def ssfm(f: AbstractFieldEquation, waves: np.ndarray, z: float, h: float
) -> np.ndarray:
r"""Split step Fourier method.
Parameters
----------
f :
The function to compute.
waves :
The value of the unknown (waves) at the considered
space step.
z :
The current value of the space variable.
h :
The step size.
Returns
-------
:
The one step euler computation results.
Notes
-----
Equations:
.. math::
\begin{align}
A_j^{N} &= \exp\Big(h\hat{\mathcal{N}}\big(A_1(z,T),
\ldots, A_j(z,T), \ldots, A_K(z,T)\big)\Big)A_j(z,T)
&\forall j=1,\ldots,K\\
A_j(z+h,T) &= \exp\big(h\mathcal{D}\big)A_j^{N} &\forall
j=1,\ldots,K
\end{align}
Implementation:
.. math::
\begin{align}
A_j^{N} &= \exp\Big(h\hat{\mathcal{N}\big(A_1(z),\ldots,
A_j(z), \ldots, A_K(z)\big)\Big) A_j(z) &\forall
j=1,\ldots,K\\
A_j(z+h) &= \mathcal{F}^{-1}\Big\{\exp\Big(h
\hat{\mathcal{D}}\Big)\mathcal{F}
\{A_j^{N}(z)\}\Big\} &\forall j=1,\ldots,K
\end{align}
where :math:`K` is the number of channels.
"""
A_N = np.zeros_like(waves)
for i in range(len(waves)):
A_N[i] = f.exp_term_non_lin(waves, i, z, h, waves[i])
for i in range(len(waves)):
waves[i] = f.exp_term_lin(A_N, i, z, h)
return waves
def rk4ip(f: AbstractFieldEquation, waves: np.ndarray, z: float,
h: float) -> np.ndarray:
r"""Runge-Kutta interaction picture method.
Parameters
----------
f :
The function to compute.
waves :
The value of the unknown (waves) at the considered
space step.
z :
The current value of the space variable.
h :
The step size.
Returns
-------
:
The one step euler computation results.
Notes
-----
Equations:
.. math::
\begin{align}
A^L_j &=\exp\Big(\frac{h}{2}\mathcal{D}\Big)A_j(z,T)
&\forall j=1,\ldots,K\\
k_{0,j} &=\exp\Big(\frac{h}{2}\mathcal{D}\Big) \Big(h
\bar{\mathcal{N}}\big(A_1(z,T), \ldots, A_j(z,T),
\ldots,A_K(z,T)\big)\Big) &\forall j=1,\ldots,K\\
k_{1,j} &=h \bar{\mathcal{N}}\Big(A^L_{1}
+ \frac{k_{0,1}}{2},\ldots, A^L_{j}
+ \frac{k_{0,j}}{2}, \ldots, A^L_{K}
+ \frac{k_{0,K}}{2}\Big) &\forall j=1,\ldots,K\\
k_{2,j} &=h \bar{\mathcal{N}}\Big(A^L_{1}
+ \frac{k_{1,1}}{2},\ldots, A^L_{j}
+ \frac{k_{1,j}}{2}, \ldots, A^L_{K}
+ \frac{k_{1,K}}{2}\Big) &\forall j=1,\ldots,K\\
k_{3,j} &=h \bar{\mathcal{N}}\Big(\exp\Big(\frac{h}{2}
\mathcal{D}\Big)(A^L_{1} + k_{2,1}, \ldots, A^L_{j}
+ k_{2,j}, \ldots, A^L_{K} + k_{2,K})\Big)
&\forall j=1,\ldots,K\\
A_j(z+h,T) &=\frac{k_{3,j}}{6}+\exp\Big(\frac{h}{2}
\mathcal{D}\Big)\Big(A^L_{j}+\frac{k_{0,j}}{6}
+\frac{k_{1,j}}{3}+\frac{k_{2,j}}{3}\Big)
\quad &\forall j=1,\ldots,K
\end{align}
Implementation:
.. math::
\begin{align}
A^L_{j} &= \mathcal{F}^{-1}\Big\{\exp\Big(\frac{h}{2}
\hat{\mathcal{D}}\Big)\mathcal{F}\{A_j(z)\}\Big\}
&\forall j=1,\ldots,K \\
k_{0,j} &= \mathcal{F}^{-1}\Big\{\exp\Big(\frac{h}{2}
\hat{\mathcal{D}}\Big)\mathcal{F}\big\{h
\hat{\mathcal{N}}\big(A_1(z), \ldots, A_j(z),
\ldots,A_K(z)\big)\big\}\Big\}
&\forall j=1,\ldots,K\\
k_{1,j} &= h \hat{\mathcal{N}}\Big(A^L_{1}
+ \frac{k_{0,1}}{2}, \ldots, A^L_{j}
+ \frac{k_{0,j}}{2}, \ldots, A^L_{K}
+ \frac{k_{0,K}}{2} \Big) &\forall j=1,\ldots,K\\
k_{2,j} &= h \hat{\mathcal{N}}\Big(A^L_{1}
+ \frac{k_{1,1}}{2}, \ldots, A^L_{j}
+ \frac{k_{1,j}}{2}, \ldots, A^L_{K}
+ \frac{k_{1,K}}{2} \Big) &\forall j=1,\ldots,K\\
k_{3,j} &= h \hat{\mathcal{N}}\Big(\mathcal{F}^{-1}
\Big\{\exp\Big(\frac{h}{2}\hat{\mathcal{D}}\Big)
\mathcal{F}\{A^L_{1} + k_{2,1}\}
\Big\} , \ldots, \nonumber \\
& \qquad \qquad \mathcal{F}^{-1}\Big\{\exp\Big(
\frac{h}{2}\hat{\mathcal{D}}\Big)\mathcal{F}
\{A^L_{j} + k_{2,j}\}\Big\} ,\ldots, \nonumber \\
& \qquad \qquad \mathcal{F}^{-1}\Big\{\exp\Big(
\frac{h}{2}\hat{\mathcal{D}} \Big)\mathcal{F}
\{A^L_{K} + k_{2,K}\}\Big\} \Big)
&\forall j=1,\ldots,K\\
A_j(z+h) &= \frac{k_{3,j}}{6}+\mathcal{F}^{-1}\Big\{
\exp\Big(\frac{h}{2}\hat{\mathcal{D}}\Big)
\mathcal{F}\big\{A^L_{j}
+\frac{k_{0,j}}{6}+\frac{k_{1,j}}{3}
+\frac{k_{2,j}}{3}\big\}\Big\} \quad
&\forall j=1,\ldots,K
\end{align}
where :math:`K` is the number of channels.
"""
h_h = 0.5 * h
A_L = np.zeros_like(waves)
k_0 = np.zeros_like(waves)
k_1 = np.zeros_like(waves)
k_2 = np.zeros_like(waves)
k_3 = np.zeros_like(waves)
for i in range(len(waves)):
A_L[i] = f.exp_term_lin(waves, i, z, h_h)
for i in range(len(waves)):
waves[i] = h * f.term_non_lin(waves, i, z)
for i in range(len(waves)):
k_0[i] = f.exp_term_lin(waves, i, z, h_h)
for i in range(len(waves)):
waves[i] = A_L[i] + 0.5*k_0[i]
for i in range(len(waves)):
k_1[i] = h * f.term_non_lin(waves, i, z)
for i in range(len(waves)):
waves[i] = A_L[i] + 0.5*k_1[i]
for i in range(len(waves)):
k_2[i] = h * f.term_non_lin(waves, i, z)
for i in range(len(waves)):
waves[i] = A_L[i] + k_2[i]
waves[i] = f.exp_term_lin(waves, i, z, h_h)
for i in range(len(waves)):
k_3[i] = h * f.term_non_lin(waves, i, z)
for i in range(len(waves)):
waves[i] = A_L[i] + (k_0[i]/6.0) + ((k_1[i]+k_2[i])/3.0)
waves[i] = (k_3[i]/6.0) + f.exp_term_lin(waves, i, z, h_h)
return waves
def ssfm_opti_super_sym(f: AbstractFieldEquation, waves: np.ndarray,
z: float, h: float) -> np.ndarray:
r"""Optimized super symmetric split step Fourier method.
Parameters
----------
f :
The function to compute.
waves :
The value of the unknown (waves) at the considered
space step.
z :
The current value of the space variable.
h :
The step size.
Returns
-------
:
The one step euler computation results.
Notes
-----
Equations:
.. math::
\begin{align}
A_j^{L} &= \exp\Big(\frac{h}{2}\mathcal{D}\Big)A_j(z, T)
&\forall j=1,\ldots,K\\
A_j^{N'} &= A_j^{L} +\frac{h}{2}\bar{\mathcal{N}}\big(
A_1^{N'},\ldots, A_{j-1}^{N'}, A_{j}^{L}, \ldots,
A_K^{L}\big) &\forall j=1,\ldots,K\\
A_j^{N} &= A_j^{N'} +\frac{h}{2}\bar{\mathcal{N}}\big(
A_1^{N'},\ldots, A_{j}^{N'}, A_{j+1}^{N}, \ldots,
A_K^{N}\big)&\forall j=K,\ldots,1\\
A_j(z+h, T) &= \exp\Big(\frac{h}{2}\mathcal{D}\Big)
A_j^{N} &\forall j=1,\ldots,K
\end{align}
Implementation:
.. math::
\begin{align}
A_j^{L} &= \mathcal{F}^{-1}\Big\{\exp\Big(\frac{h}{2}
\hat{\mathcal{D}}\Big)\mathcal{F}\{A_j(z)\}\Big\}
&\forall j=1,\ldots,K\\
A_j^{N'} &= \exp\Big(\frac{h}{2}\hat{\mathcal{N}}
\big(A_1^{N'},\ldots, A_{j-1}^{N'}, A_{j}^{L},
\ldots, A_K^{L}\big)\Big) A_j^{L}
&\forall j=1,\ldots,K\\
A_j^{N} &= \exp\Big(\frac{h}{2}\hat{\mathcal{N}}
\big(A_1^{N'},\ldots, A_{j}^{N'}, A_{j+1}^{N},
\ldots, A_K^{N}\big)\Big) A_j^{N'}
&\forall j=K,\ldots,1 \\
A_j(z+h) &= \mathcal{F}^{-1}\Big\{\exp\Big(\frac{h}{2}
\hat{\mathcal{D}}\Big)\mathcal{F}\{A_j^{N}\}\Big\}
&\forall j=1,\ldots,K
\end{align}
where :math:`K` is the number of channels.
"""
h_h = 0.5 * h
for i in range(len(waves)):
waves[i] = f.exp_term_lin(waves, i, z, h_h)
for i in range(len(waves)-1):
waves[i] = f.exp_term_non_lin(waves, i, z, h_h, waves[i])
waves[-1] = f.exp_term_non_lin(waves, len(waves)-1, z, h, waves[-1])
for i in range(len(waves)-2, -1, -1):
waves[i] = f.exp_term_non_lin(waves, i, z, h_h, waves[i])
for i in range(len(waves)):
waves[i] = f.exp_term_lin(waves, i, z, h_h)
return waves