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July  2022, 42(7): 3355-3378. doi: 10.3934/dcds.2022018

Linear stability of exact solutions for the generalized Kaup-Boussinesq equation and their dynamical evolutions

1. 

Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

2. 

School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China

3. 

College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China

*Corresponding author: Deng-Shan Wang

Received  August 2021 Revised  January 2022 Published  July 2022 Early access  March 2022

Fund Project: This work was supported by the National Natural Science Foundation of China through grant Nos. 11971067 and 12065022, and the Fundamental Research Funds for the Central Universities under Grant No. 2020NTST22

The integrability, classification of traveling wave solutions and stability of exact solutions for the generalized Kaup-Boussinesq equation are studied by prolongation structure technique and linear stability analysis. Firstly, it is proved that the generalized Kaup-Boussinesq equation is completely integrable in sense of having Lax pair. Secondly, the complete classification of exact traveling wave solutions of the generalized Kaup-Boussinesq equation are given and a family of exact solutions are proposed. Finally, the stability of these exact solutions are investigated by linear stability analysis and dynamical evolutions, and some stable traveling wave solutions are found. It is shown that the results of linear stability analysis are in excellent agreement with the results from dynamical evolutions.

Citation: Ruizhi Gong, Yuren Shi, Deng-Shan Wang. Linear stability of exact solutions for the generalized Kaup-Boussinesq equation and their dynamical evolutions. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3355-3378. doi: 10.3934/dcds.2022018
References:
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998.
[2]

R. Balakrishnan and I. I. Satija, Solitons in Bose-Einstein condensates, Pramana J. Phys., 77 (2011), 929-947.  doi: 10.1007/s12043-011-0187-z.

[3]

C. Charlier and J. Lenells, The "good" Boussinesq equation: A Riemann-Hilbert approach, Indiana Univ. Math. J., to appear 2021, arXiv: 2003.02777, 48 pp.

[4]

C. Charlier, J. Lenells and D. S. Wang, The "good" Boussinesq equation: Long-time asymptotics, Analysis & PDE, to appear 2021, arXiv: 2003.04789, 34 pp.

[5]

T. Congy, S. K. Ivanov, A. M. Kamchatnov and N. Pavloff, Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion, Chaos, 27 (2017), Paper No. 083107, 12 pp. doi: 10.1063/1.4997052.

[6]

A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commu. Pure Appl. Anal., 11 (2012), 1397-1406.  doi: 10.3934/cpaa.2012.11.1397.

[7]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299.

[8]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal., 202 (2011), 133-175.  doi: 10.1007/s00205-011-0412-4.

[9]

P. DeiftC. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628.  doi: 10.1002/cpa.3160350502.

[10]

D. Dutykh and D. Ionescu-Kruse, Effects of vorticity on the travelling waves of some shallow water two-component systems, Discrete Contin. Dyn. Syst., 39 (2019), 5521-5541.  doi: 10.3934/dcds.2019225.

[11]

G. A. ElR. H. J. Grimshaw and A. M. Kamchatnov, Wave breaking and the generation of undular bores in an integrable shallow water system, Studies Appl. Math., 114 (2005), 395-411.  doi: 10.1111/j.0022-2526.2005.01560.x.

[12]

G. A. ElR. H. J. Grimshaw and M. V. Pavlov, Integrable shallow-water equations and undular bores, Studies Appl. Math., 106 (2001), 157-186.  doi: 10.1111/1467-9590.00163.

[13]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.  doi: 10.1103/PhysRevLett.19.1095.

[14]

V. S. Gerdjikov, E. V. Doktorov and J. Yang, Adiabatic interaction of N ultrashort solitons: Universality of the complex Toda chain model, Phys. Rev. E, 64 (2001), Paper No. 056617, 15 pp. doi: 10.1103/PhysRevE.64.056617.

[15]

J. Haberlin and T. Lyons, Solitons of shallow-water models from energy-dependent spectral problems, Eur. Phys. J. Plus, 133 (2018), 16.  doi: 10.1140/epjp/i2018-11848-8.

[16]

M. A. Helal, Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics, Chaos, Solitons and Fractals, 13 (2002), 1917-1929.  doi: 10.1016/S0960-0779(01)00189-8.

[17]

R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396.  doi: 10.1016/j.wavemoti.2009.06.012.

[18]

R. Ivanov and T. Lyons, Integrable models for shallow water with energy dependent spectral problems, J. Nonlinear Math. Phys., 19 (2012), Paper No. 1240008, 17 pp. doi: 10.1142/S1402925112400086.

[19]

M. Jaulent and C. Jean, The inverse $s$-wave scattering problem for a class of potentials depending on energy, Comm. Math. Phys., 28 (1972), 177-220.  doi: 10.1007/BF01645775.

[20]

A. M. Kamchatnov, Nonlinear Periodic Waves and Their Modulations: An Introductory Course, World Scientific Publishing, 2000. doi: 10.1142/9789812792259.

[21]

A. M. KamchatnovR. A. Kraenkel and B. A. Umarov, Asymptotic soliton train solutions of Kaup-Boussinesq equations, Wave Motion, 38 (2003), 355-365.  doi: 10.1016/S0165-2125(03)00062-3.

[22]

D. J. Kaup, A higher-order water-wave equation and the method for solving it, Progr. Theoret. Phys., 54 (1975), 396-408.  doi: 10.1143/PTP.54.396.

[23]

V. B. Matveev and M. I. Yavor, Almost periodic N-soliton solutions of the nonlinear hydrodynamic Kaup equation, Ann. Inst. H. Poincar'e Sect. A, 31 (1979), 25-41. 

[24]

K. NishinaryK. Abe and J. Satsuma, A new-type of soliton behavior in a two dimensional plasma system, J. Phys. Soc. Japan, 62 (1993), 2021-2029.  doi: 10.1143/JPSJ.62.2021.

[25]

M. Pavlov, Integrable systems and metrics of constant curvature, J. Nonlinear Math. Phys., 9 (2002), 173-191.  doi: 10.2991/jnmp.2002.9.s1.15.

[26]

D. H. Sattinger and J. Szmigielski, A Riemann-Hilbert problem for an energy dependent Schrödinger operator, Inverse Problems, 12 (1996), 1003-1025.  doi: 10.1088/0266-5611/12/6/014.

[27]

K. Singla and M. Rana, Exact solutions and conservation laws of multi Kaup-Boussinesq system with fractional order, Analysis Math. Phys., 11 (2021), Paper No. 30, 15 pp. doi: 10.1007/s13324-020-00467-z.

[28]

H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys., 16 (1975), 1-7.  doi: 10.1063/1.522396.

[29]

D.-S. Wang, Integrability of the coupled KdV equations derived from two-layer fluids: Prolongation structures and Miura transformations, Nonlinear Anal., 73 (2010), 270-281.  doi: 10.1016/j.na.2010.03.021.

[30]

D.-S. Wang and J. Liu, Integrability aspects of some two-component KdV systems, Appl. Math. Lett., 79 (2018), 211-219.  doi: 10.1016/j.aml.2017.12.018.

[31]

B. WangZ. Zhang and B. Li, Soliton molecules and some hybrid solutions for the nonlinear Schrödinger equation, Chin. Phys. Lett., 37 (2020), 030501.  doi: 10.1088/0256-307X/37/3/030501.

[32]

J. Yang, Nonlinear Waves in Integrable and Non-Integrable Systems, Society for Industrial and Applied Mathematics, 2010. doi: 10.1137/1.9780898719680.

[33]

Y. S. Zhang and J. S. He, Bound-state soliton solutions of the nonlinear Schrödinger equation and their asymmetric decompositions, Chin. Phys. Lett., 36 (2019), 030201.  doi: 10.1088/0256-307X/36/3/030201.

[34]

V. E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys.-JETP, 38 (1974), 108-110; translated from Zh. Eksp. Teor. Fiz., 65 (1973), 219-225.

[35]

J. ZhouL. Tian and X. Fan, Solitary-wave solutions to a dual equation of the Kaup-Boussinesq system, Nonlinear Analy.: Real World Appl., 11 (2010), 3229-3235.  doi: 10.1016/j.nonrwa.2009.11.017.

show all references

References:
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998.
[2]

R. Balakrishnan and I. I. Satija, Solitons in Bose-Einstein condensates, Pramana J. Phys., 77 (2011), 929-947.  doi: 10.1007/s12043-011-0187-z.

[3]

C. Charlier and J. Lenells, The "good" Boussinesq equation: A Riemann-Hilbert approach, Indiana Univ. Math. J., to appear 2021, arXiv: 2003.02777, 48 pp.

[4]

C. Charlier, J. Lenells and D. S. Wang, The "good" Boussinesq equation: Long-time asymptotics, Analysis & PDE, to appear 2021, arXiv: 2003.04789, 34 pp.

[5]

T. Congy, S. K. Ivanov, A. M. Kamchatnov and N. Pavloff, Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion, Chaos, 27 (2017), Paper No. 083107, 12 pp. doi: 10.1063/1.4997052.

[6]

A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commu. Pure Appl. Anal., 11 (2012), 1397-1406.  doi: 10.3934/cpaa.2012.11.1397.

[7]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299.

[8]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal., 202 (2011), 133-175.  doi: 10.1007/s00205-011-0412-4.

[9]

P. DeiftC. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628.  doi: 10.1002/cpa.3160350502.

[10]

D. Dutykh and D. Ionescu-Kruse, Effects of vorticity on the travelling waves of some shallow water two-component systems, Discrete Contin. Dyn. Syst., 39 (2019), 5521-5541.  doi: 10.3934/dcds.2019225.

[11]

G. A. ElR. H. J. Grimshaw and A. M. Kamchatnov, Wave breaking and the generation of undular bores in an integrable shallow water system, Studies Appl. Math., 114 (2005), 395-411.  doi: 10.1111/j.0022-2526.2005.01560.x.

[12]

G. A. ElR. H. J. Grimshaw and M. V. Pavlov, Integrable shallow-water equations and undular bores, Studies Appl. Math., 106 (2001), 157-186.  doi: 10.1111/1467-9590.00163.

[13]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.  doi: 10.1103/PhysRevLett.19.1095.

[14]

V. S. Gerdjikov, E. V. Doktorov and J. Yang, Adiabatic interaction of N ultrashort solitons: Universality of the complex Toda chain model, Phys. Rev. E, 64 (2001), Paper No. 056617, 15 pp. doi: 10.1103/PhysRevE.64.056617.

[15]

J. Haberlin and T. Lyons, Solitons of shallow-water models from energy-dependent spectral problems, Eur. Phys. J. Plus, 133 (2018), 16.  doi: 10.1140/epjp/i2018-11848-8.

[16]

M. A. Helal, Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics, Chaos, Solitons and Fractals, 13 (2002), 1917-1929.  doi: 10.1016/S0960-0779(01)00189-8.

[17]

R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396.  doi: 10.1016/j.wavemoti.2009.06.012.

[18]

R. Ivanov and T. Lyons, Integrable models for shallow water with energy dependent spectral problems, J. Nonlinear Math. Phys., 19 (2012), Paper No. 1240008, 17 pp. doi: 10.1142/S1402925112400086.

[19]

M. Jaulent and C. Jean, The inverse $s$-wave scattering problem for a class of potentials depending on energy, Comm. Math. Phys., 28 (1972), 177-220.  doi: 10.1007/BF01645775.

[20]

A. M. Kamchatnov, Nonlinear Periodic Waves and Their Modulations: An Introductory Course, World Scientific Publishing, 2000. doi: 10.1142/9789812792259.

[21]

A. M. KamchatnovR. A. Kraenkel and B. A. Umarov, Asymptotic soliton train solutions of Kaup-Boussinesq equations, Wave Motion, 38 (2003), 355-365.  doi: 10.1016/S0165-2125(03)00062-3.

[22]

D. J. Kaup, A higher-order water-wave equation and the method for solving it, Progr. Theoret. Phys., 54 (1975), 396-408.  doi: 10.1143/PTP.54.396.

[23]

V. B. Matveev and M. I. Yavor, Almost periodic N-soliton solutions of the nonlinear hydrodynamic Kaup equation, Ann. Inst. H. Poincar'e Sect. A, 31 (1979), 25-41. 

[24]

K. NishinaryK. Abe and J. Satsuma, A new-type of soliton behavior in a two dimensional plasma system, J. Phys. Soc. Japan, 62 (1993), 2021-2029.  doi: 10.1143/JPSJ.62.2021.

[25]

M. Pavlov, Integrable systems and metrics of constant curvature, J. Nonlinear Math. Phys., 9 (2002), 173-191.  doi: 10.2991/jnmp.2002.9.s1.15.

[26]

D. H. Sattinger and J. Szmigielski, A Riemann-Hilbert problem for an energy dependent Schrödinger operator, Inverse Problems, 12 (1996), 1003-1025.  doi: 10.1088/0266-5611/12/6/014.

[27]

K. Singla and M. Rana, Exact solutions and conservation laws of multi Kaup-Boussinesq system with fractional order, Analysis Math. Phys., 11 (2021), Paper No. 30, 15 pp. doi: 10.1007/s13324-020-00467-z.

[28]

H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys., 16 (1975), 1-7.  doi: 10.1063/1.522396.

[29]

D.-S. Wang, Integrability of the coupled KdV equations derived from two-layer fluids: Prolongation structures and Miura transformations, Nonlinear Anal., 73 (2010), 270-281.  doi: 10.1016/j.na.2010.03.021.

[30]

D.-S. Wang and J. Liu, Integrability aspects of some two-component KdV systems, Appl. Math. Lett., 79 (2018), 211-219.  doi: 10.1016/j.aml.2017.12.018.

[31]

B. WangZ. Zhang and B. Li, Soliton molecules and some hybrid solutions for the nonlinear Schrödinger equation, Chin. Phys. Lett., 37 (2020), 030501.  doi: 10.1088/0256-307X/37/3/030501.

[32]

J. Yang, Nonlinear Waves in Integrable and Non-Integrable Systems, Society for Industrial and Applied Mathematics, 2010. doi: 10.1137/1.9780898719680.

[33]

Y. S. Zhang and J. S. He, Bound-state soliton solutions of the nonlinear Schrödinger equation and their asymmetric decompositions, Chin. Phys. Lett., 36 (2019), 030201.  doi: 10.1088/0256-307X/36/3/030201.

[34]

V. E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys.-JETP, 38 (1974), 108-110; translated from Zh. Eksp. Teor. Fiz., 65 (1973), 219-225.

[35]

J. ZhouL. Tian and X. Fan, Solitary-wave solutions to a dual equation of the Kaup-Boussinesq system, Nonlinear Analy.: Real World Appl., 11 (2010), 3229-3235.  doi: 10.1016/j.nonrwa.2009.11.017.

Figure 1.  The "Potential" curve of function $ Q(u) $ for the four roots are $ u_{4}< u_{3}< u_{2}< u_{1} $ and $ \sigma>0 $
Figure 2.  The "Potential" curve of function $ Q(u) $ for the four roots are $ u_{4}< u_{3}< u_{2}< u_{1} $ and $ \sigma<0 $
Figure 3.  The Jacobi elliptic function periodic wave solution of the generalized KB equation (2) at $ t = 0 $ based on the traveling wave solution (21): (a) the $ u $-component and (b) the $ v $-component. Here the parameters are chosen to be $ u_{1} = 1, u_{2} = 0, u_{3} = -1, u_{4} = -2, \kappa = -1, \xi_0 = 0 $, $ \sigma = 1 $
Figure 4.  The solitary wave solution of the generalized KB equation (2) at $ t = 0 $ based on the traveling wave solution (22): ($ a $) the $ u $-component of anti-bell type and ($ b $) the $ v $-component of two-hump type, where the parameters here are chosen to be $ u_1 = u_2 = 2 $, $ u_3 = 0, u_4 = -2, \kappa = -1, \xi_0 = 0 $
Figure 5.  The solitary wave solution of the generalized KB equation (2) at $ t = 0 $ based on the traveling wave solution (24): ($ a $) the $ u $-component of kink type and ($ b $) the $ v $-component of two-hump and anti-bell type. Here the parameters are chosen to be $ u_1 = u_2 = 8 $, $ u_3 = 0, u_4 = -8, \kappa = -1, \xi_0 = 0 $. (c) The "Potential" curve $ Q(u) $ in case of the roots $ u_{1} = u_{2} $
Figure 6.  The algebraic solitary wave solution of the generalized KB equation (2) at $ t = 0 $ based on the algebraic solitary wave solution (26): ($ a $) the $ u $-component of kink type and ($ b $) the $ v $-component of two-hump and anti-bell type, where the parameters are chosen to be $ u_1 = u_2 = u_3 = 5 $, $ u_4 = -10, \kappa = -1, \xi_0 = 0 $. (c) The "Potential" curve $ Q(u) $ in case of the roots $ u_{1} = u_{2} = u_{3}\neq u_{4} $
Figure 7.  The solitary wave solution of the generalized KB equation (2) at $ t = 0 $ based on the traveling wave solution (29): ($ a $) the $ u $-component of kink type and ($ b $) the $ v $-component of two-hump and anti-bell type, where the parameters are chosen to be $ u_1 = 8, u_2 = 0 $, $ u_3 = u_4 = -8, \kappa = -1, \xi_0 = 0 $. (c) The "Potential" curve $ Q(u) $ in case of the roots $ u_{3} = u_{4} $
Figure 8.  The algebraic solitary wave solution of the generalized KB equation (2) at $ t = 0 $ based on the algebraic solitary wave solution (30): ($ a $) the $ u $-component of kink type and ($ b $) the $ v $-component of two-hump and anti-bell type, where the parameters are chosen to be $ u_1 = 4 $, $ u_{2} = u_{3} = u_{4} = -2, \kappa = -1, \xi_0 = 0 $. (c) The "Potential" curve $ Q(u) $ in case of the roots $ u_{2} = u_{3} = u_{4}\neq u_{1} $
Figure 9.  The trigonometric function periodic wave solution of the generalized KB equation (2) at $ t = 0 $ based on the traveling wave solution (32): (a) the $ u $-component and (b) the $ v $-component, where the parameters are chosen to be $ u_1 = 4, u_2 = 2 $, $ u_3 = u_4 = -2, \kappa = -1, \xi_0 = 0 $. (c) The "Potential" curve $ Q(u) $ in case of the roots $ u_{3} = u_{4} $
Figure 10.  The soliton solution of the generalized KB equation (2) at $ t = 0 $ based on the traveling wave solution (33): ($ a $) the $ u $-component is a bell type bright soliton and ($ b $) the $ v $-component is two-hump and anti-bell type bright soliton with nonvanishing boundary, where the parameters are chosen to be $ u_1 = 4, u_2 = u_3 = 0, u_4 = -2, \kappa = -1, \xi_0 = 0 $. (c) The "Potential" curve $ Q(u) $ in case of the roots $ u_{2} = u_{3} $
Figure 11.  The algebraic soliton solution of the generalized KB equation (2) at $ t = 0 $ based on the algebraic solitary wave solution (34): ($ a $) the $ u $-component is a bell type bright soliton and ($ b $) the $ v $-component is a two-hump and anti-bell type bright soliton, where the parameters are chosen to be $ u_1 = 4, u_2 = u_3 = u_4 = -2, \kappa = -1, \xi_0 = 0 $. (c) The "Potential" curve $ Q(u) $ in case of the roots $ u_{2} = u_{3} = u_{4} \neq u_{1} $
Figure 12.  The distribution of spectral points for the linear eigenvalue problem (42) with parameter $ \sigma = 1 $ (a) and $ \sigma = -1 $ (b), respectively
Figure 13.  (a) The distribution of eigenvalue spectrum and (b) the eigenvectors of the soliton solution (33) in linear eigenvalue problem (39), where the red and blue represent the real parts of the eigenvector $ (u_1, v_1)^{T} $, and the green and black represent the imaginary parts of the eigenvector $ (u_1, v_1)^{T} $
Figure 14.  ($ a_1 $)-($ b_1 $) The initial profiles (at time $ t = 0 $) and ($ a_2 $)-($ b_2 $) time evolutions (at time $ t = 30 $) of the soliton solution (33), where the parameters are $ u_1 = 4, u_2 = u_3 = 0, u_4 = -2, \kappa = -1, \xi_0 = 0 $ and the 0.2% Gaussian white noise is added
Figure 15.  (a) The distribution of eigenvalue spectrum and (b) the eigenvectors of the algebraic soliton solution (34) in linear eigenvalue problem (39), where the red and blue represent the real parts of the eigenvector $ (u_1, v_1)^{T} $, and the green and black represent the imaginary parts of the eigenvector $ (u_1, v_1)^{T} $
Figure 16.  ($ a_1 $)-($ b_1 $) The initial profiles, ($ a_2 $)-($ b_2 $) time evolutions (at time $ t = 0.2 $) and ($ a_3 $)-($ b_3 $) time evolutions (at time $ t = 0.4 $) of the algebraic soliton solution (34), where the parameters are $ u_1 = 4, u_2 = u_3 = u_4 = -2, \kappa = -1, \xi_0 = 0 $ and the 0.2% Gaussian White noise is added
Table 1.  Classification of the solutions on interval $ [u_{4}, u_{3}] $
Classification of exact solutions on the interval $ [u_{4}, u_{3}] $
Distribution of the roots The roots The Exact Solutions Properties of the solution
Four different real roots $u_1, u_2, u_3, u_4 $ $u=\frac{u_{4}(u_{1}-u_{3})+u_{1}(u_{3}-u_{4}){\rm sn}^{2}(W_2, m_2)}{u_{1}-u_{3}+(u_{3}-u_{4}){\rm sn}^{2}(W_2, m_2)}$ Periodic solution
Two simple real roots and one double real root $u_1=u_2, u_3, u_4 $ $u=\frac{u_{4}(u_{2}-u_{3})+u_{2}(u_{3}-u_{4}){\rm sin}^{2}(W_{23})}{u_{2}-u_{3}+(u_{3}-u_{4}){\rm sin}^{2}(W_{23})}$ Periodic solution
$u_1, u_2=u_3, u_4 $ $u=\frac{u_{4}(u_{1}-u_{3})+u_{1}(u_{3}-u_{4}){\rm tanh}^{2}(W_{24})}{u_{1}-u_{3}+(u_{3}-u_{4}){\rm tanh}^{2}(W_{24})}$ Anti-bell soliton solution
$u_1, u_2, u_3=u_4 $ $u=u_4$ Constant solution
One simple real root and one triple real root $u_1=u_2=u_3, u_4 $ $u=\frac{u_{3}(u_{3}-u_{4})^2\xi^2+4u_{4}}{(u_{3}-u_{4})^2\xi^2+4}$ Algebraic soliton solution
$u_1, u_2=u_3=u_4 $ $u=u_4$ Constant solution
Two double real roots $u_1=u_2, u_3=u_4 $ $u=u_4$ Constant solution
One quadruple real root $u_1=u_2=u_3=u_4 $ $u=u_4$ Constant solution
Classification of exact solutions on the interval $ [u_{4}, u_{3}] $
Distribution of the roots The roots The Exact Solutions Properties of the solution
Four different real roots $u_1, u_2, u_3, u_4 $ $u=\frac{u_{4}(u_{1}-u_{3})+u_{1}(u_{3}-u_{4}){\rm sn}^{2}(W_2, m_2)}{u_{1}-u_{3}+(u_{3}-u_{4}){\rm sn}^{2}(W_2, m_2)}$ Periodic solution
Two simple real roots and one double real root $u_1=u_2, u_3, u_4 $ $u=\frac{u_{4}(u_{2}-u_{3})+u_{2}(u_{3}-u_{4}){\rm sin}^{2}(W_{23})}{u_{2}-u_{3}+(u_{3}-u_{4}){\rm sin}^{2}(W_{23})}$ Periodic solution
$u_1, u_2=u_3, u_4 $ $u=\frac{u_{4}(u_{1}-u_{3})+u_{1}(u_{3}-u_{4}){\rm tanh}^{2}(W_{24})}{u_{1}-u_{3}+(u_{3}-u_{4}){\rm tanh}^{2}(W_{24})}$ Anti-bell soliton solution
$u_1, u_2, u_3=u_4 $ $u=u_4$ Constant solution
One simple real root and one triple real root $u_1=u_2=u_3, u_4 $ $u=\frac{u_{3}(u_{3}-u_{4})^2\xi^2+4u_{4}}{(u_{3}-u_{4})^2\xi^2+4}$ Algebraic soliton solution
$u_1, u_2=u_3=u_4 $ $u=u_4$ Constant solution
Two double real roots $u_1=u_2, u_3=u_4 $ $u=u_4$ Constant solution
One quadruple real root $u_1=u_2=u_3=u_4 $ $u=u_4$ Constant solution
Table 2.  The dispersion relations under certain parameters
Cases Parameter selection Dispersion relation
A $ u_0=0 $, $ v_0=0 $ $ \omega^2=-\frac{1}{4}\sigma k^4 $
B $ u_0=0 $, $ v_0\neq0 $ $ \omega^2=v_0k^2-\frac{1}{4}\sigma k^4 $
C $ u_0\neq0 $, $ v_0\neq0 $, $ \kappa=-\frac{3}{2} $ $ \omega^2=v_0k^2+\frac{2u_0 v_1}{u_1}k^2-\frac{1}{4}\sigma k^4-\frac{3}{2}u_0^2k^2 $
D $ u_0\neq0 $, $ v_0\neq0 $, $ \kappa=-\frac{1}{2} $ $ \omega^2=v_0k^2+\frac{u_0 v_1}{u_1}k^2-\frac{1}{4}\sigma k^4-u_0 k \omega $
Cases Parameter selection Dispersion relation
A $ u_0=0 $, $ v_0=0 $ $ \omega^2=-\frac{1}{4}\sigma k^4 $
B $ u_0=0 $, $ v_0\neq0 $ $ \omega^2=v_0k^2-\frac{1}{4}\sigma k^4 $
C $ u_0\neq0 $, $ v_0\neq0 $, $ \kappa=-\frac{3}{2} $ $ \omega^2=v_0k^2+\frac{2u_0 v_1}{u_1}k^2-\frac{1}{4}\sigma k^4-\frac{3}{2}u_0^2k^2 $
D $ u_0\neq0 $, $ v_0\neq0 $, $ \kappa=-\frac{1}{2} $ $ \omega^2=v_0k^2+\frac{u_0 v_1}{u_1}k^2-\frac{1}{4}\sigma k^4-u_0 k \omega $
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