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<main>
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<article id="content">
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<header>
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<h1 class="title">Module <code>SWtools_ext_LE</code></h1>
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</header>
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<section id="section-intro">
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<p>SWtools_ext_LE_v2.py</p>
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<p>SWtools extension module implementing functions for the linear stability
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analysis of generalized nonlinear Schrödinger equations.
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The analysis proceeds
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by calculating the linearized eigenspectrum (LE), describing small-amplitude
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perturbations atop a solitary wave.</p>
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<h2 id="note">Note</h2>
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<p>This module implements two methods for solving for the LE of genarlized
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nonlinear Schrödinger equations with different restictions:</p>
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<p>LE_GNSE: this function is tailored towards a GNSE with with generic nonlinear
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functional and linear differential operator of second order in the transverse
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coordinate.</p>
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<p>LE_HONSE: this function is tailored towards a GNSE with cubic nonlinear
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functional and linear differential operator including dispersion of orders two,
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three and four.</p>
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<div class="admonition codeauthor">
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<p class="admonition-title">Codeauthor:&ensp;Oliver Melchert <a href="&#109;&#97;&#105;&#108;&#116;&#111;&#58;&#109;&#101;&#108;&#99;&#104;&#101;&#114;&#116;&#64;&#105;&#113;&#111;&#46;&#117;&#110;&#105;&#45;&#104;&#97;&#110;&#110;&#111;&#118;&#101;&#114;&#46;&#100;&#101;">&#109;&#101;&#108;&#99;&#104;&#101;&#114;&#116;&#64;&#105;&#113;&#111;&#46;&#117;&#110;&#105;&#45;&#104;&#97;&#110;&#110;&#111;&#118;&#101;&#114;&#46;&#100;&#101;</a></p>
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</div>
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</section>
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<section>
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</section>
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<section>
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</section>
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<section>
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<h2 class="section-title" id="header-functions">Functions</h2>
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<dl>
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<dt id="SWtools_ext_LE.LE_GNSE"><code class="name flex">
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<span>def <span class="ident">LE_GNSE</span></span>(<span>xi, U, kap, c2, F_fun, F1_fun)</span>
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</code></dt>
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<dd>
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<details class="source">
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<summary>
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<span>Expand source code</span>
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</summary>
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<pre><code class="python">def LE_GNSE(xi, U, kap, c2, F_fun, F1_fun):
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&#34;&#34;&#34;Linearized eigenspectrum for the GNSE.
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Solves the eigenvalue problem for the linear stability matrix of a GNSE
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with generic nonlinear functional and linear differential operator of
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second order in the transverse coordinate [P1998,K1998].
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Note
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----
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Assumes that the soliton solution is given by an even, positive,
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single-humped function U, see [P1998].
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References
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----------
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[P1998] D. E. Pelinovsky, Y. S. Kivshar, V. V. Afanasjev, Internal modes of
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envelope solitons, Physica D 116 (1) (1998) 121–142,
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https://doi.org/10.1016/S0167-2789(98)80010-9.
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[K1998] Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Internal
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modes of solitary waves, Phys. Rev. Lett. 80 (1998) 5032,
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https://doi.org/10.1103/PhysRevLett.80.5032.
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Parameters
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----------
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xi : array_like
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Disrete transverse coordinates.
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U : array_like
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Solitary wave solution.
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kap : float
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Solitary wave wavenumber.
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c2 : float
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Constant coefficient of the linear differential operator part
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F_fun : function
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Nonlinear functional.
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F1_fun : function
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Derivative of the nonlinear functional.
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&#34;&#34;&#34;
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# -- COMPOSE MATRIX DETERMINING EVP FOR THE PERTURBATION MODES
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# ... PREPARE COEFFICIENTS FOR FINITE-DIFFERENCE MATRIX
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dxi = xi[1]-xi[0]
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dxi2= dxi**2
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I = np.abs(U)**2
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# ... SET UP DIAGONALS
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D0 = -(30/12)*c2/dxi2*np.ones(xi.size)
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D1 = (16/12)*c2/dxi2*np.ones(xi.size-1)
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D2 = -( 1/12)*c2/dxi2*np.ones(xi.size-2)
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# ... SET UP SPARSE FINITE DIFFERENCE MATRICES
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Z0 = diags([np.zeros_like(U)], [0]).toarray()
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L0 = diags([D0 + kap - F_fun(I), D1, D2, D1, D2], [0,-1,-2,1,2]).toarray()
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L1 = diags([D0 + kap - F_fun(I) - 2*I*F1_fun(I), D1, D2, D1, D2], [0,-1,-2,1,2]).toarray()
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# ... SET UP AUXILIARY MATRIX
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M = np.block([
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[ Z0, L0],
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[ L1, Z0]
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])
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# ... SOLVE EIGENVALUE PROBLEM
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e_, v_ = eig(M)
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# -- FILTER FOR EIGENVALUES WITHIN GAP OF CONTINUOUS SPECTRUM
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# ... DETERMINE ONSET OF CONTINUOUS-WAVE BANDS
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Lam_max = kap
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# ... FILTER EIGENVALUES AND EIGENFUNCTIONS
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Lam_ = 1j*e_
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Lam = Lam_[ np.abs(np.imag(Lam_))&lt;kap]
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v = v_[:, np.abs(np.imag(Lam_))&lt;kap]
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# ... SEPARATE THE MODES AND MAKE INDEXING MORE INTUITIVE
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f = np.asarray([v[:xi.size,n] for n in range(Lam.size)])
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g = np.asarray([v[xi.size:,n] for n in range(Lam.size)])
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# ... GET ORDER FOR INCREASING MAGNITUDE OF IMAGINARY PART
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idx_srt = np.flip(np.argsort(np.abs(np.imag(Lam))))
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return kap, Lam[idx_srt], f[idx_srt], g[idx_srt]</code></pre>
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</details>
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<div class="desc"><p>Linearized eigenspectrum for the GNSE.</p>
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<p>Solves the eigenvalue problem for the linear stability matrix of a GNSE
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with generic nonlinear functional and linear differential operator of
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second order in the transverse coordinate [P1998,K1998].</p>
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<h2 id="note">Note</h2>
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<p>Assumes that the soliton solution is given by an even, positive,
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single-humped function U, see [P1998].</p>
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<h2 id="references">References</h2>
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<p>[P1998] D. E. Pelinovsky, Y. S. Kivshar, V. V. Afanasjev, Internal modes of
160+
envelope solitons, Physica D 116 (1) (1998) 121–142,
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<a href="https://doi.org/10.1016/S0167-2789(98)80010-9.">https://doi.org/10.1016/S0167-2789(98)80010-9.</a></p>
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<p>[K1998] Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Internal
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modes of solitary waves, Phys. Rev. Lett. 80 (1998) 5032,
164+
<a href="https://doi.org/10.1103/PhysRevLett.80.5032.">https://doi.org/10.1103/PhysRevLett.80.5032.</a></p>
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<h2 id="parameters">Parameters</h2>
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<dl>
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<dt><strong><code>xi</code></strong> :&ensp;<code>array_like</code></dt>
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<dd>Disrete transverse coordinates.</dd>
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<dt><strong><code>U</code></strong> :&ensp;<code>array_like</code></dt>
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<dd>Solitary wave solution.</dd>
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<dt><strong><code>kap</code></strong> :&ensp;<code>float</code></dt>
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<dd>Solitary wave wavenumber.</dd>
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<dt><strong><code>c2</code></strong> :&ensp;<code>float</code></dt>
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<dd>Constant coefficient of the linear differential operator part</dd>
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<dt><strong><code>F_fun</code></strong> :&ensp;<code>function</code></dt>
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<dd>Nonlinear functional.</dd>
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<dt><strong><code>F1_fun</code></strong> :&ensp;<code>function</code></dt>
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<dd>Derivative of the nonlinear functional.</dd>
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</dl></div>
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</dd>
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<dt id="SWtools_ext_LE.LE_HONSE"><code class="name flex">
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<span>def <span class="ident">LE_HONSE</span></span>(<span>xi, U, kap, coeffs)</span>
183+
</code></dt>
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<dd>
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<details class="source">
186+
<summary>
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<span>Expand source code</span>
188+
</summary>
189+
<pre><code class="python">def LE_HONSE(xi, U, kap, coeffs):
190+
&#34;&#34;&#34;Linearized eigenspectrum for a HONSE.
191+
192+
Solves the eigenvalue problem for the linear stability matrix of a GNSE
193+
with cubic nonlinear functional and linear differential operator including
194+
dispersion of orders two, three and four [T2020,M2024].
195+
196+
Note
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----
198+
The linear stability matrix for this model is generally non-selfadjoint and
199+
does not simply reduce to that of [P1998] (with higher orders of the linear
200+
derivatives included).
201+
202+
References
203+
----------
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[T2020] K. K. K. Tam, T. J. Alexander, A. Blanco-Redondo, C. M. de Sterke,
205+
Generalized dispersion Kerr solitons, Phys. Rev. A 101 (2020) 043822,
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https://doi.org/10.1103/PhysRevA.101.043822.
207+
208+
[M2024] O. Melchert, A. Demircan,
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https://doi.org/10.48550/arXiv.2504.10623.
210+
211+
[P1998] D. E. Pelinovsky, Y. S. Kivshar, V. V. Afanasjev, Internal modes of
212+
envelope solitons, Physica D 116 (1) (1998) 121–142,
213+
https://doi.org/10.1016/S0167-2789(98)80010-9.
214+
215+
Parameters
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----------
217+
xi : array_like
218+
Disrete transverse coordinates.
219+
U : array_like
220+
Solitary wave solution.
221+
kap : float
222+
Solitary wave wavenumber.
223+
coeffs : array_like
224+
Scaled dispersion parameters coeffs = [c0,c1,c2,c3,c4] defining the
225+
propagation constant (c_n = beta_n/n!).
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&#34;&#34;&#34;
227+
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# -- COMPOSE MATRIX DETERMINING EVP FOR THE PERTURBATION MODES
229+
# ... PREPARE COEFFICIENTS FOR FINITE-DIFFERENCE MATRIX
230+
dxi = xi[1]-xi[0]
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c2, c3, c4 = coeffs
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sc2, sc3, sc4 = c2/dxi**2, c3/dxi**3, c4/dxi**4
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I = np.abs(U)**2
234+
235+
# ... SET UP DIAGONALS
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D3m = ( 1j*( 1/8)*sc3 - ( 1/6)*sc4)*np.ones(xi.size-3)
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D2m = ( ( 1/12)*sc2 - 1j*sc3 + 2*sc4)*np.ones(xi.size-2)
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D1m = (-(16/12)*sc2 + 1j*(13/8)*sc3 - (13/2)*sc4)*np.ones(xi.size-1)
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D0 = ( (30/12)*sc2 + (28/3)*sc4)*np.ones(xi.size)
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D1p = (-(16/12)*sc2 - (13/8)*1j*sc3 - (13/2)*sc4)*np.ones(xi.size-1)
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D2p = ( ( 1/12)*sc2 + 1j*sc3 + 2*sc4)*np.ones(xi.size-2)
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D3p = ( -1j*( 1/8)*sc3 - ( 1/6)*sc4)*np.ones(xi.size-3)
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# ... SET UP CORRESPONDING SPARSE FINITE DIFFERENCE MATRICES
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M11 = diags([D0 + (2*np.abs(U)**2-kap), D1p, D2p, D3p, D1m, D2m, D3m], [0,-1,-2,-3,1,2,3]).toarray()
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M12 = diags([U*U],[0]).toarray()
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# ... SET UP AUXILIARY MATRIX
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M = np.block([
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[ M11, M12],
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[-np.conj(M12), -np.conj(M11)]
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])
253+
254+
# ... SOLVE EIGENVALUE PROBLEM
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e_, v_ = eig(M)
256+
257+
# -- FILTER FOR EIGENVALUES WITHIN GAP OF CONTINUOUS SPECTRUM
258+
# ... DETERMINE ONSET OF CONTINUOUS-WAVE BANDS
259+
w = FTFREQ(xi.size, d=xi[1]-xi[0])*2*np.pi
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bw = c2*w**2 + c3*w**3 + c4*w**4
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bw_max = np.max(bw)
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Lam_max = kap - bw_max
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# ... FILTER EIGENVALUES AND EIGENFUNCTIONS
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Lam_ = 1j*e_
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Lam = Lam_[ np.abs(np.imag(Lam_))&lt;Lam_max]
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v = v_[:, np.abs(np.imag(Lam_))&lt;Lam_max]
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# ... SEPARATE THE MODES AND MAKE INDEXING MORE INTUITIVE
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f = np.asarray([v[:xi.size,n] for n in range(Lam.size)])
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g = np.asarray([v[xi.size:,n] for n in range(Lam.size)])
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# ... GET ORDER FOR INCREASING MAGNITUDE OF IMAGINARY PART
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idx_srt = np.flip(np.argsort(np.abs(np.imag(Lam))))
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return Lam_max, Lam[idx_srt], f[idx_srt], g[idx_srt]</code></pre>
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</details>
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<div class="desc"><p>Linearized eigenspectrum for a HONSE.</p>
276+
<p>Solves the eigenvalue problem for the linear stability matrix of a GNSE
277+
with cubic nonlinear functional and linear differential operator including
278+
dispersion of orders two, three and four [T2020,M2024].</p>
279+
<h2 id="note">Note</h2>
280+
<p>The linear stability matrix for this model is generally non-selfadjoint and
281+
does not simply reduce to that of [P1998] (with higher orders of the linear
282+
derivatives included).</p>
283+
<h2 id="references">References</h2>
284+
<p>[T2020] K. K. K. Tam, T. J. Alexander, A. Blanco-Redondo, C. M. de Sterke,
285+
Generalized dispersion Kerr solitons, Phys. Rev. A 101 (2020) 043822,
286+
<a href="https://doi.org/10.1103/PhysRevA.101.043822.">https://doi.org/10.1103/PhysRevA.101.043822.</a></p>
287+
<p>[M2024] O. Melchert, A. Demircan,
288+
<a href="https://doi.org/10.48550/arXiv.2504.10623.">https://doi.org/10.48550/arXiv.2504.10623.</a></p>
289+
<p>[P1998] D. E. Pelinovsky, Y. S. Kivshar, V. V. Afanasjev, Internal modes of
290+
envelope solitons, Physica D 116 (1) (1998) 121–142,
291+
<a href="https://doi.org/10.1016/S0167-2789(98)80010-9.">https://doi.org/10.1016/S0167-2789(98)80010-9.</a></p>
292+
<h2 id="parameters">Parameters</h2>
293+
<dl>
294+
<dt><strong><code>xi</code></strong> :&ensp;<code>array_like</code></dt>
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<dd>Disrete transverse coordinates.</dd>
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<dt><strong><code>U</code></strong> :&ensp;<code>array_like</code></dt>
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<dd>Solitary wave solution.</dd>
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<dt><strong><code>kap</code></strong> :&ensp;<code>float</code></dt>
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<dd>Solitary wave wavenumber.</dd>
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<dt><strong><code>coeffs</code></strong> :&ensp;<code>array_like</code></dt>
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<dd>Scaled dispersion parameters coeffs = [c0,c1,c2,c3,c4] defining the
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propagation constant (c_n = beta_n/n!).</dd>
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</dl></div>
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</dd>
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<dt id="SWtools_ext_LE.LE_dump"><code class="name flex">
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<span>def <span class="ident">LE_dump</span></span>(<span>Lam_max, Lam, f, g)</span>
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</code></dt>
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<dd>
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<details class="source">
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<summary>
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<span>Expand source code</span>
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</summary>
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<pre><code class="python">def LE_dump(Lam_max, Lam, f, g):
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&#34;&#34;&#34;List discrete eigenvalues.
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Parameters
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----------
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Lam_max : float
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Continuum edge.
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Lam : array_like
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Complex-valued eigenvalues.
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f, g : array_like
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Complex-valued eigenfunctions.
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&#34;&#34;&#34;
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print(&#34;# -- DISCRETE EIGENSPECTRUM (SORTED)&#34;)
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print(&#34;# -- CONTINUUM EDGE:&#34;, Lam_max)
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for n in range(Lam.size):
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LamR, LamI = np.real(Lam[n]), np.imag(Lam[n])
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print(f&#34;n = {n} Re[Lam_n] = {LamR:+6.5F} Im[Lam_n] = {LamI:+6.5F}&#34;)</code></pre>
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</details>
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<div class="desc"><p>List discrete eigenvalues.</p>
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<h2 id="parameters">Parameters</h2>
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<dl>
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<dt><strong><code>Lam_max</code></strong> :&ensp;<code>float</code></dt>
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<dd>Continuum edge.</dd>
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<dt><strong><code>Lam</code></strong> :&ensp;<code>array_like</code></dt>
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<dd>Complex-valued eigenvalues.</dd>
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<dt><strong><code>f</code></strong>, <strong><code>g</code></strong> :&ensp;<code>array_like</code></dt>
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<dd>Complex-valued eigenfunctions.</dd>
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</dl></div>
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</dd>
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</dl>
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</section>
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<section>
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</section>
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</article>
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<nav id="sidebar">
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<div class="toc">
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<ul>
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<li><a href="#note">Note</a></li>
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</ul>
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</div>
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<ul id="index">
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<li><h3><a href="#header-functions">Functions</a></h3>
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<ul class="">
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<li><code><a title="SWtools_ext_LE.LE_GNSE" href="#SWtools_ext_LE.LE_GNSE">LE_GNSE</a></code></li>
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<li><code><a title="SWtools_ext_LE.LE_HONSE" href="#SWtools_ext_LE.LE_HONSE">LE_HONSE</a></code></li>
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<li><code><a title="SWtools_ext_LE.LE_dump" href="#SWtools_ext_LE.LE_dump">LE_dump</a></code></li>
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</ul>
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</li>
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</ul>
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</nav>
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</main>
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<footer id="footer">
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<p>Generated by <a href="https://pdoc3.github.io/pdoc" title="pdoc: Python API documentation generator"><cite>pdoc</cite> 0.11.6</a>.</p>
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