락스 쌍 (Lax pair)

수학노트
http://bomber0.myid.net/ (토론)님의 2012년 6월 19일 (화) 11:55 판
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개요
  • 해밀턴 역학에서 보존량을 얻기 위해 유용한 방법
  • spectral parameter
  • \(H(q,p)\)
    • the coordinates \(q=(q_1,\cdots,q_N)\)
    • the momenta \(p=(p_1,\cdots,p_N)\)
    • \(\{q_i,p_i\}=\delta_{ij}\)
    • the equation of motion
      \(\dot{q}_i=\{q_i,H\}=\partial H/\partial p_i\)
      \(\dot{p}_i=\{q_i,H\}-\partial H/\partial q_i\)
  • For an integrable system, sometimes there exists a Lax pair
  • a pair of \(N\times N\) matrices L(x,p) and M(x,p) such that the Lax equation \(\dot{L}=\{L,M\}\) is equivalent to the Hamiltonian equations of the mechanical system
    \(\dot{q}_i=\{q_i,H\}=\partial H/\partial p_i\)
    \(\dot{p}_i=\{q_i,H\}-\partial H/\partial q_i\)
  • integrals of motion can be derived from the trace of powers of L

 

 

 

Lax pairs with spectral parameters
  • spectral curve
    \(\{(k,z)\in\mathbb{C}\times\mathbb{C}:\det(kI-L(z))=0\}\)
  • integrals of motion
    \(\operatorname{tr} L(z)=\sum_{n}L_{n}z^{n} \)
  • for examples, look at Introduction to classical integrable systems, chapter 3 http://goo.gl/LaawC

 

 

 

isospectral deformation
  • L is an isospectral deformation of L(0) if  L(t) has the same eigenvalues for all t
  • \(L(t)v(t)=\lambda v(t)\)
  • Record their derivative by a matrix
    v'(t)=B(t)v(t)
  • Differentiate \(L(t)v(t)=\lambda v(t)\)
    L'(t)v(t)+L(t)v'(t)=\lambda v'(t)
    L'(t)v'(t)=[B(t),L(t)]v(t)
    L'(t)=[B[t],L(t)
  • So B(t) and L(t) are a Lax pair

 

 

examples
  • \(u_t=\frac{1}{4}u_{xxx}+\frac{3}{2}uu_x\)
  • Sturm-Liouville operator
    • \(L=\partial^2+u\)
  • \(B=\partial_{x}^3+\frac{3}{2}u\partial_{x}+\frac{3}{4}u_{x}\)
  • equation
    \(u_{t}=[B,L]=\frac{1}{4}u_{xxx}+\frac{3}{2}uu_x\)

 

 

 

examples : KdV equation
  • \(u_t=6uu_x-u_{xxx}\)
  • Sturm-Liouville operator
    • \(L=-\partial^2+u\)
  • \(A=4\partial^3-3(u\partial +\partial u)\)
  • KdV equation
    • \(\dot{u}=[L,A]\)

 

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