What is it about?

Nuclear fusion is viewed by physicists as the Holy Grail of "clean", renewable energy. The program of controlled thermonuclear fusion since the 1950's has posed many extremely complicated theoretical problems of stability and control of plasma during the thermonuclear reaction. In this stability theory, kinetic phenomena and turbulent processes in a plasma are confined by a magnetic field. One of the most intriguing and promising approaches was the study of plasma stability in toroidal systems—TOKAMAKS - where a powerful magnetic field should confine plasma in the shape of a torus. Soviet physicists B. Kadomtsev and V. Petviashvili have studied ion-sound waves in the theory of collective phenomena in plasma, and obtained in 1970 a famous two-dimensional integrable nonlinear partial differential equation known now as the Kadomtsev-Petviashvili (KP) equation: 3 u_yy = (4u_t - 12u_x u - u_xxx )_x . This equation is the basic actor of our work.

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Why is it important?

The KP equation is a natural generalization of the Korteweg and de Vries (KdV) equation–a one-dimensional model equation or a laboratory for all non-linear phenomena. More precisely the KP equation originates from a study of the evolution model for long ion-acoustic waves of small amplitude propagating in plasmas under the effect of long transverse perturbations. In the absence of transverse dynamics, this problem is described by the KdV equation. The KP equation is widely accepted as a natural extension of the classical KdV equation to two spatial dimensions. Moreover the KP equation admits another extremely important application. Its so-called dispersionless version widely used in non-linear acoustics and in the numerical computations of LASER beams parameters. KP equation, like KdV, is completely integrable: it admits an infinite number of commuting flows and may be considered an infinite-dimensional analogue of a classical Hamiltonian system of integrable differential equations which can be solved explicitly. The so-called KP hierarchy of flows that we obtain is important by its wide spectrum of applications from pure mathematics to numerical models of the weather forecast.

Perspectives

Integrability can be studied in the framework of involved and very important modern algebraic structures: mathematicians and theoretical physicists extend the notion of integrability by considering more and more complicated initial conditions, constraints, and non-trivial admissible base function algebras. Our paper an attempt to go beyond the “usual" classes of functions, extending these classes and considering non-commutative function algebras with various symmetry and commutativity constraints. We use our framework to derive generalized KP equations, for which we can produce a class of solutions that depend smoothly on generalized initial conditions. Such integrable models and solutions are potentially important in modern Quantum Field Theory where they fill a part of its «integrable sector».

Enrique Reyes
Universidad de Santiago de Chile

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This page is a summary of: On (t2, t3)-Zakharov–Shabat equations of generalized Kadomtsev–Petviashvili hierarchies, Journal of Mathematical Physics, September 2022, American Institute of Physics, DOI: 10.1063/5.0093238.
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