.. _sphx_glr_api_gallery_fields_solitons:

Solitons
----------

The Korteweg-de Vries, nonlinear Schrodinger, and Sine-Gordon equations:
shape-preserving traveling waves, elastic collisions, wavepacket spreading,
and topologically protected kinks -- several animated frame by frame.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Korteweg and de Vries&#x27;s 1895 equation,">

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  .. image:: /api/gallery/fields/solitons/images/thumb/sphx_glr_plot_kdv_elastic_collision_thumb.png
    :alt:

  :doc:`/api/gallery/fields/solitons/plot_kdv_elastic_collision`

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      <div class="sphx-glr-thumbnail-title">An elastic KdV soliton-soliton collision</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Trying to resolve the Fermi-Pasta-Ulam-Tsingou recurrence puzzle, Norman Zabusky and Martin Kruskal simulated the continuum (KdV) limit of a nonlinear oscillator chain in 1965 and watched a single generic disturbance -- not an exact multi-soliton solution -- &quot;fission&quot; into a rank-ordered train of solitary waves that then collided and emerged unscathed. kdv_evolve integrates the Korteweg-de Vries equation">

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  .. image:: /api/gallery/fields/solitons/images/thumb/sphx_glr_plot_kdv_soliton_fission_thumb.png
    :alt:

  :doc:`/api/gallery/fields/solitons/plot_kdv_soliton_fission`

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      <div class="sphx-glr-thumbnail-title">Zabusky-Kruskal fission: coining "soliton"</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In 1834, John Scott Russell watched a solitary heap of water detach from a boat&#x27;s bow wave on the Union Canal and roll on for miles &quot;without change of form or diminution of speed.&quot; Sixty years later, Korteweg and de Vries explained this with the equation now bearing their names,">

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  .. image:: /api/gallery/fields/solitons/images/thumb/sphx_glr_plot_kdv_soliton_translation_thumb.png
    :alt:

  :doc:`/api/gallery/fields/solitons/plot_kdv_soliton_translation`

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      <div class="sphx-glr-thumbnail-title">Russell's wave of translation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Akira Hasegawa and Fred Tappert showed in 1973 that the same balance of nonlinearity and dispersion behind Russell&#x27;s water wave -- now governed by the focusing nonlinear Schrodinger equation instead of KdV,">

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  .. image:: /api/gallery/fields/solitons/images/thumb/sphx_glr_plot_nls_optical_soliton_thumb.png
    :alt:

  :doc:`/api/gallery/fields/solitons/plot_nls_optical_soliton`

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      <div class="sphx-glr-thumbnail-title">A dispersion-free optical soliton</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Erwin Schrodinger&#x27;s 1926 wave equation,">

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  .. image:: /api/gallery/fields/solitons/images/thumb/sphx_glr_plot_nls_wavepacket_spreading_thumb.png
    :alt:

  :doc:`/api/gallery/fields/solitons/plot_nls_wavepacket_spreading`

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      <div class="sphx-glr-thumbnail-title">Schrodinger's wavepacket spreading</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Yakov Frenkel and Tatiana Kontorova modeled a crystal dislocation as a chain of atoms in a periodic substrate potential -- a chain of coupled pendula; in the continuum limit this becomes the Sine-Gordon equation">

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  .. image:: /api/gallery/fields/solitons/images/thumb/sphx_glr_plot_sine_gordon_kink_thumb.png
    :alt:

  :doc:`/api/gallery/fields/solitons/plot_sine_gordon_kink`

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      <div class="sphx-glr-thumbnail-title">A topologically protected Sine-Gordon kink</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Gardner, Greene, Kruskal, and Miura&#x27;s 1967 inverse scattering transform (IST) explained why solitons collide elastically, and showed that KdV was only the first of a broad class of exactly integrable equations -- the Sine-Gordon equation">

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  .. image:: /api/gallery/fields/solitons/images/thumb/sphx_glr_plot_sine_gordon_kink_antikink_ist_thumb.png
    :alt:

  :doc:`/api/gallery/fields/solitons/plot_sine_gordon_kink_antikink_ist`

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      <div class="sphx-glr-thumbnail-title">An exact kink-antikink collision from inverse scattering</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/fields/solitons/plot_kdv_elastic_collision
   /api/gallery/fields/solitons/plot_kdv_soliton_fission
   /api/gallery/fields/solitons/plot_kdv_soliton_translation
   /api/gallery/fields/solitons/plot_nls_optical_soliton
   /api/gallery/fields/solitons/plot_nls_wavepacket_spreading
   /api/gallery/fields/solitons/plot_sine_gordon_kink
   /api/gallery/fields/solitons/plot_sine_gordon_kink_antikink_ist