Inverse scattering transform: Difference between revisions

Revision as of 01:52, 31 March 2024

Revision as of 00:56, 4 April 2024

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\end{align}</math>

The <em>dependency constants</em> <math display="inline"> \gamma_{j}(t)</math> relate the right and left Jost functions and right and left normalization constants.{{sfn|Aktosun|2009}}{{rp|4965-4966}}

:<math>\gamma_{j}(t)=\frac{f_\psi_{L}(x,ki\kappa_{j},t)}{f_\psi_{R}(x,ki\kappa_{j},t)}=(-1)^{N-j} \frac{c_{Rj}(t)}{c_{Lj}(t)}</math>

The Lax <math display="inline">M</math> differential operator generates an eigenfunction which can be expressed as a time-dependent linear combination of other eigenfunctions.{{sfn|Aktosun|2009}}{{rp|4967}}

:<math>\partial_{t}\psi_{L}(k,x,t)-M\psi_{L}(x,k,t)=

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c_{Lj}(t)&=c_{Lj}(0)e^{+4\kappa^{3}_{j}t}, \ j=1, \ldots, N \\

c_{Rj}(t)&=c_{Rj}(0)e^{-4\kappa^{3}_{j}t}, \ j=1, \ldots, N \end{align}</math>

===Inverse scattering transform===

The <em>Marchenko kernel</em> is <math display="inline">F(x,t)</math>.{{sfn|Aktosun|2009}}{{rp|4968-4969}}