코테베그-드 브리스 방정식(KdV equation)

수학노트
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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)



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

  • 1844 러셀이 관찰과 실험을 통해 솔리톤을 발견
  • 1895 코테베그와 드 브리스가 1-솔리톤의 해석적 해를 구함
    • 러셀의 발견을 모형화하고 미분방정식을 도입
  • 1965 자부스키와 크루스칼의 수치해석적 연구
    • 두 솔리톤(1-soliton)의 상호작용
    • 크기가 다른 두 솔리톤이 깔끔하게 상호작용한다는 사실을 발견
  • John Scott Russell and the solitary wave
  • 수학사 연표



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말뭉치

  1. The derivation of the KdV is given in KdV Equation Derivation.[1]
  2. The KdV equation posesses travelling wave solutions.[1]
  3. We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions.[2]
  4. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg-de Vries equation as the number of solitons tends to infinity.[2]
  5. Zabusky and Kruskal (1965) subsequently studied the continuum limit of the Fermi-Pasta-Ulam Experiment and, surprisingly, obtained the Korteweg-de Vries equation.[3]
  6. An important step in the solution of the KdV equation was provided by Gardner et al.[3]
  7. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation.[4]
  8. Our main interest was to analyse which explicit scheme among the four performs well when implemented to the KdV equation to produce the best soliton solution.[5]
  9. Accuracy, consistency and Fourier stability in regard to the four explicit schemes for the Korteweg-de Vries equation are discussed.[5]
  10. The KdV equation has infinitely many integrals of motion (Miura, Gardner & Kruskal 1968), which do not change with time.[6]
  11. They also made the connection to earlier numerical experiments by Fermi, Pasta, Ulam, and Tsingou by showing that the KdV equation was the continuum limit of the FPUT system.[6]
  12. Kruskal, R.M. Miura, "Method for solving the Korteweg–de Vries equation" Phys.[7]
  13. Moreover, it is shown that the Modified Korteweg-de Vries equation has new families of the solution.[8]
  14. The Korteweg–de Vries Equation, Posed in a Quarter-Plane.[9]
  15. (2018) Uniform null controllability of a linear KdV equation using two controls.[10]
  16. Well-posedness of a nonlinear boundary value problem for the Korteweg–de Vries equation on a bounded domain.[10]
  17. An example of non-decreasing solution for the KdV equation posed on a bounded interval.[10]
  18. Nonhomogeneous boundary value problems for the Korteweg-de Vries equation on a bounded domain.[10]
  19. In this note, we shall summarize some main features of the KdV equation.[11]
  20. The assumptions of this paper imply that the usual spectral and nonlinearity assumptions for the derivation of the KdV equation are met.[12]
  21. Moreover, the mechanism for the emergence of the KdV equation is simplified, reducing it to a single condition.[12]
  22. The KdV equation was first derived in the context of water waves in shallow water.[12]
  23. The approach here is to obtain the KdV equation by modulating the basic state.[12]
  24. Carlos E. Kenig, Gustavo Ponce, Luis Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun.[13]
  25. In this paper, HPTM is applied to find the solution of fifth order KdV equation.[14]
  26. Note that the below figure show that the coupled solution of KDV equation is not only the function of time and space but also an increasing function of the fractional order derivative, which are and .[15]

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