.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/gallery_03/g_model_KdV.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_examples_gallery_03_g_model_KdV.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_gallery_03_g_model_KdV.py:


Korteweg-deVries equation -- Interaction of solitons
====================================================

This example solves the Korteweg-deVries (KdV) equation, given by

.. math::
    \partial_t u = - \delta^2 \partial_x^3 u  -u \partial_x u,

wherein the evolution of the real-valued field :math:`u = u(t, x)` is
determined by the parameter :math:`\delta=0.022`.

The propagation scenario below considers the initial condition

.. math::
    u_0(x) = \cos(\pi x), \quad 0\leq x \leq 2,

and uses a pseudospectral integrating factor method, i.e. the "Runge-Kutta in
the interaction picture" method implemented in the `py-fmas` libragy, to
reproduce the data shown in Fig. 2 of Ref. [1].

References:
    [1] N. J. Zabusky, M. D. Kruskal, Interaction of "Solitons" in a
    Collisionless Plasma and the Recurrence of Initial States, Phys. Rev. Lett.
    15 (1965) 240, https://doi.org/10.1103/PhysRevLett.15.240.

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

.. GENERATED FROM PYTHON SOURCE LINES 29-113



.. image:: /auto_examples/gallery_03/images/sphx_glr_g_model_KdV_001.png
    :alt: $u(x,t)$
    :class: sphx-glr-single-img





.. code-block:: default

    import fmas
    import numpy as np
    import matplotlib as mpl
    import matplotlib.pyplot as plt
    import matplotlib.colors as col
    from fmas.config import FTFREQ, FT, IFT, C0
    from fmas.solver import IFM_RK4IP


    def plot_evolution_KdV(t, x, u):

        def _setColorbar(im, refPos):
            """colorbar helper"""
            x0, y0, w, h = refPos.x0, refPos.y0, refPos.width, refPos.height
            cax = f.add_axes([x0, y0+1.02*h, w, 0.03*h])
            cbar = f.colorbar(im, cax=cax, orientation='horizontal')
            cbar.ax.tick_params(color='k',
                                labelcolor='k',
                                bottom=False,
                                direction='out',
                                labelbottom=False,
                                labeltop=True,
                                top=True,
                                size=4,
                                pad=0
                                )

            cbar.set_ticks((-1,-0.5,0,0.5,1))
            cbar.ax.tick_params(which="minor", bottom=False, top=False )
            return cbar

        f, ax1 = plt.subplots(1, 1, sharey=True, figsize=(2.5,4))
        plt.subplots_adjust(left=0.2, right=0.9, bottom=0.12, top=0.84)
        cmap=mpl.cm.get_cmap('coolwarm')

        im1 = ax1.pcolorfast(x, t, u[:-1,:-1],
                             norm=col.Normalize(vmin=-1,vmax=1),
                             cmap=cmap
                             )
        cbar1 = _setColorbar(im1,ax1.get_position())
        cbar1.ax.set_title(r"$u(x,t)$",color='k',y=3.5)
        ax1.xaxis.set_ticks_position('bottom')
        ax1.yaxis.set_ticks_position('left')
        ax1.set_xlim(x[0],x[-1])
        ax1.set_xticks((0,0.5,1.,1.5,2))
        ax1.set_ylim([0.,t.max()])
        ax1.set_xlabel(r"Distance $x$")
        ax1.set_ylabel(r"Time $t$")

        plt.show()


    def main():
        # -- DEFINE SIMULATION PARAMETERS
        x_min = 0.
        x_max = 2.
        Nx = 512
        t_min = 0.
        t_max = 6.0
        Nt = 30000
        n_skip = 10
        delta = 0.022

        # -- INITIALIZATION STAGE
        # ... COMPUTATIONAL DOMAIN
        x = np.linspace(x_min, x_max, Nx, endpoint=False)
        k = FTFREQ(x.size,d=x[1]-x[0])*2*np.pi
        # ... KORTEWEG DEVRIES MODEL 
        Lk = -1j*k*k*k*delta*delta
        Nk_fun = lambda uk: 0.5j*k*FT(IFT(uk)**2)
        # ... SOLVER BASED ON INTEGRATING FACTOR METHOD 
        solver = IFM_RK4IP(Lk, Nk_fun)
        # ... INITIAL CONDITION
        u_0x = np.cos(np.pi*x)
        solver.set_initial_condition(k, FT(u_0x))

        # -- RUN SIMULATION
        solver.propagate(z_range = t_max, n_steps = Nt, n_skip = n_skip)

        plot_evolution_KdV( solver.z, x, np.real(solver.utz))


    if __name__=='__main__':
        main()


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

   **Total running time of the script:** ( 0 minutes  6.980 seconds)


.. _sphx_glr_download_auto_examples_gallery_03_g_model_KdV.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_model_KdV.py <g_model_KdV.py>`



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

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


.. only:: html

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

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