Operator Algebras, Non-Linear Equations and Darboux-Like Transforms

Abstract

Nonlinear evolutionary equations such as

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWG1bWaaSbaaSqaaiaadshaaeqaaOGa % eyypa0JaaGOnaiaadwhacaWG1bWaaSbaaSqaaiaadIhaaeqaaOGaey % OeI0IaamyDamaaBaaaleaacaWG4bGaamiEaiaadIhaaeqaaOGaaiil % aaaa!4BFA! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${u_t} = 6u{u_x} - {u_{xxx}},$$

(1)

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWG1bWaaSbaaSqaaiaadshaaeqaaOGa % eyypa0JaamyDamaaBaaaleaacaWG4bGaamiEaaqabaGccqGHXcqSda % abdaqaaiaadwhaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGc % caWG1bGaaiilaaaa!4E20! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${u_t} = {u_{xx}} \pm {\left| u \right|^2}u,$$ EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWG1bWaaSbaaSqaaiaadshacaWG0baa % beaakiabgkHiTiaadwhadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaey % ypa0Jaci4CaiaacMgacaGGUbGaamyDaaaa!4B31! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${u_{tt}} - {u_{xx}} = \sin u$$ EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacqGHXcqScaaIZaGaamyDamaaBaaaleaa % caWG5bGaamyEaaqabaGccqGHRaWkdaWcaaqaaiabgkGi2cqaaiabgk % Gi2kaadIhaaaWaaiWaaeaacaWG1bWaaSbaaSqaaiaadshaaeqaaOGa % ey4kaSIaamyDamaaBaaaleaacaWG4bGaamiEaiaadIhaaeqaaOGaey % 4kaSIaaGOnaiaadwhadaqhaaWcbaGaamiEaaqaaiaaikdaaaaakiaa % wUhacaGL9baacqGH9aqpcaaIWaaaaa!5961! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ \pm 3{u_{yy}} + \frac{\partial }{{\partial x}}\left\{ {{u_t} + {u_{xxx}} + 6u_x^2} \right\} = 0$$ appear in problems of different nature and play an important role in various fields of Physics and Mathematics. But it was only in 1967 when Gardner, Green, Kruskal and Miura [1] discovered the so-called inverse scattering method which at first allowed to solve the Cauchy problem for the KdV equation (1); this was followed by the Lax, Zakharov and Shabat works which enabled to solve many other equations. There exist now different modifications of this method which considerably extend the list of integrable equations as well as the classes of functions describing the solutions. This lecture is devoted to one of them.