General Higher-Order Breather and Hybrid Solutions of the Fokas System
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摘要
Fokas system is the simplest(2+1)-dimensional extension of the nonlinear Schr?dinger(NLS) equation(Eq.(2), Inverse Problems 10(1994) L19-L22). By appropriately limiting on soliton solutions generated by the Hirota bilinear method, the explicit forms of n-th breathers and semi-rational solutions for the Fokas system are derived. The obtained first-order breather exhibits a range of interesting dynamics. For high-order breather, it has more rich dynamical behaviors. The first-order and the second-order breather solutions are given graphically. Using the long wave limit in soliton solutions, rational solutions are obtained, which are used to analyze the mechanism of the rogue wave and lump respectively. By taking a long waves limit of a part of exponential functions in f and g appeared in the bilinear form of the Fokas system, many interesting hybrid solutions are constructed. The hybrid solutions illustrate various superposed wave structures involving rogue waves, lumps, solitons, and periodic line waves. Their rather complicated dynamics are revealed.
Fokas system is the simplest(2+1)-dimensional extension of the nonlinear Schr?dinger(NLS) equation(Eq.(2), Inverse Problems 10(1994) L19-L22). By appropriately limiting on soliton solutions generated by the Hirota bilinear method, the explicit forms of n-th breathers and semi-rational solutions for the Fokas system are derived. The obtained first-order breather exhibits a range of interesting dynamics. For high-order breather, it has more rich dynamical behaviors. The first-order and the second-order breather solutions are given graphically. Using the long wave limit in soliton solutions, rational solutions are obtained, which are used to analyze the mechanism of the rogue wave and lump respectively. By taking a long waves limit of a part of exponential functions in f and g appeared in the bilinear form of the Fokas system, many interesting hybrid solutions are constructed. The hybrid solutions illustrate various superposed wave structures involving rogue waves, lumps, solitons, and periodic line waves. Their rather complicated dynamics are revealed.
Fokas system is the simplest(2+1)-dimensional extension of the nonlinear Schr?dinger(NLS) equation(Eq.(2), Inverse Problems 10(1994) L19-L22). By appropriately limiting on soliton solutions generated by the Hirota bilinear method, the explicit forms of n-th breathers and semi-rational solutions for the Fokas system are derived. The obtained first-order breather exhibits a range of interesting dynamics. For high-order breather, it has more rich dynamical behaviors. The first-order and the second-order breather solutions are given graphically. Using the long wave limit in soliton solutions, rational solutions are obtained, which are used to analyze the mechanism of the rogue wave and lump respectively. By taking a long waves limit of a part of exponential functions in f and g appeared in the bilinear form of the Fokas system, many interesting hybrid solutions are constructed. The hybrid solutions illustrate various superposed wave structures involving rogue waves, lumps, solitons, and periodic line waves. Their rather complicated dynamics are revealed.
引文
[1]M.J.Ablowitz and P.A.Clarkson,Nonlinear Evolution Equations and Inverse Scattering.Cambridge University Press,Cambridge(1991).
[2]G.Biondini and G.Kovaˇciˇc,J.Math.Phys.55(2014)031506
[3]B.Prinari,G.Biondini,and A.D.Trubatch,Stud.Appl.Math.126(2015)245.
[4]X.Q.Zhao,L.M.Wang,and J.S.Wei,Chaos,Solitons and Fractals 28(2006)448.
[5]M.Senthilvelan,Appl.Math.Comput.123(2001)381.
[6]J.S.He,H.R.Zhang,L.H.Wang,et al.,Phys.Rev.E87(2013)052914.
[7]J.S.He,S.W.Xu,and K.Porsezian,Phys.Rev.E 86(2012)066603.
[8]M.Lakshmanan and P.Kaliappan,J.Math.Phys.49(1983)795.
[9]H.Liu and C.Yue,Nonlinear Dyn.89(2017)1989.
[10]Y.Ohta and J.K.Yang,Phys.Rev.E 86(2012)036604.
[11]Y.Ohta and J.K.Yang,J.Phys.A 46(2013)105202.
[12]C.Qian,J.G.Rao,Y.B.Liu,and J.S.He,Chin.Phys.Lett.33(2016)110201.
[13]Y.L.Cao,J.S.He,and D.Mihalache,Nonlinear Dyn.91(2018)2593.
[14]Y.L.Cao,J.G.Rao,and D.Mihalache,Appl.Math.Lett.80(2018)27.
[15]W.Malflied and W.Hureman,Phys.Scr.54(1996)569.
[16]E.G.Fan,Phys,Lett.A 277(2000)212.
[17]X.P.Zeng,Z.D.Dai,and D.L.Li,Chaos,Solitons and Fractals 42(2009)657.
[18]M.L.Wang and X.Z.Li,Chaos,Solitons and Fractals24(2005)1257.
[19]X.Y.Li,S.Yang,and M.L.Wang,Chaos,Solitons and Fractals 25(2005)629.
[20]A.S.Fokas,J.Math.Phys.41(2000)4188.
[21]M.Dimakos and A.S.Fokas,J.Math.Phys.56(2015)2093.
[22]P.Razborova,L.Moraru,and A.Biswas,Romanian J.Phys.59(2014)658.
[23]J.H.He and X.H.Wu,Chaos,Solitons and Fractals 30(2006)700.
[24]J.L.Zhang,M.L.Wang,Y.M.Wang,and Z.D.Fang,Phys.Lett.A 350(2006)103.
[25]A.S.Fokas,Inverse Problems 10(1994)L19.
[26]J.C.Chen and Y.Chen,J.Nonlinear Math.Phys.21(2014)454.
[27]R.Radha and M.Lakshmanan,Chaos,Solitons and Fractals 8(1997)17.
[28]Javier Villarroel,J.Prada,and P.G.Est′evez,Stud.Appl.Math.122(2009)395.
[29]R.Radha and M.Lakshmanan,Chaos,Solitons and Fractals 10(1999)1821.
[30]P.G.Est′evez,J.Math.Phys.40(1999)1406.
[31]J.G.Rao,L.H.Wang,Y.Zhang,and J.S.He,Commun.Theor.Phys.64(2015)605.
[32]Q.X.Xing,Z.W.Wu,D.Mihalache,and J.S.He,Nonlinear Dyn.89(2017)2299.
[33]Q.X.Xing,L.H.Wang,D.Mihalache,et al.,Chaos 27(2017)053102.
[34]J.S.He,L.H.Wang,and L.J.Li,Phys.Rev.E 89(2014)062917.
[35]J.Lin,X.W.Jin,X.L.Gao,and S.Y.Lou,Commun.Theor.Phys.70(2018)119.
[36]N.Vishnu Priya,M.Senthilvelan,and M.Lakshmanan,Phys.Rev.E 88(2013)022918.
[37]S.H.Chen and D.Mihalache,J.Phys.A:Math.Theor.48(2015)215202.
[38]V.S.Shchesnovich and E.V.Doktorov,Phys.Rev.E 55(1997)7626.
[39]Z.D.Dai,J.Huang,M.R.Jiang,and S.H.Wang,Chaos,Solitons and Fractals 26(2005)1189.
[40]W.X.Ma,J.Geom.Phys.133(2018)10.
[41]G.Mu,Z.Y.Qin,and R.Grimshaw,SIAM J.Appl.Math.75(2015)1.
[42]M.Tajiri and T.Arai,Phys.Rev.E 60(1999)2297.
[43]R.Hirota,The Direct Method in Soliton Theory,Cambridge University Press,Cambridge(2004).
[44]J.G.Rao,Y.S.Zhang,A.S.Fokas,and J.S.He,Nonlinearity 31(2018)4090.
[45]J.G.Rao,Y.Cheng,and J.S.He,Stud.Appl.Math.139(2017)568.
[46]M.J.Ablowitz and J.Satsuma,J.Math.Phys.19(1978)2180.
[47]J.Satsuma and M.J.Ablowitz,J.Math.Phys.20(1979)1496.
[2]G.Biondini and G.Kovaˇciˇc,J.Math.Phys.55(2014)031506
[3]B.Prinari,G.Biondini,and A.D.Trubatch,Stud.Appl.Math.126(2015)245.
[4]X.Q.Zhao,L.M.Wang,and J.S.Wei,Chaos,Solitons and Fractals 28(2006)448.
[5]M.Senthilvelan,Appl.Math.Comput.123(2001)381.
[6]J.S.He,H.R.Zhang,L.H.Wang,et al.,Phys.Rev.E87(2013)052914.
[7]J.S.He,S.W.Xu,and K.Porsezian,Phys.Rev.E 86(2012)066603.
[8]M.Lakshmanan and P.Kaliappan,J.Math.Phys.49(1983)795.
[9]H.Liu and C.Yue,Nonlinear Dyn.89(2017)1989.
[10]Y.Ohta and J.K.Yang,Phys.Rev.E 86(2012)036604.
[11]Y.Ohta and J.K.Yang,J.Phys.A 46(2013)105202.
[12]C.Qian,J.G.Rao,Y.B.Liu,and J.S.He,Chin.Phys.Lett.33(2016)110201.
[13]Y.L.Cao,J.S.He,and D.Mihalache,Nonlinear Dyn.91(2018)2593.
[14]Y.L.Cao,J.G.Rao,and D.Mihalache,Appl.Math.Lett.80(2018)27.
[15]W.Malflied and W.Hureman,Phys.Scr.54(1996)569.
[16]E.G.Fan,Phys,Lett.A 277(2000)212.
[17]X.P.Zeng,Z.D.Dai,and D.L.Li,Chaos,Solitons and Fractals 42(2009)657.
[18]M.L.Wang and X.Z.Li,Chaos,Solitons and Fractals24(2005)1257.
[19]X.Y.Li,S.Yang,and M.L.Wang,Chaos,Solitons and Fractals 25(2005)629.
[20]A.S.Fokas,J.Math.Phys.41(2000)4188.
[21]M.Dimakos and A.S.Fokas,J.Math.Phys.56(2015)2093.
[22]P.Razborova,L.Moraru,and A.Biswas,Romanian J.Phys.59(2014)658.
[23]J.H.He and X.H.Wu,Chaos,Solitons and Fractals 30(2006)700.
[24]J.L.Zhang,M.L.Wang,Y.M.Wang,and Z.D.Fang,Phys.Lett.A 350(2006)103.
[25]A.S.Fokas,Inverse Problems 10(1994)L19.
[26]J.C.Chen and Y.Chen,J.Nonlinear Math.Phys.21(2014)454.
[27]R.Radha and M.Lakshmanan,Chaos,Solitons and Fractals 8(1997)17.
[28]Javier Villarroel,J.Prada,and P.G.Est′evez,Stud.Appl.Math.122(2009)395.
[29]R.Radha and M.Lakshmanan,Chaos,Solitons and Fractals 10(1999)1821.
[30]P.G.Est′evez,J.Math.Phys.40(1999)1406.
[31]J.G.Rao,L.H.Wang,Y.Zhang,and J.S.He,Commun.Theor.Phys.64(2015)605.
[32]Q.X.Xing,Z.W.Wu,D.Mihalache,and J.S.He,Nonlinear Dyn.89(2017)2299.
[33]Q.X.Xing,L.H.Wang,D.Mihalache,et al.,Chaos 27(2017)053102.
[34]J.S.He,L.H.Wang,and L.J.Li,Phys.Rev.E 89(2014)062917.
[35]J.Lin,X.W.Jin,X.L.Gao,and S.Y.Lou,Commun.Theor.Phys.70(2018)119.
[36]N.Vishnu Priya,M.Senthilvelan,and M.Lakshmanan,Phys.Rev.E 88(2013)022918.
[37]S.H.Chen and D.Mihalache,J.Phys.A:Math.Theor.48(2015)215202.
[38]V.S.Shchesnovich and E.V.Doktorov,Phys.Rev.E 55(1997)7626.
[39]Z.D.Dai,J.Huang,M.R.Jiang,and S.H.Wang,Chaos,Solitons and Fractals 26(2005)1189.
[40]W.X.Ma,J.Geom.Phys.133(2018)10.
[41]G.Mu,Z.Y.Qin,and R.Grimshaw,SIAM J.Appl.Math.75(2015)1.
[42]M.Tajiri and T.Arai,Phys.Rev.E 60(1999)2297.
[43]R.Hirota,The Direct Method in Soliton Theory,Cambridge University Press,Cambridge(2004).
[44]J.G.Rao,Y.S.Zhang,A.S.Fokas,and J.S.He,Nonlinearity 31(2018)4090.
[45]J.G.Rao,Y.Cheng,and J.S.He,Stud.Appl.Math.139(2017)568.
[46]M.J.Ablowitz and J.Satsuma,J.Math.Phys.19(1978)2180.
[47]J.Satsuma and M.J.Ablowitz,J.Math.Phys.20(1979)1496.