Solutions to the modified Korteweg–de Vries equation

This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix diff...

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Published inReviews in mathematical physics Vol. 26; no. 7; p. 1430006
Main Authors Zhang, Da-Jun, Zhao, Song-Lin, Sun, Ying-Ying, Zhou, Jing
Format Journal Article
LanguageEnglish
Published Singapore World Scientific Publishing Company 01.08.2014
World Scientific Publishing Co. Pte., Ltd
Subjects
Breathers
Differential equations
Korteweg-Devries equation
Mathematical analysis
Matrix algebra
Matrix methods
Solitary waves
The modified Korteweg–de Vries equation
breathers
dynamics
Wronskian
rational solutions
Online AccessGet full text
ISSN0129-055X
1793-6659
DOI10.1142/S0129055X14300064

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Abstract This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.
AbstractList This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.
Author Zhang, Da-Jun
Sun, Ying-Ying
Zhao, Song-Lin
Zhou, Jing
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  fullname: Sun, Ying-Ying
  organization: Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China
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  givenname: Jing
  surname: Zhou
  fullname: Zhou, Jing
  organization: Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China
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Cites_doi 10.1088/0305-4470/36/17/313
10.1023/A:1006910004292
10.1143/JPSJ.71.2649
10.1016/S0378-4371(98)00466-X
10.1103/PhysRevLett.25.1108
10.1007/s11232-011-0072-4
10.1143/JPSJ.26.1305
10.1016/S0378-4371(02)00995-0
10.1090/S0002-9947-1991-1029000-7
10.1088/0305-4470/17/7/009
10.1016/0375-9601(88)90541-5
10.1017/CBO9780511543043
10.1143/JPSJ.41.1817
10.1007/BF02430638
10.1063/1.530497
10.1143/JPSJ.46.359
10.1016/0375-9601(83)90159-7
10.1103/PhysRevA.67.013804
10.1143/JPSJ.61.4336
10.1090/S0002-9947-04-03726-2
10.1063/1.523550
10.1143/PTPS.59.64
10.1016/0375-9601(83)90764-8
10.1143/JPSJ.38.673
10.1143/JPSJ.33.1456
10.1063/1.1664701
10.1175/JPO2652.1
10.3792/pja/1195519590
10.1016/0034-4877(91)90025-I
10.1137/1.9781611970883
10.1103/PhysRevE.52.5574
10.1143/JPSJ.53.950
10.1103/PhysRevLett.19.1095
10.1007/s11232-010-0046-y
10.1143/JPSJ.37.1631
10.1088/0253-6102/55/4/02
10.1143/JPSJ.34.1289
10.1016/S0378-4371(97)00260-4
10.1016/j.physleta.2009.07.004
10.1103/PhysRevLett.27.1192
10.1088/0253-6102/57/6/03
10.1088/0253-6102/61/4/03
10.1016/S0960-0779(00)00137-5
10.1016/j.physrep.2012.10.006
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breathers
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Wronskian
rational solutions
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References rf5
rf23
rf45
rf4
rf22
rf44
rf7
rf25
Kamke E. (rf39) 1977
rf47
rf6
rf24
rf46
rf9
rf41
rf8
rf40
rf21
rf43
rf20
rf42
rf27
rf49
rf26
rf48
rf29
rf28
rf12
rf34
rf11
rf33
rf13
rf35
rf30
rf51
rf10
rf32
rf31
rf19
rf16
rf38
rf15
rf37
rf18
rf17
Pratt L. J. (rf14) 1992; 23
rf3
rf2
References_xml – ident: rf48
  doi: 10.1088/0305-4470/36/17/313
– ident: rf10
  doi: 10.1023/A:1006910004292
– ident: rf40
  doi: 10.1143/JPSJ.71.2649
– ident: rf12
  doi: 10.1016/S0378-4371(98)00466-X
– ident: rf18
  doi: 10.1103/PhysRevLett.25.1108
– ident: rf25
  doi: 10.1007/s11232-011-0072-4
– ident: rf7
  doi: 10.1143/JPSJ.26.1305
– ident: rf21
  doi: 10.1016/S0378-4371(02)00995-0
– ident: rf26
  doi: 10.1090/S0002-9947-1991-1029000-7
– ident: rf28
  doi: 10.1088/0305-4470/17/7/009
– ident: rf31
  doi: 10.1016/0375-9601(88)90541-5
– ident: rf38
  doi: 10.1017/CBO9780511543043
– ident: rf42
  doi: 10.1143/JPSJ.41.1817
– ident: rf15
  doi: 10.1007/BF02430638
– ident: rf19
  doi: 10.1063/1.530497
– ident: rf27
  doi: 10.1143/JPSJ.46.359
– ident: rf47
  doi: 10.1016/0375-9601(83)90159-7
– ident: rf4
  doi: 10.1103/PhysRevA.67.013804
– ident: rf6
  doi: 10.1143/JPSJ.61.4336
– ident: rf32
  doi: 10.1090/S0002-9947-04-03726-2
– ident: rf43
  doi: 10.1063/1.523550
– ident: rf45
  doi: 10.1143/PTPS.59.64
– ident: rf30
  doi: 10.1016/0375-9601(83)90764-8
– ident: rf44
  doi: 10.1143/JPSJ.38.673
– ident: rf22
  doi: 10.1143/JPSJ.33.1456
– ident: rf2
  doi: 10.1063/1.1664701
– ident: rf16
  doi: 10.1175/JPO2652.1
– ident: rf17
– ident: rf24
  doi: 10.3792/pja/1195519590
– ident: rf29
  doi: 10.1016/0034-4877(91)90025-I
– ident: rf41
  doi: 10.1137/1.9781611970883
– ident: rf11
  doi: 10.1103/PhysRevE.52.5574
– volume-title: Handbook on Ordinary Differential Equations
  year: 1977
  ident: rf39
– ident: rf9
  doi: 10.1143/JPSJ.53.950
– volume: 23
  start-page: 91
  year: 1992
  ident: rf14
  publication-title: J. Phys. Oceangr.
– ident: rf3
  doi: 10.1103/PhysRevLett.19.1095
– ident: rf34
  doi: 10.1007/s11232-010-0046-y
– ident: rf8
  doi: 10.1143/JPSJ.37.1631
– ident: rf49
– ident: rf35
  doi: 10.1088/0253-6102/55/4/02
– ident: rf23
  doi: 10.1143/JPSJ.34.1289
– ident: rf20
  doi: 10.1016/S0378-4371(97)00260-4
– ident: rf33
  doi: 10.1016/j.physleta.2009.07.004
– ident: rf37
  doi: 10.1103/PhysRevLett.27.1192
– ident: rf46
  doi: 10.1088/0253-6102/57/6/03
– ident: rf51
  doi: 10.1088/0253-6102/61/4/03
– ident: rf13
  doi: 10.1016/S0960-0779(00)00137-5
– ident: rf5
  doi: 10.1016/j.physrep.2012.10.006
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Snippet This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we...
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SubjectTerms Breathers
Differential equations
Korteweg-Devries equation
Mathematical analysis
Matrix algebra
Matrix methods
Solitary waves
Title Solutions to the modified Korteweg–de Vries equation
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