The Boussinesq equation: Lax pair, Bäcklund transformation, symmetry group transformation and consistent Riccati expansion solvability

Liu Ping; Xu Heng-Rui; Yang Jian-Rong

doi:10.7498/aps.69.20191316

Abstract

The Boussinesq equation is a very important equation in fluid mechanics and some other disciplines. A Lax pair of the Boussinesq equation is proposed. With the help of the truncated Painlevé expansion, auto-Bäcklund transformation of the Boussinesq equation and Bäcklund transformation between the Boussinesq equation and the Schwarzian Boussinesq equation are demonstrated. Nonlocal symmetries of the Boussinesq equation are discussed. One-parameter subgroup invariant solutions and one-parameter group transformations are obtained. The consistent Riccati expansion solvability of the Boussinesq equation is proved and some interaction structures between soliton-cnoidal waves are obtained by consistent Riccati expansion.

Keywords:

Authors and contacts

1.
School of Electronic and Information Engineering, University of Electronic Science and Technology of China, Zhongshan Institute, Zhongshan 528402, China
2.
School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China
3.
School of Physics and Electronic Information, Shangrao Normal University, Shangrao 334001, China

Corresponding author: Liu Ping, liuping49@126.com

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References

[1]	Boussinesq J 1872 J. Math. Pures Appl. 17 55
[2]	Ursell F 1953 Proc. Cambridge Philos. Soc. 49 685 Google Scholar
[3]	Daripa P 1998 J. Comput. Appl. Math. 100 161 Google Scholar
[4]	Guo B, Gao Z, Lin J 2016 Commun. Theor. Phys. 64 589
[5]	Liu W 2009 Z. Naturforschung A 64 709 Google Scholar
[6]	Benny D J, Luke J C 1964 J. Math. Phys. 43 309 Google Scholar
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[9]	Weiss J 1983 J. Math. Phys. 24 1405 Google Scholar
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[12]	Liu Y K, Li B 2016 Chin. J. Phys. 54 718 Google Scholar
[13]	Gao X N, Lou SY, Tang X Y 2013 J. High Energy Phys. 5 029
[14]	Olver P J 1993 Applications of Lie Group to Differential Equations (2nd ed.) (New York: Springer) pp75–238
[15]	Liu P, Zeng B Q, Ren B 2015 Commun. Theor. Phys. 63 413 Google Scholar
[16]	Liu Y K, Li B 2017 Chin. Phys. Lett. 34 010202 Google Scholar
[17]	Liu P, Zeng B Q, Deng B B, Yang J R 2015 AIP Adv. 5 087162 Google Scholar
[18]	Liu P, Wang Y X, Ren B, Li J H 2016 Commun. Theor. Phys. 66 595 Google Scholar
[19]	焦小玉, 贾曼, 安红利 2019 物理学报 68 140201 Google Scholar Jiao X Y, Jia M, An H L 2019 Acta Phys. Sin. 68 140201 Google Scholar
[20]	Lou S Y 2015 Stud. Appl. Math. 134 372 Google Scholar

Cited By

图 1 满足(47)式的碰撞波解(49)式. 自由参数为{n = 0.2, m = 0.5, a₁ = 1, a₃ = 1, k₁ = 1, k₂ = 1, ω₂ = 1, α = –0.8, β = 1}

Figure 1. The solution (49) with Formula (47). The free parameters are {n = 0.2, m = 0.5, a₁ = 1, a₃ = 1, k₁ = 1, k₂ = 1, ω₂ = 1, α = –0.8, β = 1}.

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图 2 满足(47)式的碰撞波解(49)式. 自由参数为 {n = 0.2, m = 0.9, a₁ = 1, a₃ = 1, k₁ = 1, k₂ = 1, ω₂ = 1, α = –0.8, β = 1}

Figure 2. The solution (49) with Formula (47). The free parameters are {n = 0.2, m = 0.9, a₁ = 1, a₃ = 1, k₁ = 1, k₂ = 1, ω₂ = 1, α = –0.8, β = 1}.

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图 3 u的密度函数图. 图(a)的参数与图1相同, 图(b)的参数与图2相同

Figure 3. The density of u. The parameters of the Fig. (a) are the same as those of Figure 1 and the parameters of the Fig. (b) are the same as those of Figure 2.

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图 4 参数关系满足(48)式的碰撞波解(49)式的演化图. 自由参数为 {n = 0.4, a₁ = 1, a₂ = 1, a₃ = 2.2, k₁ = 1, k₂ = –0.22, ω₂ = 1, α = –400, β = 80}

Figure 4. The interaction solution (49) with parameter satisfying Formula (48). The free parameters are chosen as {n = 0.4, a₁ = 1, a₂ = 1, a₃ = 2.2, k₁ = 1, k₂ = –0.22, ω₂ = 1, α = –400, β = 80}.

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图 5 参数关系满足(48)式的碰撞波解(49)式. 自由参数为{n = 0.6, a₁ = 2, a₂ = 1, a₃ = 4, k₁ = 1, k₂ = –0.12, ω₂ = 0.1, α = –14, β = 6}

Figure 5. The interaction solution (49) with parameter satisfying Formula (48). The free parameters are selected as {n = 0.6, a₁ = 2, a₂ = 1, a₃ = 4, k₁ = 1, k₂ = –0.12, ω₂ = 0.1, α = –14, β = 6}.

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图 6 u的密度函数图. 图(a)对应图4, 图(b)对应图5

Figure 6. The density of u. The Fig. (a) is related to Fig. 4 and the Fig. (b) is corresponding to Fig. 5.

DownLoad: Full-Size Img PowerPoint

[1]	Boussinesq J 1872 J. Math. Pures Appl. 17 55
[2]	Ursell F 1953 Proc. Cambridge Philos. Soc. 49 685 Google Scholar
[3]	Daripa P 1998 J. Comput. Appl. Math. 100 161 Google Scholar
[4]	Guo B, Gao Z, Lin J 2016 Commun. Theor. Phys. 64 589
[5]	Liu W 2009 Z. Naturforschung A 64 709 Google Scholar
[6]	Benny D J, Luke J C 1964 J. Math. Phys. 43 309 Google Scholar
[7]	Himonas A A, Mantzavinos D 2015 J. Differ. Equations 258 3107 Google Scholar
[8]	Li S, Zhang W, Bu X 2017 J. Math. Anal. Appl. 449 96 Google Scholar
[9]	Weiss J 1983 J. Math. Phys. 24 1405 Google Scholar
[10]	Guo B X, Lin J 2013 Int. J. Mod. Phys. B 30 1640013
[11]	Guo B X, Gao Z J, Lin J 2016 Commun. Theor. Phys. 66 589 Google Scholar
[12]	Liu Y K, Li B 2016 Chin. J. Phys. 54 718 Google Scholar
[13]	Gao X N, Lou SY, Tang X Y 2013 J. High Energy Phys. 5 029
[14]	Olver P J 1993 Applications of Lie Group to Differential Equations (2nd ed.) (New York: Springer) pp75–238
[15]	Liu P, Zeng B Q, Ren B 2015 Commun. Theor. Phys. 63 413 Google Scholar
[16]	Liu Y K, Li B 2017 Chin. Phys. Lett. 34 010202 Google Scholar
[17]	Liu P, Zeng B Q, Deng B B, Yang J R 2015 AIP Adv. 5 087162 Google Scholar
[18]	Liu P, Wang Y X, Ren B, Li J H 2016 Commun. Theor. Phys. 66 595 Google Scholar
[19]	焦小玉, 贾曼, 安红利 2019 物理学报 68 140201 Google Scholar Jiao X Y, Jia M, An H L 2019 Acta Phys. Sin. 68 140201 Google Scholar
[20]	Lou S Y 2015 Stud. Appl. Math. 134 372 Google Scholar

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Vol. 69, Issue 1 - 2020-01-05

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