"코테베그-드 브리스 방정식(KdV equation)"의 두 판 사이의 차이

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==개요==
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* 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)
  
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==러셀(John Scott Russell)의 관찰 ==
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*  Using a wave tank, he demonstrated four facts
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** First, solitary waves have a hyperbolic secant shape.
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** Second, a sufficiently large initial mass of water produces two or more independent solitary waves.
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** Third, solitary waves cross each other “without change of any kind.”
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** 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.
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==코테베그-드 브리스 방정식 (KdV equation)==
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* <math>u_{xxx}=u_t+6uu_x</math>
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* 1-soliton 해의 유도
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<math>u(x,t)=f(x-ct)</math>로 두자.
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<math>f'''= 6ff'-cf'</math>
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<math>f''=3f^2-cf+b</math>
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<math>f''f'=(3f^2-cf+b)f'</math>
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<math>\frac{1}{2}(f')^2=f^3-\frac{c}{2}f^2+bf+a</math>
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==역사==
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* 1844 러셀이 관찰과 실험을 통해 솔리톤을 발견
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* 1895 코테베그와 드 브리스가 1-솔리톤의 해석적 해를 구함
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** 러셀의 발견을 모형화하고 미분방정식을 도입
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* 1965 자부스키와 크루스칼의 수치해석적 연구
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** 두 솔리톤(1-soliton)의 상호작용
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** 크기가 다른 두 솔리톤이 깔끔하게 상호작용한다는 사실을 발견
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* [http://www.ma.hw.ac.uk/%7Echris/scott_russell.html John Scott Russell and the solitary wave]
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* [[수학사 연표]]
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==메모==
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* Bihlo, Alexander, Xavier Coiteux-Roy, and Pavel Winternitz. “The Korteweg-de Vries Equation and Its Symmetry-Preserving Discretization.” arXiv:1409.4340 [math-Ph], September 15, 2014. http://arxiv.org/abs/1409.4340.
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* http://docs.google.com/viewer?a=v&q=cache:dWzyEHjy6JsJ:kft.umcs.lublin.pl/kmur/download/prezentacje/solitons_my.ppt+soliton+ppt&hl=ko&gl=us&pid=bl&srcid=ADGEESi5cLc2o4aGrXBSQM9i6u_2MalwSshBjfJzoGv4FsWRYcdUPcXNvQhwXLG6RpQsnwlT0f5-UGFkKVJr14cvsGjY2zDOhqLc1bwORnRHVYCsbv08l5dgO9xFhgNO8D1Vg29R4SAJ&sig=AHIEtbRDvlbVm-kiG23Az3C2olliRZdB8Q
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* [http://people.seas.harvard.edu/%7Ejones/solitons/pdf/025.pdf http://people.seas.harvard.edu/~jones/solitons/pdf/025.pdf]
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* http%3A%2F%2Fkft.umcs.lublin.pl%2Fkmur%2Fdownload%2Fprezentacje%2Fsolitons_my.ppt
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==관련된 항목들==
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* [[파동 방정식|파동방정식]]
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* [[사인-고든 방정식]]
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==사전 형태의 자료==
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* [http://ko.wikipedia.org/wiki/%EC%86%94%EB%A6%AC%ED%86%A4 http://ko.wikipedia.org/wiki/솔리톤]
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* http://en.wikipedia.org/wiki/John_Scott_Russell
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==리뷰, 에세이, 강의노트==
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* Kaya, Doğan. “Partial Differential Equations That Lead to Solitons.” In Encyclopedia of Complexity and Systems Science, edited by Robert A. Meyers Ph. D, 6453–59. Springer New York, 2009. http://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_380.
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* [http://kasmana.people.cofc.edu/SOLITONPICS/index.html An Introduction to Solitons] ,Alex Kasman
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== 노트 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q601796 Q601796]
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===말뭉치===
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# The derivation of the KdV is given in KdV Equation Derivation.<ref name="ref_f38025f4">[https://wikiwaves.org/Introduction_to_KdV Introduction to KdV]</ref>
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# The KdV equation posesses travelling wave solutions.<ref name="ref_f38025f4" />
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# We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions.<ref name="ref_bc1e2f89">[https://link.springer.com/10.1134/S0040577920030058 Primitive solutions of the Korteweg–de Vries equation]</ref>
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# These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg-de Vries equation as the number of solitons tends to infinity.<ref name="ref_bc1e2f89" />
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# Zabusky and Kruskal (1965) subsequently studied the continuum limit of the Fermi-Pasta-Ulam Experiment and, surprisingly, obtained the Korteweg-de Vries equation.<ref name="ref_58abd854">[https://mathworld.wolfram.com/Korteweg-deVriesEquation.html Korteweg-de Vries Equation -- from Wolfram MathWorld]</ref>
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# An important step in the solution of the KdV equation was provided by Gardner et al.<ref name="ref_58abd854" />
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# 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.<ref name="ref_69d70c80">[https://projecteuclid.org/euclid.die/1356039428 Faminskii : Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation]</ref>
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# 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.<ref name="ref_20dbd443">[http://www.ccsenet.org/journal/index.php/mas/article/view/46132 Applying Explicit Schemes to the Korteweg-de Vries Equation]</ref>
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# Accuracy, consistency and Fourier stability in regard to the four explicit schemes for the Korteweg-de Vries equation are discussed.<ref name="ref_20dbd443" />
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# The KdV equation has infinitely many integrals of motion (Miura, Gardner & Kruskal 1968), which do not change with time.<ref name="ref_b6b49cfe">[https://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation Korteweg–de Vries equation]</ref>
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# 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.<ref name="ref_b6b49cfe" />
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# Kruskal, R.M. Miura, "Method for solving the Korteweg–de Vries equation" Phys.<ref name="ref_597e3e33">[https://encyclopediaofmath.org/wiki/Korteweg%E2%80%93de_Vries_equation Korteweg-de Vries equation]</ref>
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# Moreover, it is shown that the Modified Korteweg-de Vries equation has new families of the solution.<ref name="ref_d20a5b6d">[https://journals.jps.jp/doi/10.1143/JPSJ.34.1289 The Modified Korteweg-de Vries Equation]</ref>
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# The Korteweg–de Vries Equation, Posed in a Quarter-Plane.<ref name="ref_9b52147b">[https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/shallowwater-waves-the-kortewegdevries-equation-and-solitons/618C02D914F1E42FBD213FB3F93D6E3A Shallow-water waves, the Korteweg-deVries equation and solitons]</ref>
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# (2018) Uniform null controllability of a linear KdV equation using two controls.<ref name="ref_17953313">[https://epubs.siam.org/doi/abs/10.1137/S0363012997327501 Exact Boundary Controllability of the Korteweg--de Vries Equation]</ref>
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# Well-posedness of a nonlinear boundary value problem for the Korteweg–de Vries equation on a bounded domain.<ref name="ref_17953313" />
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# An example of non-decreasing solution for the KdV equation posed on a bounded interval.<ref name="ref_17953313" />
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# Nonhomogeneous boundary value problems for the Korteweg-de Vries equation on a bounded domain.<ref name="ref_17953313" />
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# In this note, we shall summarize some main features of the KdV equation.<ref name="ref_958a2a23">[https://www.researchgate.net/publication/277156623_A_Summary_of_the_Korteweg-de_Vries_Equation (PDF) A Summary of the Korteweg-de Vries Equation]</ref>
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# The assumptions of this paper imply that the usual spectral and nonlinearity assumptions for the derivation of the KdV equation are met.<ref name="ref_efb1c474">[https://royalsocietypublishing.org/doi/10.1098/rspa.2012.0707 A universal form for the emergence of the Korteweg–de Vries equation]</ref>
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# Moreover, the mechanism for the emergence of the KdV equation is simplified, reducing it to a single condition.<ref name="ref_efb1c474" />
 +
# The KdV equation was first derived in the context of water waves in shallow water.<ref name="ref_efb1c474" />
 +
# The approach here is to obtain the KdV equation by modulating the basic state.<ref name="ref_efb1c474" />
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# Carlos E. Kenig, Gustavo Ponce, Luis Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun.<ref name="ref_df349c67">[http://www.numdam.org/item/AIHPC_2015__32_5_1071_0/ The Korteweg–de Vries equation at <math> {H}^{-1}</math> regularity]</ref>
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# In this paper, HPTM is applied to find the solution of fifth order KdV equation.<ref name="ref_8d036782">[https://www.degruyter.com/view/journals/nleng/6/2/article-p89.xml Solution of Fifth-order Korteweg and de Vries Equation by Homotopy perturbation Transform Method using He’s Polynomial]</ref>
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# 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 .<ref name="ref_9e9ae0a7">[https://www.hindawi.com/journals/aaa/2013/947986/ The Time-Fractional Coupled-Korteweg-de-Vries Equations]</ref>
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===소스===
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<references />
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[[분류:적분가능모형]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q601796 Q601796]
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===Spacy 패턴 목록===
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* [{'LOWER': 'korteweg'}, {'OP': '*'}, {'LOWER': 'de'}, {'LOWER': 'vries'}, {'LEMMA': 'equation'}]
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* [{'LOWER': 'kdv'}, {'LEMMA': 'equation'}]

2021년 2월 17일 (수) 05:02 기준 최신판

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



메모



관련된 항목들


사전 형태의 자료



리뷰, 에세이, 강의노트


노트

위키데이터

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]

소스

  1. ↑ 1.0 1.1 Introduction to KdV
  2. ↑ 2.0 2.1 Primitive solutions of the Korteweg–de Vries equation
  3. ↑ 3.0 3.1 Korteweg-de Vries Equation -- from Wolfram MathWorld
  4. ↑ Faminskii : Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation
  5. ↑ 5.0 5.1 Applying Explicit Schemes to the Korteweg-de Vries Equation
  6. ↑ 6.0 6.1 Korteweg–de Vries equation
  7. ↑ Korteweg-de Vries equation
  8. ↑ The Modified Korteweg-de Vries Equation
  9. ↑ Shallow-water waves, the Korteweg-deVries equation and solitons
  10. ↑ 10.0 10.1 10.2 10.3 Exact Boundary Controllability of the Korteweg--de Vries Equation
  11. ↑ (PDF) A Summary of the Korteweg-de Vries Equation
  12. ↑ 12.0 12.1 12.2 12.3 A universal form for the emergence of the Korteweg–de Vries equation
  13. ↑ The Korteweg–de Vries equation at \( {H}^{-1}\) regularity
  14. ↑ Solution of Fifth-order Korteweg and de Vries Equation by Homotopy perturbation Transform Method using He’s Polynomial
  15. ↑ The Time-Fractional Coupled-Korteweg-de-Vries Equations

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'korteweg'}, {'OP': '*'}, {'LOWER': 'de'}, {'LOWER': 'vries'}, {'LEMMA': 'equation'}]
  • [{'LOWER': 'kdv'}, {'LEMMA': 'equation'}]
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