락스 쌍 (Lax pair)

이 항목의 스프링노트 원문주소

• 락스 쌍 (Lax pair)
  

 

 

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
  
  • \(\dot{u}=[L,A]\)

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
•  

 

 

메모

• Does the existence of a Lax pair imply integrability?
  
• http://iopscience.iop.org/0266-5611/25/12/123007
  
• Curves and Lax pairs -many examples
  

 

 

 

 

물리학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
• 발음사전 http://www.forvo.com/search/
• 한국물리학회 물리용어
  
  • http://www.kps.or.kr/home/kor/morgue/dic/default.asp?globalmenu=6&localmenu=2
  • http://www.kps.or.kr/home/kor/morgue/dic/word_list.asp?globalmenu=6&localmenu=2&lang=english
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

 

 

사전 형태의 자료

•  
  
• http://en.wikipedia.org/wiki/Lax_pair
• http://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian)
• http://en.wikipedia.org/wiki/Poisson_bracket

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/

 

 

•  
  

 

 

관련논문

• How to find the Lax pair from the Yang-Baxter equation M. Q. Zhang, 1991
  
• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

 

 

관련도서

• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/mainSearch.do?query=
  • http://book.daum.net/search/mainSearch.do?query=

 

 

블로그

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