In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ...
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
Mark Jay Ablowitz (born June 5, 1945, New York) is a professor in the department of Applied Mathematics at the University of Colorado at Boulder, Colorado. He was born in New York City.
Education
Ablowitz received his Bachelor of Science degree ...
, David J. Kaup,
Alan C. Newell, and Harvey Segur from their publication in
Studies in Applied Mathematics
The journal ''Studies in Applied Mathematics'' is published by Wiley–Blackwell on behalf of the Massachusetts Institute of Technology.
It features scholarly articles on mathematical applications in allied fields, notably computer science, m ...
: .
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, an ...
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 algebraic ...
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 formall ...
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 (1563–1 ...
References
*
Integrable systems
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