In
mathematics, the AKNS system is an
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
of
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
, introduced by and named after
Mark J. Ablowitz, David J. Kaup,
Alan C. Newell
Alan C. Newell (born 5 November 1941 in Dublin) is an Irish/American mathematician and Regents Professor at the University of Arizona. He was awarded a Guggenheim Fellowship in 1976 and in 2004 the John von Neumann Lecture
The John von Neu ...
, and Harvey Segur from their publication in
Studies in Applied Mathematics: .
Definition
The AKNS system is a pair of two partial
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
s for two
complex-valued function
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
s ''p'' and ''q'' of 2 variables ''t'' and ''x'':
:
:
If ''p'' and ''q'' are
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s this reduces to the
nonlinear Schrödinger equation
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...
.
Huygens' principle applied to the
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally ...
gives rise to the AKNS hierarchy.
[Fabio A. C. C. Chalub and Jorge P. Zubelli,]
Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies
See also
*
Huygens principle
Huygens (also Huijgens, Huigens, Huijgen/Huygen, or Huigen) is a Dutch patronymic surname, meaning "son of Hugo". Most references to "Huygens" are to the polymath Christiaan Huygens. Notable people with the surname include:
* Jan Huygen (1563– ...
References
*
Integrable systems
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