Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 32E (Integrable Systems)

Insert the finite exponential expansion into the Gelfand-Levitan-Marchenko equation. The integral contribution isComparing coefficients of $e^{-c_{n}y}$ givesFor this coefficient comparison take the $c_n$ distinct; repeated exponents can first be combined. The linear system supplies the coefficients whenever its matrix is invertible.

In the inverse scattering transform convention $u_t-6u{u}_x+u_{xxx}=0$, the Korteweg-De Vries equation solution is $u=-2dK(x,x)/dx$. The reflectionless scattering data evolve as ${\beta }_n(t)={\beta }_n(0)e^{8c_n^{3}t}$, with the $c_n$ fixed. For one exponential,Differentiating its diagonal givesFor $\beta >0$, this is the smooth solitonIt travels at speed $4c^2$. If $\beta =0$ the solution is zero; if $\beta <0$ the denominator vanishes somewhere, so the formula is not a globally smooth one-soliton. Positivity is the necessary qualification for a regular soliton.