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Investigation of interaction between solitons and their background small amplitude waves has been an interesting topic in numerical study for more than three decades. A classical soliton accompanied with oscillatory tails to infinite extent in space, is an interesting quasi-soliton, which has been revealed in experimental study and really observed. However, analytical solution of such a special quasi-soliton structure is rarely considered. In this paper, two branches of soliton-cnoidal wave solution as well as the two-soliton solution of the Korteweg-de Vries (KdV) equation are obtained by the generalized tanh expansion method. The exact relation between the soliton-cnoidal wave solution and the classical soliton solution of the KdV equation is established. By choosing suitable wave parameters, the quasi-soliton behavior of the soliton-cnoidal wave solution is revealed. It is found that with modulus of the Jacobi elliptic function approaching to zero asymptotically, the oscillating tails can be minimized and the soliton core of the soliton-cnoidal wave turns closer to the classical soliton solution. In addition, the quasi-soliton structure is revealed in a plasma physics system. By the reductive perturbation approach, the KdV equation modeling ion acoustic waves in an ideal homogeneous magnetized plasma is derived. It is confirmed that the waveform of the quasi-soliton is significantly influenced by the electron distribution, temperature ratio of ion to electron, magnetic field strength, and magnetic field direction. Interestingly, the amplitude of the quasi-soliton keeps constant due to the -independence of nonlinear coefficient A. The width of the soliton core and the wavelength of the surrounded periodic wave become constant with the further increase of . The explicit soliton-cnoidal wave solution with quasi-soliton behavior obtained here is applicable to many physical scenarios. For instance, the quasi-soliton structure can be viewed as a classical soliton with perturbations, and can correct the classical soliton in both theoretical and experimental study.
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Keywords:
- Korteweg-de Vries equation /
- soliton-cnoidal wave /
- quasi-soliton behavior /
- ion acoustic wave
[1] Lax P D 1968 Commun. Pur. Appl. Math. 21 467
[2] Miura R, Gardner C, Kruskal M 1968 J. Math. Phys. 9 1204
[3] Hirota R 1971 Phys. Rev. Lett. 27 1192
[4] Weiss J, Tabor M, Carnevale G 1983 J. Math. Phys. 25 522
[5] Dauxois T, Peyrard M 2006 Physics of Solitons (Cambridge: Cambridge University Press)
[6] Jeffrey A, Kakutani T 1972 SIAM Rev. 14 582
[7] Bandyopadhyay P, Prasad G, Sen A, Kaw P K 2008 Phys. Rev. Lett. 101 065006
[8] Cheng X P, Li J Y, Xue J R 2011 Acta Phys. Sin. 60 110204(in Chinese) [程雪苹, 李金玉, 薛江蓉 2011 60 110204]
[9] Mao J J, Yang J R, Li C Y 2012 Acta Phys. Sin. 61 020206(in Chinese) [毛杰健, 杨建荣, 李超英 2012 61 020206]
[10] Lou S Y, Hu X R, Chen Y 2012 J. Phys. A: Math. Theor. 45 155209
[11] Lou S Y 2015 Stud. Appl. Math. 134 372
[12] Tang X Y, Hao X Z, Liang Z F 2017 Comp. Math. Appl. 74 1311
[13] Tang X Y, Liang Z F, Wang J Y 2015 J. Phys. A: Math. Theor. 48 285204
[14] Gao X N, Lou S Y, Tang X Y 2013 JHEP 05 029
[15] Chen C L, Lou S Y 2013 Chin. Phys. Lett. 30 110202
[16] Cheng X P, Lou S Y, Chen C L, Tang X Y 2014 Phys. Rev. E 89 043202
[17] Ren B, Cheng X P, Lin J 2016 Nonlinear Dyn. 86 1855
[18] Ren B 2017 Commun. Nonlinear Sci. Numer. Simul. 42 456
[19] Hao X Z, Liu Y P, Tang X Y, Li Z B 2016 Comp. Math. Appl. 72 2405
[20] Wang J Y, Cheng X P, Tang X Y, Yang J R, Ren B 2014 Phys. Plasmas 21 032111
[21] Deeskow P, Schamel H, Rao N N, Yu M Y, Varma R K, Shukla P K 1987 Phys. Fluids 30 2703
[22] Keane A J, Mushtaq A, Wheatland M S 2011 Phys. Rev. E 83 066407
[23] Davis R E, Acrivos A 1967 J. Fluid Mech. 29 593
[24] Farmer D M, Smith J D 1980 Deep-sea Rea. 27A 239
[25] Akylas T R, Grimshaw R H J 1992 J. Fluid Mech. 242 279
[26] Wang J Y, Tang X Y, Lou S Y, Gao X N, Jia M 2014 Europhys. Lett. 108 20005
[27] Williams G, Kourakis I 2013 Plasma Phys. Controlled Fusion 55 055005
[28] Singh S V, Devanandhan S, Lakhina G S, Bharuthram R 2013 Phys. Plasmas 20 012306
[29] Saini N S, Kourakis I 2010 Plasma Phys. Controlled Fusion 52 075009
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[1] Lax P D 1968 Commun. Pur. Appl. Math. 21 467
[2] Miura R, Gardner C, Kruskal M 1968 J. Math. Phys. 9 1204
[3] Hirota R 1971 Phys. Rev. Lett. 27 1192
[4] Weiss J, Tabor M, Carnevale G 1983 J. Math. Phys. 25 522
[5] Dauxois T, Peyrard M 2006 Physics of Solitons (Cambridge: Cambridge University Press)
[6] Jeffrey A, Kakutani T 1972 SIAM Rev. 14 582
[7] Bandyopadhyay P, Prasad G, Sen A, Kaw P K 2008 Phys. Rev. Lett. 101 065006
[8] Cheng X P, Li J Y, Xue J R 2011 Acta Phys. Sin. 60 110204(in Chinese) [程雪苹, 李金玉, 薛江蓉 2011 60 110204]
[9] Mao J J, Yang J R, Li C Y 2012 Acta Phys. Sin. 61 020206(in Chinese) [毛杰健, 杨建荣, 李超英 2012 61 020206]
[10] Lou S Y, Hu X R, Chen Y 2012 J. Phys. A: Math. Theor. 45 155209
[11] Lou S Y 2015 Stud. Appl. Math. 134 372
[12] Tang X Y, Hao X Z, Liang Z F 2017 Comp. Math. Appl. 74 1311
[13] Tang X Y, Liang Z F, Wang J Y 2015 J. Phys. A: Math. Theor. 48 285204
[14] Gao X N, Lou S Y, Tang X Y 2013 JHEP 05 029
[15] Chen C L, Lou S Y 2013 Chin. Phys. Lett. 30 110202
[16] Cheng X P, Lou S Y, Chen C L, Tang X Y 2014 Phys. Rev. E 89 043202
[17] Ren B, Cheng X P, Lin J 2016 Nonlinear Dyn. 86 1855
[18] Ren B 2017 Commun. Nonlinear Sci. Numer. Simul. 42 456
[19] Hao X Z, Liu Y P, Tang X Y, Li Z B 2016 Comp. Math. Appl. 72 2405
[20] Wang J Y, Cheng X P, Tang X Y, Yang J R, Ren B 2014 Phys. Plasmas 21 032111
[21] Deeskow P, Schamel H, Rao N N, Yu M Y, Varma R K, Shukla P K 1987 Phys. Fluids 30 2703
[22] Keane A J, Mushtaq A, Wheatland M S 2011 Phys. Rev. E 83 066407
[23] Davis R E, Acrivos A 1967 J. Fluid Mech. 29 593
[24] Farmer D M, Smith J D 1980 Deep-sea Rea. 27A 239
[25] Akylas T R, Grimshaw R H J 1992 J. Fluid Mech. 242 279
[26] Wang J Y, Tang X Y, Lou S Y, Gao X N, Jia M 2014 Europhys. Lett. 108 20005
[27] Williams G, Kourakis I 2013 Plasma Phys. Controlled Fusion 55 055005
[28] Singh S V, Devanandhan S, Lakhina G S, Bharuthram R 2013 Phys. Plasmas 20 012306
[29] Saini N S, Kourakis I 2010 Plasma Phys. Controlled Fusion 52 075009
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