Inverse scattering transform: Difference between revisions

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The Korteweg–de Vries equation is a nonlinear, dispersive, evolution [[partial differential equation]] for a [[function (mathematics)|function]] ''u''; of two [[real number|real]] variables, one space variable ''x'' and one time variable ''t'' :

:<math> \frac{\partialu_t u}{\partial- t6uu_x + u_{xxx}- 6\,= u\,0 </math>

\frac{\partial u}{\partial x}+

\frac{\partial^3 u}{\partial x^3} =0,\,</math>

with <math> \frac{\partialu_t u}{\partial t}</math> and <math> \frac{\partialu_x u}{\partial x}</math> denoting [[partial derivative]]s with respect to ''t'' and ''x'', respectively.

To solve the initial value problem for this equation where <math>u(x,0)</math> is a known function of ''x'', one associates to this equation the Schrödinger eigenvalue equation

:<math> \fracpsi_{\partial^2 \psixx}{\partial x^2}-u(x,t)\psi=\lambda\psi.</math>

where <math>\psi</math> is an unknown function of ''t'' and ''x'' and ''u'' is the solution of the Korteweg–de Vries equation that is unknown except at <math>t=0</math>. The constant <math>\lambda</math> is an eigenvalue.

From the Schrödinger equation we obtain

:<math> u=\frac{1}{\psi} \fracpsi_{\partial^2 \psi}{\partial x^2xx} - \lambda.</math>

Substituting this into the Korteweg–de Vries equation and integrating gives the equation

:<math> \frac{\partialpsi_t \psi}{\partial+ t}+\fracpsi_{\partial^3 \psixxx}{\partial x^3}-3(u-\lambda)

\frac{\partialpsi_x \psi}{\partial= x}=C\psi+D\psi\int \frac{dx1}{\psi^2} dx</math>

where ''C'' and ''D'' are constants.