how light behaves when it propagates through nonlinear optical fibers, or within planar waveguides [2]. It also shows up when studying hot rubidium vapors [3], and in the peculiar world of Bose–Einstein condensates that are confined to highly anisotropic, cigar-shaped traps, all within the mean-field regime [4].

Beyond that, you'll find this equation popping up in studies of small-amplitude gravity waves on the surface of deep, utterly inviscid water [2]. It's there for Langmuir waves in hot plasmas [2], and for the propagation of plane-diffracted wave beams in those areas of the ionosphere where things are focusing [5]. It even creeps into the study of Davydov's alpha-helix solitons, which are apparently responsible for energy transport along molecular chains [6]. The list goes on. More broadly, the NLSE is one of those universal equations that pops up when you're trying to describe the evolution of packets of quasi-monochromatic waves that are changing slowly in weakly nonlinear media exhibiting dispersion [2]. Just remember: unlike the linear Schrödinger equation, the NLSE never describes the actual time evolution of a quantum state. [Citation needed] The 1D NLSE is, for what it's worth, an example of an integrable model.

Now, in the hallowed halls of quantum mechanics, the 1D NLSE is just a particular instance of the classical nonlinear Schrödinger field. This, in turn, is considered a classical limit of a quantum Schrödinger field. Conversely, when you take that classical Schrödinger field and canonically quantize it, you get a quantum field theory. This theory, despite its name, is linear. It describes bosonic point particles that interact via a delta-function—they either push each other away or pull each other in when they occupy the same point. In fact, if you have a finite number of these particles, this quantum field theory is precisely equivalent to the Lieb–Liniger model. Both the quantum and the classical 1D nonlinear Schrödinger equations are, rather conveniently, integrable. It's worth noting the limit of infinite repulsion strength, where the Lieb–Liniger model transforms into the Tonks–Girardeau gas – also known as the hard-core Bose gas, or the impenetrable Bose gas. In this specific scenario, the bosons can be converted into a system of one-dimensional, noninteracting, spinless fermions through a change of variables that’s essentially a continuum version of the Jordan–Wigner transformation [7, 8].

The nonlinear Schrödinger equation itself is a simplified, 1+1-dimensional version of the Ginzburg–Landau equation. That equation was first put down in 1950 by Ginzburg and Landau in their work on superconductivity. The explicit form of the NLSE, however, was penned by R. Y. Chiao, E. Garmire, and C. H. Townes in 1964 [10], as part of their research into optical beams.

If you venture into multiple dimensions, the equation gets more complex. The second spatial derivative is replaced by the Laplacian. In dimensions higher than one, the equation loses its integrability. It then becomes capable of exhibiting phenomena like collapse and wave turbulence [9].

Definition

The nonlinear Schrödinger equation is a nonlinear partial differential equation. It finds its place in both classical and quantum mechanics.

Classical equation

In its dimensionless form, the classical field equation reads as follows [10]:

$i \frac{\partial \psi}{\partial t} = -\frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + \kappa |\psi|^2 \psi$

Here, $\psi(x, t)$ represents the complex field.

This equation originates from the Hamiltonian [10]:

$H = \int dx \left[ \frac{1}{2} |\partial_x \psi|^2 + \frac{\kappa}{2} |\psi|^4 \right]$

with the Poisson brackets defined as:

$\{\psi(x), \psi(y)\} = \{\psi^*(x), \psi^*(y)\} = 0$ $\{\psi^*(x), \psi(y)\} = i \delta(x-y)$

Unlike its linear ancestor, it never describes the time evolution of a quantum state. [Citation needed]

The case where $\kappa$ is negative is termed "focusing." This scenario allows for bright soliton solutions – those that are localized in space and fade towards infinity. It also permits breather solutions. This particular case can be precisely solved exactly using the inverse scattering transform, a feat accomplished by Zakharov & Shabat in 1972. The alternative, where $\kappa$ is positive, is known as the "defocusing" NLS. This version yields dark soliton solutions, characterized by a constant amplitude at infinity and a localized dip in amplitude. [11]

Quantum mechanics

To obtain the quantized version, you simply swap the Poisson brackets for commutators:

$[\psi(x), \psi(y)] = [\psi^*(x), \psi^*(y)] = 0$ $[\psi^*(x), \psi(y)] = -\delta(x-y)$

And you must normal order the Hamiltonian:

$H = \int dx \left[ \frac{1}{2} \partial_x \psi^\dagger \partial_x \psi + \frac{\kappa}{2} \psi^\dagger \psi^\dagger \psi \psi \right]$

The quantum version was solved by Bethe ansatz by Lieb and Liniger. The thermodynamic behavior was later described by Chen-Ning Yang. Quantum correlation functions were also evaluated by Korepin in 1993 [8]. This model possesses higher conservation laws, which Davies and Korepin expressed in terms of local fields in 1989 [12].

Solution

The nonlinear Schrödinger equation is integrable in one dimension. Zakharov and Shabat (1972) cracked it using the inverse scattering transform. The associated linear system of equations is known as the Zakharov–Shabat system:

$\phi_x = J\phi \Lambda + U\phi$ $\phi_t = 2J\phi \Lambda^2 + 2U\phi \Lambda + (JU^2 - JU_x)\phi$

where

$\Lambda = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}, \quad J = i\sigma_z = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \quad U = i \begin{pmatrix} 0 & q \\ r & 0 \end{pmatrix}$

The nonlinear Schrödinger equation emerges as the compatibility condition of the Zakharov–Shabat system:

$\phi_{xt} = \phi_{tx} \quad \Rightarrow \quad U_t = -JU_{xx} + 2JU^2 U \quad \Leftrightarrow \quad \begin{cases} iq_t = q_{xx} + 2qrq \\ ir_t = -r_{xx} - 2qrr \end{cases}$

By setting $q = r^*$ or $q = -r^*$ , one obtains the nonlinear Schrödinger equation with either attractive or repulsive interactions, respectively.

An alternative approach involves using the Zakharov–Shabat system directly and applying the following Darboux transformation:

$\phi \to \phi[1] = \phi \Lambda - \sigma \phi$ $U \to U[1] = U + [J, \sigma]$ $\sigma = \varphi \Omega \varphi^{-1}$

This transformation leaves the system invariant.

Here, $\varphi$ is another invertible matrix solution to the Zakharov–Shabat system, distinct from $\phi$ , and $\Omega$ is the spectral parameter:

$\varphi_x = J\varphi \Omega + U\varphi$ $\varphi_t = 2J\varphi \Omega^2 + 2U\varphi \Omega + (JU^2 - JU_x)\varphi$

Starting with the trivial solution $U=0$ and iterating this process allows one to construct solutions involving $n$ solitons. This can also be achieved through direct numerical simulation, for instance, using the split-step method [13]. This method has been implemented on both CPU and GPU architectures [14, 15].

Applications

Fiber optics

In the field of optics, the nonlinear Schrödinger equation is a key component of the Manakov system, which models wave propagation in fiber optics. The field $\psi$ represents a wave, and the NLSE describes how this wave travels through a nonlinear medium. The second-order derivative accounts for dispersion, while the $\kappa$ term signifies the nonlinearity. This equation is fundamental to understanding numerous nonlinear effects within optical fibers, including, but not limited to, self-phase modulation, four-wave mixing, second-harmonic generation, stimulated Raman scattering, optical solitons, and the behavior of ultrashort pulses.

Water waves

The image displays a hyperbolic secant (sech) envelope soliton for surface waves on deep water. The blue line represents the water waves themselves, while the red line shows the envelope soliton.

In the context of water waves, the nonlinear Schrödinger equation serves to describe the evolution of the envelope of modulated wave groups. In a seminal paper from 1968, Vladimir E. Zakharov detailed the Hamiltonian structure of water waves. Within the same publication, Zakharov demonstrated that for wave groups that are slowly modulated, the wave amplitude adheres to the nonlinear Schrödinger equation, albeit approximately [16]. The specific value of the nonlinearity parameter $\kappa$ is contingent upon the relative depth of the water. For deep water, where the water depth is significantly larger than the wave length of the water waves, $\kappa$ is negative, and envelope solitons can indeed form. Furthermore, the group velocity of these envelope solitons can be accelerated by external time-dependent water flows [17].

However, in shallow water conditions, where wavelengths are less than 4.6 times the water depth, the nonlinearity parameter $\kappa$ becomes positive. In this regime, wave groups with envelope solitons do not exist. While shallow water can support surface-elevation solitons, or waves of translation, these phenomena are not governed by the nonlinear Schrödinger equation.

The nonlinear Schrödinger equation is widely believed to play a crucial role in explaining the formation of rogue waves [18].

The complex field $\psi$ , as it appears in the nonlinear Schrödinger equation, is intrinsically linked to the amplitude and phase of the water waves. Consider a slowly modulated carrier wave with a surface elevation $\eta$ of the form:

$\eta = a(x_0, t_0) \cos[k_0 x_0 - \omega_0 t_0 - \theta(x_0, t_0)]$

Here, $a(x_0, t_0)$ and $\theta(x_0, t_0)$ represent the slowly modulated amplitude and phase, respectively. $\omega_0$ and $k_0$ are the constant angular frequency and wavenumber of the carrier waves, which must satisfy the dispersion relation $\omega_0 = \Omega(k_0)$ . In this formulation, we have:

$\psi = a \exp(i\theta)$

Consequently, the modulus of $\psi$ , $|\psi|$ , corresponds to the wave amplitude $a$ , and its argument, $\arg(\psi)$ , represents the phase $\theta$ .

The relationship between the physical coordinates $(x_0, t_0)$ and the transformed coordinates $(x, t)$ used in the nonlinear Schrödinger equation is given by:

$x = k_0 [x_0 - \Omega'(k_0) t_0], \quad t = k_0^2 [-\Omega''(k_0)] t_0$

Thus, $(x, t)$ defines a coordinate system that moves with the group velocity $\Omega'(k_0)$ of the carrier waves. The curvature of the dispersion relation, $\Omega''(k_0)$ , which signifies group velocity dispersion, is invariably negative for water waves under gravitational influence, regardless of the water depth.

For waves on the surface of deep water, the coefficients relevant to the nonlinear Schrödinger equation are:

$\kappa = -2k_0^2, \quad \Omega(k_0) = \sqrt{gk_0} = \omega_0$ So, $\Omega'(k_0) = \frac{1}{2} \frac{\omega_0}{k_0}, \quad \Omega''(k_0) = -\frac{1}{4} \frac{\omega_0}{k_0^2}$ where $g$ is the acceleration due to gravity.

When expressed in the original $(x_0, t_0)$ coordinates, the nonlinear Schrödinger equation for water waves takes this form [19]:

$i\,\partial_{t_0}A + i\,\Omega'(k_0)\,\partial_{x_0}A + \frac{1}{2}\Omega''(k_0)\,\partial_{x_0x_0}A - \nu\,|A|^2\,A = 0$

Here, $A = \psi^*$ (the complex conjugate of $\psi$ ) and:

$\nu = \kappa \, k_0^2 \, \Omega''(k_0)$

Therefore, for deep water waves:

$\nu = \frac{1}{2} \omega_0 k_0^2$

Vortices

Hasimoto (1972) established a significant connection between the work of da Rios (1906) on vortex filaments and the nonlinear Schrödinger equation. Subsequently, Salman (2013) leveraged this correspondence to demonstrate that breather solutions can also manifest in the context of vortex filaments [20].

Galilean invariance

The nonlinear Schrödinger equation exhibits Galilean invariance in a specific manner:

Given a solution $\psi(x, t)$ , a new solution can be generated by substituting $x$ with $x + vt$ throughout $\psi(x, t)$ and appending a phase factor of $e^{-iv(x+vt/2)}$ :

$\psi(x,t) \mapsto \psi_{[v]}(x,t) = \psi(x+vt, t) \, e^{-iv(x+vt/2)}$

Gauge equivalent counterpart

The NLSE (1) is gauge equivalent to the following isotropic Landau–Lifshitz equation (LLE), also known as the Heisenberg ferromagnet equation:

$\vec{S}_t = \vec{S} \wedge \vec{S}_{xx}$

It is important to note that this equation allows for several integrable and non-integrable generalizations in 2+1 dimensions, such as the Ishimori equation and others.

Zero-curvature formulation

The NLSE is equivalent to the condition that the curvature of a specific su(2)-connection on $\mathbb{R}^2$ is zero [21].

More precisely, using coordinates $(x, t)$ on $\mathbb{R}^2$ , the connection components $A_{\mu}$ are defined as:

$A_x = \begin{pmatrix} i\lambda & i\varphi^{*} \\ i\varphi & -i\lambda \end{pmatrix}$ $A_t = \begin{pmatrix} 2i\lambda^2 - i|\varphi|^2 & 2i\lambda \varphi^{*} + \varphi_x^{*} \\ 2i\lambda \varphi - \varphi_x & -2i\lambda^2 + i|\varphi|^2 \end{pmatrix}$

where $\sigma_i$ are the Pauli matrices. The zero-curvature equation,

$\partial_t A_x - \partial_x A_t + [A_x, A_t] = 0$

is then equivalent to the NLSE:

$i\varphi_t + \varphi_{xx} + 2|\varphi|^2 \varphi = 0$

This zero-curvature equation is so named because it implies that the curvature, defined as $F_{\mu \nu} = [\partial_{\mu} - A_{\mu}, \partial_{\nu} - A_{\nu}]$ , is identically zero.

The pair of matrices $A_x$ and $A_t$ are also recognized as a Lax pair for the NLSE. This is in the sense that the zero-curvature equation recovers the partial differential equation itself, rather than them satisfying Lax's equation directly.