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The [[Lax pair|Lax differential operators]], <math display="inline">L</math> and <math display="inline">M</math>, are linear ordinary differential operators with coefficients that may contain the function <math display="inline">u(x,t)</math> or its derivatives. The [[self-adjoint operator]] <math display="inline">L</math> has a time derivative <math display="inline">L_{t}</math> and generates a <em>eigenvalue (spectral) equation</em> with [[eigenfunction]]s <math display="inline">\psi</math> and time-constant [[eigenvalues and eigenvectors|eigenvalues]] (<em>[[Spectral theory|spectral parameters]]</em>) <math display="inline">\lambda</math>.{{sfn|Aktosun|2009}}{{rp|4963}}{{sfn|Drazin|Johnson|1989}}{{rp|98}}
: <math> L(\psi)=\lambda \psi , \ </math> and <math display="inline"> \ L_{t}(\psi) \overset{def}{=}(L(\psi))_{t}-L(\psi_{t})</math>
The operator <math display="inline">M</math> describes how the eigenfunctions evolve over time, and generates a new eigenfunction <math display="inline">\
: <math>\
The Lax operators combine to form a multiplicative operator, not a differential operator, of the eigenfuctions <math display="inline">\psi</math>.{{sfn|Aktosun|2009}}{{rp|4963}}
: <math>(L_{t}+LM-ML)\psi=0</math>
The Lax operators are chosen to make the multiplicative operator equal to the nonlinear differential equation.{{sfn|Aktosun|2009}}{{rp|4963}}
: <math>L_{t}+LM-ML=u_{t}+N(u)=0</math>
The [[AKNS system|AKNS differential operators]], developed by Ablowitz, Kaup, Newell, and Segur, are an alternative to the Lax differential operators and achieve a similar result.{{sfn|Aktosun|2009}}{{rp|4964}}{{sfn|Ablowitz|Kaup|Newell|Segur|1973}}{{sfn|Ablowitz|Kaup|Newell|Segur|1974}}
===Direct scattering transform===
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