gpe.exact_solutions

gpe.exact_solutions#

This file contains a set of exact solutions useful for testing.

Classes#

`HarmonicOscillator`	Modified 1D harmonic oscillator potential with analytic solution.
`HarmonicOscillators`	Modified 2D harmonic oscillator potential with analytic solution.
`BrightSoliton`	Analytic moving bright soliton (for attractive interactions).
`TravellingWaves`	Analytic traveling wave solutions for the GPE without a
`DarkBrightSolitons`	State with HO potential for backward compatibility.

Functions#

`sn`(u, m)	Jacobi elliptic function sn.
`K`(m)	Jacobi elliptic integral

Module Contents#

class HarmonicOscillator(Nx=46, Lx=17.0, sigma=1.0, g=1.0, n0=1.0, **kw)[source]#

Bases: gpe.bec.StateBase

Modified 1D harmonic oscillator potential with analytic solution.

Here we add a piece to the harmonic oscillator potential to ensure that a Gaussian is an eigenstate, even in the presence of interactions.

w = 1.0[source]#

n0 = 1.0[source]#

sigma = 1.0[source]#

get_psi_exact(r)[source]#

get_Vext_r(r)[source]#

property psi_exact[source]#

get_Vext_GPU()[source]#: Return the external potential. This default version is zero.

class HarmonicOscillators(Nxyz=(46, 46), Lxyz=(17.0, 17.0), sigmas=(1.0, 1.0), g=1.0, n0=1.0, **kw)[source]#

Bases: gpe.bec.StateBase

Modified 2D harmonic oscillator potential with analytic solution.

Here we add a piece to the harmonic oscillator potential to ensure that a Gaussian is an eigenstate, even in the presence of interactions.

ws[source]#

n0 = 1.0[source]#

sigmas = (1.0, 1.0)[source]#

get_psi_exact(xyz=None)[source]#

get_Vext_GPU()[source]#: Return the external potential. This default version is zero.

property psi_exact[source]#

class BrightSoliton(Nx=2**4 * 3**3, Lx=58.0, sigma=1.0, g=None, n0=1.0, v=0.0, **kw)[source]#

Bases: gpe.bec.StateTwist_x

Analytic moving bright soliton (for attractive interactions).

sigma = 1.0[source]#

n0 = 1.0[source]#

v = 0.0[source]#

property x[source]#: Flat x abscissa as a numpy array.

property psi0[source]#

sn(u, m)[source]#: Jacobi elliptic function sn.

K(m)[source]#: Jacobi elliptic integral

class TravellingWaves(Nx=64, Lx=10.0, n0=0.1, n1=1.0, m=1.0, g=1.0, hbar=1.0, v_p=0.0, v_x=None, twist=None, **kw)[source]#

Bases: gpe.bec.StateTwist_x

Analytic traveling wave solutions for the GPE without a potential.

The solutions are characterized by the following parameters:

n0, n1float: Minimum and maximum density of the soliton. The amplitude is a=n1-n0.
Lxfloat: The period of the wave, and the length of our box. This is specified in terms of Lx = 2lK(m_), where m_=al^2.
v_pfloat: Phase velocity.
v_xfloat: Velocity of the frame. If None, this is set to be the phase velocity so that the solution should be stationary.
twistfloat: Twisted boundary condition. If None, this is computed by integrating the continuity equation.

v_unit = 1.0[source]#

v_p = 0.0[source]#

_n0 = 0.1[source]#

_m[source]#

_C[source]#

mu = 1.05[source]#

_twist = None[source]#

n_exact(x=None)[source]#

psi_exact(x=None, Lx=None)[source]#

Return the exact solution.

Parameters:

x (array, float) – Abscissa and box size - used by the constructor for getting the exact solution before the state has been properly initialized.
Lx (array, float) – Abscissa and box size - used by the constructor for getting the exact solution before the state has been properly initialized.

static get_m(_a, _L)[source]#: Return the solution to _L = 2*sqrt(m/_a)*K(m).

class DarkBrightSolitons(kappa=1.0, bs=[-1.0, -2.0, -3.0], lambdas=(0.0, 0.0), **kw)[source]#

Bases: gpe.bec2.State

State with HO potential for backward compatibility.

kappa = 1.0[source]#

bs[source]#

lambdas = (0.0, 0.0)[source]#

Cs[source]#

get_initial_state(x0=0.0, eta=1.0, v=1.0, t=0.0, mu=1.0)[source]#