.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/fields/solitons/plot_nls_optical_soliton.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_fields_solitons_plot_nls_optical_soliton.py: A dispersion-free optical soliton ===================================== Akira Hasegawa and Fred Tappert showed in 1973 that the same balance of nonlinearity and dispersion behind Russell's water wave -- now governed by the focusing nonlinear Schrodinger equation instead of KdV, .. math:: i\partial_t\psi + \tfrac12\partial_x^2\psi + |\psi|^2\psi = 0, -- lets light pulses in an optical fiber propagate as solitons, self-correcting against the pulse-spreading dispersion that otherwise limits every long-distance optical line. :func:`~physicskit.fields.solitons.nls_bright_soliton` constructs the exact envelope solution, .. math:: \psi(x,t) = A\,\mathrm{sech}\big(A(x-x_0-vt)\big)\, e^{\,i\left[v(x-x_0) + (A^2-v^2)t/2\right]}, and :func:`~physicskit.fields.solitons.nls_evolve` propagates it via split-step Fourier integration (here at rest, :math:`v=0`, amplitude :math:`A=1`), showing the envelope :math:`|\psi|` is unchanged after "fiber" propagation where an ordinary pulse would visibly spread; the same run can be watched frame by frame with :func:`~physicskit.fields.solitons.nls_evolve_frames`. .. GENERATED FROM PYTHON SOURCE LINES 31-37 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.fields import animate_field_1d, nls_bright_soliton, nls_evolve, nls_evolve_frames, plot_field_1d .. GENERATED FROM PYTHON SOURCE LINES 38-40 A fiber-optic bright soliton (focusing NLS: g > 0) -------------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 40-45 .. code-block:: Python N, L = 1024, 80.0 x = np.linspace(-L / 2, L / 2, N, endpoint=False) psi0 = nls_bright_soliton(x, t=0.0, A=1.0) .. GENERATED FROM PYTHON SOURCE LINES 46-48 Propagate it down 2000 dt of "fiber" ------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 48-51 .. code-block:: Python psi = nls_evolve(psi0, x, dt=0.001, steps=2000, g=1.0) .. GENERATED FROM PYTHON SOURCE LINES 52-55 Nonlinearity (self-phase modulation) exactly cancels dispersion: the envelope :math:`|\psi(x,t)|` is unchanged, unlike an ordinary pulse governed by dispersion alone. .. GENERATED FROM PYTHON SOURCE LINES 55-66 .. code-block:: Python shape_error = np.max(np.abs(np.abs(psi) - np.abs(psi0))) fig, ax = plot_field_1d(x, np.abs(psi0), label="t = 0") plot_field_1d(x, np.abs(psi), ax=ax, label="t = 2 (fiber units)") ax.set_title(f"envelope shape error: {shape_error:.1e}") fig.tight_layout() print(f"envelope |psi| shape error after propagation: {shape_error:.2e}") print(f"peak envelope amplitude: t=0 -> {np.abs(psi0).max():.4f}, after -> {np.abs(psi).max():.4f}") .. image-sg:: /api/gallery/fields/solitons/images/sphx_glr_plot_nls_optical_soliton_001.png :alt: envelope shape error: 1.2e-04 :srcset: /api/gallery/fields/solitons/images/sphx_glr_plot_nls_optical_soliton_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none envelope |psi| shape error after propagation: 1.18e-04 peak envelope amplitude: t=0 -> 1.0000, after -> 0.9999 .. GENERATED FROM PYTHON SOURCE LINES 67-72 Animating the dispersion-free envelope -------------------------------------------- :func:`~physicskit.fields.solitons.nls_evolve_frames` records the same propagation as a sequence of snapshots, showing the shape-preserving envelope travel down the "fiber" rather than comparing only two instants. .. GENERATED FROM PYTHON SOURCE LINES 72-78 .. code-block:: Python frames, times = nls_evolve_frames(psi0, x, dt=0.001, steps_per_frame=50, n_frames=40, g=1.0) anim = animate_field_1d(x, frames, times, ylabel="|psi(x, t)|") plt.show() .. container:: sphx-glr-animation .. raw:: html .. GENERATED FROM PYTHON SOURCE LINES 79-83 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("nls_optical_soliton.gif", writer="pillow", fps=20) .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.620 seconds) .. _sphx_glr_download_api_gallery_fields_solitons_plot_nls_optical_soliton.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_nls_optical_soliton.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_nls_optical_soliton.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_nls_optical_soliton.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_