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An equivalence transformation of Whitham-Broer-Kaup equations with variable coefficients (VCWBK) is obtainedby using modified Clarkson-Kruskal direct method. Further, the relationship between the solutions of VCWBK equationsand ones of the corresponding WBK equations with constant coefficients is obtained. In addition, by applying directsymmetry method, some symmetries and similarity reductions of the corresponding WBK equations with constantcoefficients are derived. Using an auxiliary function to solve some special cases, we obtain some new exact solutionsof VCWBK equations, including rational solutions, hyperbolic function solutions, trigonometric function solutions, andJacobi elliptic function solutions.
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Keywords:
- Whitham-Broer-Kaup equations with variable coefficients /
- modified Clarkson-Kruska direct method /
- similarity reductions /
- exact solutions
[1] Yu Y D, Ma H C 2010 Appl. Math. Comput. 215 3534
[2] Fan E G, Zhang H Q 1998 Acta Phys. Sin. 47 353 (in Chinese) [范恩贵, 张鸿庆1998 物理学报47 353]
[3] Dong Z Z, Chen Y, Lang H Y 2010 Chin. Phys. B 19 090205
[4] Li D S, Zhang H Q 2003 Acta Phys. Sin. 52 1569 (in Chinese) [李德生, 张鸿庆2003 物理学报52 1569]
[5] Chen Y M, Ma S H, Ma Z Y 2013 Chin. Phys. B 22 050510
[6] Bekir A, Ayhan B, Özer M N 2013 Chin. Phys. B 22 010202
[7] Clarkson P A, Kruskal M D 1989 J. Math. Phys. 30 2201
[8] Lou S Y 1990 Phys. Lett. A 151 133
[9] Yan Z L, Zhou J P 2010 Commun. Theor. Phys. 54 965
[10] Yan Z L, Liu X Q 2005 Commun. Theor. Phys. 44 479
[11] Zhang Z Y, Yong X L, Chen Y F 2008 J. Nonlinear Math. Phys. 15 383
[12] Emmanuel Y, Peng y Z 2006 Acta. J. Theor. Phys. 45 197
[13] Yan Z Y, Zhang H Q 2001 Phys. Lett. A 285 355
[14] Mohammed Khalfallah 2009 Math. Comput. Model. 49 666
[15] Tian Y H, Chen H L, Liu X Q 2010 Appl. Math. Comput. 215 3509
[16] Zhang L H, Liu X Q, Bai C L 2007 Commun. Theor. Phys. (Beijing, China) 48 405
[17] Bai C L, Bai C J, Zhao H 2005 Z. Naturforsch. 60a 211
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[1] Yu Y D, Ma H C 2010 Appl. Math. Comput. 215 3534
[2] Fan E G, Zhang H Q 1998 Acta Phys. Sin. 47 353 (in Chinese) [范恩贵, 张鸿庆1998 物理学报47 353]
[3] Dong Z Z, Chen Y, Lang H Y 2010 Chin. Phys. B 19 090205
[4] Li D S, Zhang H Q 2003 Acta Phys. Sin. 52 1569 (in Chinese) [李德生, 张鸿庆2003 物理学报52 1569]
[5] Chen Y M, Ma S H, Ma Z Y 2013 Chin. Phys. B 22 050510
[6] Bekir A, Ayhan B, Özer M N 2013 Chin. Phys. B 22 010202
[7] Clarkson P A, Kruskal M D 1989 J. Math. Phys. 30 2201
[8] Lou S Y 1990 Phys. Lett. A 151 133
[9] Yan Z L, Zhou J P 2010 Commun. Theor. Phys. 54 965
[10] Yan Z L, Liu X Q 2005 Commun. Theor. Phys. 44 479
[11] Zhang Z Y, Yong X L, Chen Y F 2008 J. Nonlinear Math. Phys. 15 383
[12] Emmanuel Y, Peng y Z 2006 Acta. J. Theor. Phys. 45 197
[13] Yan Z Y, Zhang H Q 2001 Phys. Lett. A 285 355
[14] Mohammed Khalfallah 2009 Math. Comput. Model. 49 666
[15] Tian Y H, Chen H L, Liu X Q 2010 Appl. Math. Comput. 215 3509
[16] Zhang L H, Liu X Q, Bai C L 2007 Commun. Theor. Phys. (Beijing, China) 48 405
[17] Bai C L, Bai C J, Zhao H 2005 Z. Naturforsch. 60a 211
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