코테베그-드 브리스 방정식(KdV equation)

수학노트
Pythagoras0 (토론 | 기여)님의 2012년 11월 1일 (목) 04:30 판 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로)
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개요
  • any localized nonlinear wave which interacts with another (arbitrary) local disturbance and always regains asymptotically its exact initial shape and velocity (allowing for a possible phase shift)
  • Solitons were discovered experimentally (Russell 1844)
  • analytically (Korteweg & de Vries, 1895)
    • modelling of Russell's discovery
    • 1-soliton solution
  • numerically (Zabusky & Kruskal 1965).
    • interaction of two 1-soliton solutions
    • they discovered that solitons of different sizes interact cleanly

 

 

==러셀(John Scott Russell)의 관찰 

  • Using a wave tank, he demonstrated four facts
    • First, solitary waves have a hyperbolic secant shape.
    • Second, a sufficiently large initial mass of water produces two or more independent solitary waves.
    • Third, solitary waves cross each other “without change of any kind.”
    • Finally, a wave of height h and traveling in a channel of depth d has a velocity given by the expression (where g is the acceleration of gravity), implying that a large amplitude solitary wave travels faster than one of low amplitude.

 

 

 

==코테베그-드 브리스 방정식 (KdV equation)

  • \(u_{xxx}=u_t+6uu_x\)
  • 1-soliton 해의 유도

\(u(x,t)=f(x-ct)\)로 두자.

\(f'''= 6ff'-cf'\)

\(f''=3f^2-cf+b\)

\(f''f'=(3f^2-cf+b)f'\)

\(\frac{1}{2}(f')^2=f^3-\frac{c}{2}f^2+bf+a\)

 

 

==역사

 

 

==메모

 

 

==관련된 항목들

 

 

수학용어번역

 

 

==사전 형태의 자료

 

 

==리뷰

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