Please guide me to a source which builds the idea from the basics.
As for some brief introductions to the inverse scattering transform per se, please see the links below.
http://en.wikipedia.org/wiki/Inverse_scattering_transform
http://math.univ-lille1.fr/~cempi/activites_scientifiques/FR/groupe_travail/files/El_IST_Lille2014.pdf
http://www.math.uwaterloo.ca/~karigian/papers/ist.pdf
See the good book on IST, Solitons and the Inverse Scattering Transform by Ablowitz
DOI: http://dx.doi.org/10.1137/1.9781611970883.fm
You may also consider the paper by Ablowitz and Clarkson:
Ablowitz, M. J. and P. A. Clarkson (1991), Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society lecture Notes 149, Cambridge University press.
When you have developed a richer understanding, may I encourage you share it by contributing to http://www.scholarpedia.org/article/Inverse_scattering_transform_method?
The purpose of the inverse scattering transform is to solve nonlinear evolution equations. Many important equations are integrable by this method: Korteweg-de Vries equation, nonlinear Schrödinger equation, sine-Gordon equation, self-dual Yang-Mills etc.
The original paper from Lax gives much insight:
"Integrals of Nonlinear Equations of Evolution and Solitary Waves" Comm. Pure Appl. Math 21(1968)467
Let me deposit my five cents in. This transformation, from the entire vast class of nonlinear dynamic systems, separates only those systems that are truly linear, and only hide behind the attributes of nonlinear dynamic systems. Who can say that the Riccati equation is non-linear if there is a change of variables that reduces this equation to a linear one? But this change of variables itself is not easy - one has to pay by raising the order of the linear differential equation, which remains in the final as a penalty. Let's reverse the current situation in reverse order: there is a linear differential equation, albeit with variable coefficients, which is not very comfortable, but still - the superposition principle is valid for it, and it is always possible to find quadrature, that is, a solution in integral form, just do not be lazy and try hard. As a reward, we obtain the general solution of a nonlinear equation.
The natural question is this: is it possible, always by taking up all the linear systems, to build an isomorphism into the total class of nonlinear dynamical systems? Then it would turn out that there are truly no non-linear dynamic systems in the world, and nature just only confuses playing with us to make it harder for people to comprehend her. And therefore, maybe there are no algebraic curves in geometry, except for an ellipse, parabola, hyperbola and a straight line, since all other curves are the result of applying some isomorphism.
Fortunately or unfortunately, the set of such systems, reducible to linear, almost empty. This is a set of measure zero in the space of all dynamic systems. Although, it is likely that, say, stability in the microworld is due to the fact that the microworld falls into the class of dynamic systems of measure-zero. There is no place for free will, not for the chaos that frees the world from boredom and predictability.
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