What is the Physics problem ?

Physical situation

The code offers to solve a typical non linear Schrödinger / Gross-Pitaevskii equation of the type :

\[i\partial_{t}\psi = -\frac{1}{2}\nabla^2\psi+V\psi+g|\psi|^2\psi\]

In this particular instance, we solve in the formalism of the propagation of a pulse of light in a non linear medium. Within the paraxial approximation, the propagation equation for the field \(E\) in V/m solved is:

\[ i\partial_{z}E = -\frac{1}{2k_0}\nabla_{\perp}^2 E + \frac{D_0}{2}\partial^2_t E -\frac{k_0}{2}\delta n(r) E - n_2 \frac{k_0}{2n}c\epsilon_0|E|^2E \]

Here, the constants are defined as followed :

• \(k_0\) : is the electric field wavenumber in \(m^{-1}\)
• \(D_0\) : is the group velocity dispersion (GVD) in \(s^2/m\)
• \(\delta n(\mathbf{r})\) : the "potential" i.e a local change in linear index of refraction. Dimensionless.
• \(n_2\) : the non linear index of refraction in \(m^2/W\).
• \(n\) is the linear index of refraction. In our case 1.
• \(c,\epsilon_0\) : the speed of light and electric permittivity of vacuum.

In all generality, the interaction term can be non-local i.e \(n_2=n_2(\mathbf{r})\). This means usually that the response will be described as a convolution by some non-local kernel:

\[ n_2(\mathbf{r})|E|^2(\mathbf{r})=n_2\int_{\mathbb{R}^2}\mathrm{d}\mathbf{r}' K(\mathbf{r}-\mathbf{r}')|E|^2(\mathbf{r}'), \]

where \(K(\mathbf{r})\) is the non-local kernel, typically the Green function of some diffusion equation.

Please note that all of the code works with the "God given" units i.e SI units !