|
|
| Line 1: |
Line 1: |
| In [[mathematics]], in the theory of [[integrable systems]], a '''Lax pair''' is a pair of time-dependent matrices or [[operator (mathematics)|operator]]s that satisfy a corresponding [[differential equation]], called the ''Lax equation''. Lax pairs were introduced by [[Peter Lax]] to discuss [[soliton]]s in [[continuous media]]. The [[inverse scattering transform]] makes use of the Lax equations to solve such systems.
| | Friends contact him Royal. Her spouse and her selected to reside in Delaware but she needs to move simply because of her family members. The preferred hobby for him and his children is to generate and now he is trying to make money with it. His day occupation is a cashier and his salary has been really satisfying.<br><br>My webpage - extended auto warranty ([http://Mirim.ir/index.php?do=/profile-202/info/ click the following webpage]) |
| | |
| ==Definition==
| |
| A Lax pair is a pair of matrices or operators <math>L(t), P(t)</math> dependent on time and acting on a fixed [[Hilbert space]], and satisfying '''Lax's equation''':
| |
| | |
| :<math>\frac{dL}{dt}=[P,L]</math>
| |
| | |
| where <math>[P,L]=PL-LP</math> is the [[commutator]].
| |
| Often, as in the example below, <math>P</math> depends on <math>L</math> in a prescribed way, so this is a nonlinear equation for <math>L</math> as a function of <math>t</math>.
| |
| | |
| ==Isospectral property==
| |
| It can then be shown that the [[eigenvalue]]s and more generally the [[Operator spectrum|spectrum]] of ''L'' are independent of ''t''. The matrices/operators ''L'' are said to be ''[[isospectral]]'' as <math>t</math> varies.
| |
| | |
| The core observation is that the matrices <math>L(t)</math> are all similar by virtue of
| |
| | |
| :<math>L(t)=U(t,s) L(s) U(t,s)^{-1}</math>
| |
| | |
| where <math>U(t,s)</math> is the solution of the [[Cauchy problem]]
| |
| | |
| :<math> \frac{d}{dt} U(t,s) = P(t) U(t,s), \qquad U(s,s) = I,</math>
| |
| | |
| where ''I'' denotes the identity matrix. Note that if ''L(t)'' is [[self-adjoint]] and ''P(t)'' is [[skew-adjoint]], then ''U(t,s)'' will be [[unitary operator|unitary]].
| |
| | |
| In other words, to solve the eigenvalue problem ''Lψ = λψ'' at time ''t'', it is possible to solve the same problem at time 0 where L is generally known better, and to propagate the solution with the following formulas:
| |
| :<math>\lambda(t)=\lambda(0)</math> (no change in spectrum)
| |
| :<math>\frac{\partial \psi}{\partial t}=P \psi.</math>
| |
| | |
| === Link with the inverse scattering method ===
| |
| The above property is the basis for the inverse scattering method. In this method, ''L'' and ''P'' act on a [[functional space]] (thus ''ψ = ψ(t,x)''), and depend on an unknown function ''u(t,x)'' which is to be determined. It is generally assumed that ''u(0,x)'' is known, and that ''P'' does not depend on ''u'' in the scattering region where <math>\Vert x \Vert\to \infty</math>.
| |
| The method then takes the following form:
| |
| # Compute the spectrum of <math>L(0)</math>, giving <math>\lambda</math> and <math>\psi(0,x)</math>,
| |
| # In the scattering region where <math>P</math> is known, propagate <math>\psi</math> in time by using <math>\frac{\partial \psi}{\partial t}(t,x)=P \psi(t,x)</math> with initial condition <math>\psi(0,x)</math>,
| |
| # Knowing <math>\psi</math> in the scattering region, compute <math>L(t)</math> and/or <math>u(t,x)</math>.
| |
| | |
| ==Example==
| |
| The [[Korteweg–de Vries equation]] is
| |
| :<math>u_t=6uu_x-u_{xxx}.\,</math> | |
| It can be reformulated as the Lax equation
| |
| :<math>L_t=[P,L]\,</math>
| |
| with
| |
| :<math>L=-\partial_{x}^2+u\,</math> (a [[Sturm–Liouville operator]])
| |
| :<math>P= -4\partial_{x}^3+3(u\partial_{x}+\partial_{x} u)\,</math>
| |
| where all derivatives act on all objects to the right. This accounts for the infinite number of first integrals of the KdV equation.
| |
| | |
| ==Equations with a Lax pair==
| |
| Further examples of systems of equations that can be formulated as a Lax pair include:
| |
| | |
| * [[Benjamin–Ono equation]]
| |
| * One dimensional cubic [[non-linear Schrödinger equation]]
| |
| * [[Davey-Stewartson system]]
| |
| * [[Kadomtsev–Petviashvili equation]]
| |
| * [[Korteweg–de Vries equation]]
| |
| * [[KdV hierarchy]]
| |
| * [[Modified Korteweg-de Vries equation]]
| |
| * [[Sine-Gordon equation]]
| |
| * [[Toda lattice]]
| |
| | |
| ==References==
| |
| * {{citation|first=P.|last= Lax|title=Integrals of nonlinear equations of evolution and solitary waves|journal=Comm. Pure Applied Math.|volume=21|year=1968|pages= 467–490|doi=10.1002/cpa.3160210503|issue=5 }} [http://archive.org/details/integralsofnonli00laxp archive]
| |
| * P. Lax and R.S. Phillips, ''Scattering Theory for Automorphic Functions'', (1976) Princeton University Press.
| |
| | |
| [[Category:Differential equations]]
| |
| [[Category:Automorphic forms]]
| |
| [[Category:Spectral theory]]
| |
| [[Category:Exactly solvable models]]
| |
Friends contact him Royal. Her spouse and her selected to reside in Delaware but she needs to move simply because of her family members. The preferred hobby for him and his children is to generate and now he is trying to make money with it. His day occupation is a cashier and his salary has been really satisfying.
My webpage - extended auto warranty (click the following webpage)