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Finding explicit solutions of nonlinear partial differential equation is one of the most important problems in mathematical physics. And it is very difficult to find interaction solutions among different types of nonlinear excitations except for soliton-soliton interactions. It is known that Painlev analysis is an important method to investigate the integrable property of a given nonlinear evolution equation, and the truncated Painlev expansion method is a straight way to provide auto-Bcklund transformation and analytic solution, furthermore, it can also be used to obtain nonlocal symmetries. Symmetry group theory plays an important role in constructing explicit solutions, whether the equations are integrable or not. By applying the nolocal symmetry method, many new exact group invariant solution can be obtained. This method is greatly valid for constructing various interaction solutions between different types of excitations, for example, solitons, cnoidal waves, Painlev waves, Airy waves, Bessel waves, etc. It has been revealed that many more integrable systems are consistent tanh expansion (CTE) solvable and possess quite similar interaction solutions which can be described by the same determining equation with different constant constraints. In this paper, the (2+1)-dimensional higher-order Broer-Kaup (HBK) system is studied by the nonlocal symmetry method and CTE method. By using the nonlocal symmetry method, the residual symmetries of (2+1)-dimensional higher order Broer-Kaup system can be localized to Lie point symmetries after introducing suitable prolonged systems, and symmetry groups can also be obtained from the Lie point symmetry approach via the localization of the residual symmetries. By developing the truncated Painlev analysis, we use the CTE method to solve the HBK system. It is found that the HBK system is not only integrable under some nonstandard meaning but also CTE solvable. Some interaction solutions among solitons and other types of nonlinear waves which may be explicitly expressed by the Jacobi elliptic functions and the corresponding elliptic integral are constructed. To leave it clear, we give out four types of soliton+cnoidal periodic wave solutions. In order to study their dynamic behaviors, corresponding images are explicitly given.
[1] Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Phys. Rev. Lett. 19 1095
[2] Ablowitz M J, Kaup D J, Newell A C, Segur H 1973 Phys. Rev. Lett. 30 1262
[3] Weiss J, Tabor M, Carnevale G 1983 J. Math. Phys. 24 522
[4] Conte R 1989 Phys. Lett. A 140 383
[5] Yan Z Y 2015 Nonlinear Dyn. 82 119
[6] Liu H Z, Li J B, Liu L 2010 Nonlinear Dyn. 59 497
[7] Fan E G 2000 Acta Phys. Sin. 49 1409 (in Chinese)[范恩贵2000 49 1409]
[8] Yan Z Y, Zhang H Q 2000 Acta Phys. Sin. 49 2113 (in Chinese)[闫振亚, 张鸿庆2000 49 2113]
[9] Zhang H P, Chen Y, Li B 2009 Acta Phys. Sin. 58 7393 (in Chinese)[张焕萍, 陈勇, 李彪2009 58 7393]
[10] Lou S Y, Hu X B 1997 J. Phys. A:Math. Gen. 30 L95
[11] Galas F 1992 J. Phys. A:Math. Gen. 25 L981
[12] Hu X R, Lou S Y, Chen Y 2012 Phys. Rev. E 85 056607
[13] Hu X R, Chen Y 2015 Chin. Phys. B 24 090203
[14] Huang L L, Chen Y 2016 Chin. Phys. B 25 060201
[15] Huang L L, Chen Y, Ma Z Y 2016 Commun. Theor. Phys. 66 189
[16] Tang X Y, Lou S Y 2003 J. Math. Phys. 44 4000
[17] Qian X M, Lou S Y, Hu X B 2004 J. Phys. A:Math. Gen. 37 2401
[18] Fan E G, Zhang J 2002 Phys. Lett. A 305 383
[19] Fan E G 2000 Phys. Lett. A 265 353
[20] Wang Y H 2014 Appl. Math. Lett. 38 100
[21] Cheng W G, Li B, Chen Y 2015 Commun. Nonlinear Sci. Numer. Simulat. 29 198
[22] Yang D, Lou S Y, Yu W F 2013 Commun. Theor. Phys. 60 387
[23] Chen C L, Lou S Y 2014 Commun. Theor. Phys. 61 545
[24] Lou S Y, Cheng X P, Tang X Y 2014 Chin. Phys. Lett. 31 070201
[25] Lou S Y, Hu X B 1997 J. Math. Phys. 38 6401
[26] Lin J, Li H M 2002 Z. Naturforsch. 57a 929
[27] Li D S, Gao F, Zhang H Q 2004 Chaos Solitons Fract. 20 1021
[28] Shin H J 2004 J. Phys. A:Math. Gen. 37 8017
[29] Shin H J 2005 Phys. Rev. E 71 036628
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[1] Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Phys. Rev. Lett. 19 1095
[2] Ablowitz M J, Kaup D J, Newell A C, Segur H 1973 Phys. Rev. Lett. 30 1262
[3] Weiss J, Tabor M, Carnevale G 1983 J. Math. Phys. 24 522
[4] Conte R 1989 Phys. Lett. A 140 383
[5] Yan Z Y 2015 Nonlinear Dyn. 82 119
[6] Liu H Z, Li J B, Liu L 2010 Nonlinear Dyn. 59 497
[7] Fan E G 2000 Acta Phys. Sin. 49 1409 (in Chinese)[范恩贵2000 49 1409]
[8] Yan Z Y, Zhang H Q 2000 Acta Phys. Sin. 49 2113 (in Chinese)[闫振亚, 张鸿庆2000 49 2113]
[9] Zhang H P, Chen Y, Li B 2009 Acta Phys. Sin. 58 7393 (in Chinese)[张焕萍, 陈勇, 李彪2009 58 7393]
[10] Lou S Y, Hu X B 1997 J. Phys. A:Math. Gen. 30 L95
[11] Galas F 1992 J. Phys. A:Math. Gen. 25 L981
[12] Hu X R, Lou S Y, Chen Y 2012 Phys. Rev. E 85 056607
[13] Hu X R, Chen Y 2015 Chin. Phys. B 24 090203
[14] Huang L L, Chen Y 2016 Chin. Phys. B 25 060201
[15] Huang L L, Chen Y, Ma Z Y 2016 Commun. Theor. Phys. 66 189
[16] Tang X Y, Lou S Y 2003 J. Math. Phys. 44 4000
[17] Qian X M, Lou S Y, Hu X B 2004 J. Phys. A:Math. Gen. 37 2401
[18] Fan E G, Zhang J 2002 Phys. Lett. A 305 383
[19] Fan E G 2000 Phys. Lett. A 265 353
[20] Wang Y H 2014 Appl. Math. Lett. 38 100
[21] Cheng W G, Li B, Chen Y 2015 Commun. Nonlinear Sci. Numer. Simulat. 29 198
[22] Yang D, Lou S Y, Yu W F 2013 Commun. Theor. Phys. 60 387
[23] Chen C L, Lou S Y 2014 Commun. Theor. Phys. 61 545
[24] Lou S Y, Cheng X P, Tang X Y 2014 Chin. Phys. Lett. 31 070201
[25] Lou S Y, Hu X B 1997 J. Math. Phys. 38 6401
[26] Lin J, Li H M 2002 Z. Naturforsch. 57a 929
[27] Li D S, Gao F, Zhang H Q 2004 Chaos Solitons Fract. 20 1021
[28] Shin H J 2004 J. Phys. A:Math. Gen. 37 8017
[29] Shin H J 2005 Phys. Rev. E 71 036628
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