American Institute of Mathematical Sciences

October  2020, 13(10): 2917-2926. doi: 10.3934/dcdss.2020217

Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

 1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China 2 African Institute for Mathematical Sciences, Muizenberg, Cape Town, South Africa 3 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 4 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Yixia Shi

Received  February 2019 Revised  June 2019 Published  October 2020 Early access  December 2019

Fund Project: Part of this work was done during L. Zhang's research visit to African Institute for Mathematical Sciences South Africa

In this paper, we consider the traveling wave solutions of the one-dimensional Serre-Green-Naghdi (SGN) equations which are proposed to model dispersive nonlinear long water waves in a one-layer flow over flat bottom. We decouple the traveling wave system of SGN equations into two ordinary differential equations. By studying the bifurcations and phase portraits of each bifurcation set of one equation, we obtain the exact traveling wave solutions of SGN equations for the variable $u(x, t)$ which represents average horizontal velocity of water wave. For the compacted orbits intersecting with the singular line in phase plane, we obtained two families of solutions: a family of smooth traveling wave solutions including periodic wave solutions and solitary wave solutions, and a family of compacted singular solutions which have continuous first-order derivative but discontinuous second-order derivative.

Citation: Lijun Zhang, Yixia Shi, Maoan Han. Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2917-2926. doi: 10.3934/dcdss.2020217
References:
 [1] R. Aitbayev, P. W. Bates and H. Lu, Mathematical studies of Poisson-Nernst-Planck model for membrane channels: Finite ion size effects without electroneutrality boundary conditions, J. Comp. Appl. Math., 362 (2019), 510-527.  doi: 10.1016/j.cam.2018.10.037. [2] P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes, Eur. J. Mech. B Fluids, 30 (2011), 589-597.  doi: 10.1016/j.euromechflu.2011.02.005. [3] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, New York, 1971. [4] S. N. Chow and J. K. Hale, Method of Bifurcation Theory, Springer-Verlag, New York, 1981. [5] D. Clamond, D. Dutykh and A. Galligo, Algebraic method for constructing singular steady solitary waves: A case study, Proc. A., 472 (2016), 18 pp. doi: 10.1098/rspa.2016.0194. [6] D. Clamond, D. Dutykh and D. Mitsotakis, Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics, Commun. Nonlinear Sci. Numer. Simul., 45 (2017), 245-257.  doi: 10.1016/j.cnsns.2016.10.009. [7] S. Deng, B. Guo and T. Wang, Some traveling wave solitons of the Green-Naghdi system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 575-585.  doi: 10.1142/S0218127411028623. [8] D. Dutykh, M. Hoefer and D. Mitsotakis, Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations, Theor. Comput. Fluid Dyn., 32 (2018), 371-397.  doi: 10.1007/s00162-018-0455-3. [9] N Favrie and S Gavrilyuk, A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves, Nonlinearity, 30 (2017), 2718-2736.  doi: 10.1088/1361-6544/aa712d. [10] A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves, Proc. Roy. Soc. London Ser. A, 338 (1974), 43-55.  doi: 10.1098/rspa.1974.0072. [11] A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.  doi: 10.1017/S0022112076002425. [12] M. A. Han, L. J. Zhang, Y. Wang and C. M. Khalique, The effects of the sigular lines on the traveling wave solutios of modified dispersive water wave equation, Nonlinear Anal. Real World Appl., 47 (2019), 236-250.  doi: 10.1016/j.nonrwa.2018.10.012. [13] C. H. He, Y. N. Tang and J. L. Ma, New interaction solutions for the (3+1)-dimensional Jimbo-Miwa equation, Compu. Math. Appl., 76 (2018), 2141-2147.  doi: 10.1016/j.camwa.2018.08.012. [14] H. Kalisch, Z. Khorsand and D. Mitsotakis, Mechanical balance laws for fully nonlinear and weakly dispersive water waves, Phys. D, 333 (2016), 243-253.  doi: 10.1016/j.physd.2016.03.001. [15] D. Lannes and F. Marche, A new class of fully nonlinear and weakly dispersive Green-Naghdi models for effcient 2D simulations, J. Comput. Phys., 282 (2015), 238-268.  doi: 10.1016/j.jcp.2014.11.016. [16] J. Li, Singular nonlinear travelling wave equations: Bifurcations and exact solutions, Science Press, 2013. [17] F. S. Li and G. W. Du, General Energy Decay for a Degenerate Viscoelastic Petrovsky-Type Plate Equation with Boundary Feedback, J. Appl. Anal. Compu., 8 (2018), 390-401. [18] F. S. Li and Q. Y. Gao, Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Compu., 274 (2016), 383-392.  doi: 10.1016/j.amc.2015.11.018. [19] F. S. Li and J. L. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions, J. Math. Anal. Appl., 385 (2012), 1005-1014.  doi: 10.1016/j.jmaa.2011.07.018. [20] X. Y. Li and Q. L. Zhao, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys. 121 (2017), 123–137. doi: 10.1016/j.geomphys.2017.07.010. [21] Y. Liu, H. H. Dong and Y. Zhang, Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows, Anal. Math. Phys., 9 (2019), 465-481.  doi: 10.1007/s13324-018-0209-9. [22] H. Liu and L. Zhang, Symmetry reductions and exact solutions to systems of nonlinear partial differential equations, Phys. Scr., 94 (2019), 015202. [23] C. N. Lu, L. Y. Xie and H. W. Yang, Analysis of Lie symmetries with conservation laws and solutions for the generalized (3+1)-dimensional time fractional Camassa-Holm-Kadomtsev-Petviashvili equation, Compu. Math. Appl., 77 (2019), 3154-3171.  doi: 10.1016/j.camwa.2019.01.022. [24] S. Popinet, A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations, J. Compu. Phys., 302 (2015), 336-358.  doi: 10.1016/j.jcp.2015.09.009. [25] L. J. Zhang and C. M. Khalique, Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 759-772.  doi: 10.3934/dcdss.2018048. [26] D. Mitsotakis, D. Dutykh, A. Assylbekuly and D. Zhakebayev, On weakly singular and fully nonlinear travelling shallow capillarygravity waves in the critical regime, Phys. Lett. A, 381 (2017), 1719-1726.  doi: 10.1016/j.physleta.2017.03.041. [27] L. J. Zhang, Y. Wang, C. M. Khalique and Y. Z. Bai, Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation, J. Appl. Anal. Comput., 8 (2018), 1938-1958. [28] J. M. Zhang, L. J. Zhang and Y. Z. Bai, Stability and bifurcation analysis on a predator prey system with the weak allee effect, Mathematics, 7 (2019), 1-15.  doi: 10.3390/math7050432. [29] H. Q. Zhao and W. X. Ma, Mixed lumpkink solutions to the KP equation, Compu. Math. Appl., 74 (2017), 1399-1405.  doi: 10.1016/j.camwa.2017.06.034. [30] X. X. Zheng, Y. D. Shang and X. M. Peng, Orbital stability of solitary waves of the coupled Klein-Gordon-Zakharov equations, Math. Methods Appl. Sci., 40 (2017), 2623-2633.  doi: 10.1002/mma.4187.

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References:
 [1] R. Aitbayev, P. W. Bates and H. Lu, Mathematical studies of Poisson-Nernst-Planck model for membrane channels: Finite ion size effects without electroneutrality boundary conditions, J. Comp. Appl. Math., 362 (2019), 510-527.  doi: 10.1016/j.cam.2018.10.037. [2] P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes, Eur. J. Mech. B Fluids, 30 (2011), 589-597.  doi: 10.1016/j.euromechflu.2011.02.005. [3] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, New York, 1971. [4] S. N. Chow and J. K. Hale, Method of Bifurcation Theory, Springer-Verlag, New York, 1981. [5] D. Clamond, D. Dutykh and A. Galligo, Algebraic method for constructing singular steady solitary waves: A case study, Proc. A., 472 (2016), 18 pp. doi: 10.1098/rspa.2016.0194. [6] D. Clamond, D. Dutykh and D. Mitsotakis, Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics, Commun. Nonlinear Sci. Numer. Simul., 45 (2017), 245-257.  doi: 10.1016/j.cnsns.2016.10.009. [7] S. Deng, B. Guo and T. Wang, Some traveling wave solitons of the Green-Naghdi system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 575-585.  doi: 10.1142/S0218127411028623. [8] D. Dutykh, M. Hoefer and D. Mitsotakis, Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations, Theor. Comput. Fluid Dyn., 32 (2018), 371-397.  doi: 10.1007/s00162-018-0455-3. [9] N Favrie and S Gavrilyuk, A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves, Nonlinearity, 30 (2017), 2718-2736.  doi: 10.1088/1361-6544/aa712d. [10] A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves, Proc. Roy. Soc. London Ser. A, 338 (1974), 43-55.  doi: 10.1098/rspa.1974.0072. [11] A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.  doi: 10.1017/S0022112076002425. [12] M. A. Han, L. J. Zhang, Y. Wang and C. M. Khalique, The effects of the sigular lines on the traveling wave solutios of modified dispersive water wave equation, Nonlinear Anal. Real World Appl., 47 (2019), 236-250.  doi: 10.1016/j.nonrwa.2018.10.012. [13] C. H. He, Y. N. Tang and J. L. Ma, New interaction solutions for the (3+1)-dimensional Jimbo-Miwa equation, Compu. Math. Appl., 76 (2018), 2141-2147.  doi: 10.1016/j.camwa.2018.08.012. [14] H. Kalisch, Z. Khorsand and D. Mitsotakis, Mechanical balance laws for fully nonlinear and weakly dispersive water waves, Phys. D, 333 (2016), 243-253.  doi: 10.1016/j.physd.2016.03.001. [15] D. Lannes and F. Marche, A new class of fully nonlinear and weakly dispersive Green-Naghdi models for effcient 2D simulations, J. Comput. Phys., 282 (2015), 238-268.  doi: 10.1016/j.jcp.2014.11.016. [16] J. Li, Singular nonlinear travelling wave equations: Bifurcations and exact solutions, Science Press, 2013. [17] F. S. Li and G. W. Du, General Energy Decay for a Degenerate Viscoelastic Petrovsky-Type Plate Equation with Boundary Feedback, J. Appl. Anal. Compu., 8 (2018), 390-401. [18] F. S. Li and Q. Y. Gao, Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Compu., 274 (2016), 383-392.  doi: 10.1016/j.amc.2015.11.018. [19] F. S. Li and J. L. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions, J. Math. Anal. Appl., 385 (2012), 1005-1014.  doi: 10.1016/j.jmaa.2011.07.018. [20] X. Y. Li and Q. L. Zhao, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys. 121 (2017), 123–137. doi: 10.1016/j.geomphys.2017.07.010. [21] Y. Liu, H. H. Dong and Y. Zhang, Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows, Anal. Math. Phys., 9 (2019), 465-481.  doi: 10.1007/s13324-018-0209-9. [22] H. Liu and L. Zhang, Symmetry reductions and exact solutions to systems of nonlinear partial differential equations, Phys. Scr., 94 (2019), 015202. [23] C. N. Lu, L. Y. Xie and H. W. Yang, Analysis of Lie symmetries with conservation laws and solutions for the generalized (3+1)-dimensional time fractional Camassa-Holm-Kadomtsev-Petviashvili equation, Compu. Math. Appl., 77 (2019), 3154-3171.  doi: 10.1016/j.camwa.2019.01.022. [24] S. Popinet, A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations, J. Compu. Phys., 302 (2015), 336-358.  doi: 10.1016/j.jcp.2015.09.009. [25] L. J. Zhang and C. M. Khalique, Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 759-772.  doi: 10.3934/dcdss.2018048. [26] D. Mitsotakis, D. Dutykh, A. Assylbekuly and D. Zhakebayev, On weakly singular and fully nonlinear travelling shallow capillarygravity waves in the critical regime, Phys. Lett. A, 381 (2017), 1719-1726.  doi: 10.1016/j.physleta.2017.03.041. [27] L. J. Zhang, Y. Wang, C. M. Khalique and Y. Z. Bai, Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation, J. Appl. Anal. Comput., 8 (2018), 1938-1958. [28] J. M. Zhang, L. J. Zhang and Y. Z. Bai, Stability and bifurcation analysis on a predator prey system with the weak allee effect, Mathematics, 7 (2019), 1-15.  doi: 10.3390/math7050432. [29] H. Q. Zhao and W. X. Ma, Mixed lumpkink solutions to the KP equation, Compu. Math. Appl., 74 (2017), 1399-1405.  doi: 10.1016/j.camwa.2017.06.034. [30] X. X. Zheng, Y. D. Shang and X. M. Peng, Orbital stability of solitary waves of the coupled Klein-Gordon-Zakharov equations, Math. Methods Appl. Sci., 40 (2017), 2623-2633.  doi: 10.1002/mma.4187.
The phase portraits of system (9) in each bifurcation set for $A>0$. (1) $c>\frac{B}{A}-\frac{3}{2}\sqrt{Ag}$; (2) $c = \frac{B}{A}-\frac{3}{2}\sqrt{Ag}$; (3) $c<\frac{B}{A}-\frac{3}{2}\sqrt{Ag}$
Phase orbit of (12) and the corresponding bounded traveling wave solutions. (1) a compact orbit of (12) intersecting with the singular line; (2) a smooth periodic traveling wave solution; (3) compacton
Phase orbit of (12) and the corresponding bounded traveling wave solutions. (1) a compact orbit of (12) intersecting with the singular line; (2) a smooth periodic traveling wave solution; (3) compacton
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