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m →Direct scattering transform: Corrected eigenvalue equation. |
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===Direct scattering transform===
The solutions to this differential equation
: <math display="inline">L(\psi)=-\psi_{xx}+u(x,0)\psi= \lambda \psi</math>
may include <em>scattering solutions</em> with a continuous range of eigenvalues (<em>continuous spectrum</em>) and <em>[[bound state|bound-state]]</em> solutions with discrete eigenvalues (<em>discrete spectrum</em>). The scattering data includes transmission coefficients <math display="inline">T(k,0)</math>, left reflection coefficient <math display="inline">R_{L}(k,0)</math>, right reflection coefficient <math display="inline">R_{R}(k,0)</math>, discrete eigenvalues <math display="inline">-\kappa^{2}_{1}, \ldots,-\kappa^{2}_{N}</math>, and left and right bound-state <em>normalization (norming) constants</em>.{{sfn|Aktosun|2009}}{{rp|4960}}
: <math>c(0)_{Lj}=\left( \int^{\infty}_{-\infty} \ \psi^{2}_{L}(ik_{j},x,0) \ dx \right)^{-1/2} \ j=1, \dots, N </math>
: <math> c(0)_{Rj}=\left( \int^{\infty}_{-\infty} \ \psi^{2}_{R}(ik_{j},x,0) \ dx \right)^{-1/2} \ j=1, \dots, N </math>
===Scattering data time evolution===
The spatially asymptotic left <math display="inline">\psi_{L}(k,x,t)</math> and right <math display="inline">\psi_{R}(k,x,t)</math> [[Jost function]]s simplify this step.{{sfn|Aktosun|2009}}{{rp|4965-4966}}
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