.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/gallery_02/g_molecules_01.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_auto_examples_gallery_02_g_molecules_01.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_gallery_02_g_molecules_01.py:


Superimposed fundamental solitons
=================================

This examples demonstrates the generation of two-frequency soliton molecules,
using the forward model for the analytic signal [1,2], in `py-fmas` implemented
as :class:`FMAS`.

In particular, this example shows how soliton molecules are generated from two
initially superimposed fundamental solitons at distinctly different frequencies
[3]. The exmample reproduces the propagation scenario shown in Fig. S10 of the
supplementary material to [3].

References:

    [1] Sh. Amiranashvili, A. Demircan, Hamiltonian structure of
    propagation equations for ultrashort optical pulses, Phys. Rev. E 10
    (2010) 013812, http://dx.doi.org/10.1103/PhysRevA.82.013812.

    [2] Sh. Amiranashvili, A. Demircan, Ultrashort Optical Pulse Propagation in
    terms of Analytic Signal, Adv. Opt. Tech. 2011 (2011) 989515,
    http://dx.doi.org/10.1155/2011/989515.

    [3] O. Melchert, S. Willms, S. Bose, A. Yulin, B. Roth, F. Mitschke, U.
    Morgner, I. Babushkin, A. Demircan, Soliton Molecules with Two Frequencies,
    Phys. Rev. Lett. 123 (2019) 243905,
    https://doi.org/10.1103/PhysRevLett.123.243905.

.. codeauthor:: Oliver Melchert <melchert@iqo.uni-hannover.de>

.. GENERATED FROM PYTHON SOURCE LINES 31-85



.. image:: /auto_examples/gallery_02/images/sphx_glr_g_molecules_01_001.png
    :alt: $|u|^2/{\rm{max}}\left(|u|^2\right)$, $|u_\omega|^2/{\rm{max}}\left(|u_\omega|^2\right)$
    :class: sphx-glr-single-img





.. code-block:: default

    import fmas
    import numpy as np
    from fmas.grid import Grid
    from fmas.models import FMAS
    from fmas.solver import IFM_RK4IP
    from fmas.analytic_signal import AS
    from fmas.tools import change_reference_frame, plot_evolution
    from fmas.propagation_constant import PropConst


    def define_beta_fun():
        r"""Custom refractive index.
        """
        p = np.poly1d((9.653881, -39.738626, 16.8848987, -2.745456)[::-1])
        q = np.poly1d((1.000000,  -9.496406,  4.2206250, -0.703437)[::-1])
        n_idx = lambda w: p(w)/q(w)     # (-)
        c0 = 0.29979                    # (micron/fs)
        return lambda w: n_idx(w)*w/c0  # (1/micron)

    def main():

        t_max = 2000.           # (fs)
        t_num = 2**14           # (-)
        z_max =  0.06e6         # (micron)
        z_num = 25000           # (-)
        z_skip = 50             # (-)
        chi = 1.0               # (micron^2/W)
        c0 = 0.29979            # (micron/fs)

        # -- PROPAGATION CONSTANT
        beta_fun = define_beta_fun()
        pc = PropConst(beta_fun)

        # -- INITIALIZE DATA-STRUCTURES AND ALGORITHMS
        grid = Grid( t_max = t_max, t_num = t_num, z_max = z_max, z_num = z_num)
        model = FMAS(w=grid.w, beta_w = beta_fun(grid.w), chi = chi )
        solver = IFM_RK4IP( model.Lw, model.Nw, user_action = model.claw)

        # -- PREPARE INITIAL CONDITION AND RUN SIMULATION
        w01, t01, A01 = 1.178, 30.0,  0.0248892 # (rad/fs), (fs), (sqrt(W))
        w02, t02, A02 = 2.909, 30.0, 0.0136676  # (rad/fs), (fs), (sqrt(W))
        A_0t_fun = lambda t, A0, t0, w0: np.real(A0/np.cosh(t/t0)*np.exp(1j*w0*t))
        E_0t = A_0t_fun(grid.t, A01, t01, w01) + A_0t_fun(grid.t, A02, t02, w02)
        solver.set_initial_condition( grid.w, AS(E_0t).w_rep)
        solver.propagate( z_range = z_max, n_steps = z_num, n_skip = z_skip)

        # -- SHOW RESULTS IN MOVING FRAME OF REFERENCE
        v0 = 0.0749641870819 # (micron/fs)
        utz = change_reference_frame(solver.w, solver.z, solver.uwz, v0)
        plot_evolution( solver.z, grid.t, utz, t_lim=(-100,150), w_lim=(0.3,3.8))


    if __name__=='__main__':
        main()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 2 minutes  31.420 seconds)


.. _sphx_glr_download_auto_examples_gallery_02_g_molecules_01.py:


.. only :: html

 .. container:: sphx-glr-footer
    :class: sphx-glr-footer-example



  .. container:: sphx-glr-download sphx-glr-download-python

     :download:`Download Python source code: g_molecules_01.py <g_molecules_01.py>`



  .. container:: sphx-glr-download sphx-glr-download-jupyter

     :download:`Download Jupyter notebook: g_molecules_01.ipynb <g_molecules_01.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_