Traveling wave solutions of a highly nonlinear shallow water equation

Previous Article
Wave breaking of periodic solutions to the Fornberg-Whitham equation
DCDS Home
This Issue
Next Article
On the universality of the incompressible Euler equation on compact manifolds

March 2018, 38(3): 1567-1604. doi: 10.3934/dcds.2018065

Traveling wave solutions of a highly nonlinear shallow water equation

Anna Geyer ^1,, and Ronald Quirchmayr ^2,

1.	Delft University of Technology, Delft Institute of Applied Mathematics, Faculty of EEMCS, Mekelweg 4,2628 CD Delft, The Netherlands
2.	KTH Royal Institute of Technology, Department of Mathematics, Lindstedtsvägen 25,100 44 Stockholm, Sweden

* Corresponding author: Anna Geyer

A. Geyer acknowledges the support of the Austrian Science Fund (FWF) project J3452 "Dynamical Systems Methods in Hydrodynamics". R. Quirchmayr acknowledges the support of the Austrian Science Fund (FWF), Grant W1245.

Received August 2017 Revised October 2017 Published December 2017

Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.

Keywords: Shallow water equation, traveling waves, phase plane analysis.

Mathematics Subject Classification: Primary: 35Q35, 58F17; Secondary: 34C37.

Citation: Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065

References:

[1]	T. B. Benjamin and J. E. Feir, The disintegration of wavetrains in deep water, J. Fluid Mech., 27 (1967), 417-430.
[2]	J. L. Bona, P. E. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 411 (1987), 395-412. doi: 10.1098/rspa.1987.0073.
[3]	G. Brüll, M. Ehrnström, A. Geyer and L. Pei, Symmetric solutions of evolutionary partial differential equations, Nonlinearity, 30 (2017), 3932-3950.
[4]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[5]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.
[6]	A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. : Oceans, 117 (2012), C05029. doi: 10.1029/2012JC007879.
[7]	A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. doi: 10.1175/JPO-D-13-0174.1.
[8]	A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, 81 SIAM Philadelphia, 2011.
[9]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.
[10]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.
[11]	A. Constantin and W. Strauss, Stability of peakons, Commun. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.
[12]	A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x.
[13]	A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.
[14]	B. Deconinck and T. Kapitula, The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. Sect. A Gen. At. Solid State Phys., 374 (2010), 4018-4022. doi: 10.1016/j.physleta.2010.08.007.
[15]	A. Degasperis and M. Procesi, Asymptotic integrability, In A. Degasperis and G. Gaeta, editors, Symmetry and Perturbation Theory, pages 23–37, World Scientific, Singapore, 1999.
[16]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006.
[17]	N. Duruk Mutlubas and A. Geyer, Orbital stability of solitary waves of moderate amplitude in shallow water, J. Differ. Equations, 255 (2013), 254-263. doi: 10.1016/j.jde.2013.04.010.
[18]	M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not., 2009 (2009), 4578-4596.
[19]	A. Gasull and A. Geyer, Traveling surface waves of moderate amplitude in shallow water, Nonlinear Anal. Theory, Methods Appl., 102 (2014), 105-119. doi: 10.1016/j.na.2014.02.005.
[20]	A. Geyer, Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude, J. Nonlinear Math. Phys., 22 (2015), 545-551. doi: 10.1080/14029251.2015.1129492.
[21]	A. Geyer and V. Mañosa, Singular solutions for a class of traveling wave equations, Nonlinear Anal. Real World Appl., 31 (2016), 57-76. doi: 10.1016/j.nonrwa.2016.01.009.
[22]	J. Guggenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983. doi: 10.1007/978-1-4612-1140-2.
[23]	D. Henry, Equatorially trapped nonlinear water waves in a $β$-plane approximation with centripetal forces, J. Fluid Mech. , 804 (2016), R1, 11 pp.
[24]	E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975.
[25]	V. M. Hur, Analyticity of rotational flows beneath solitary water waves, Int. Math. Res. Not. IMRN, 2012 (2012), 2550-2570.
[26]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.
[27]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739.
[28]	J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2.
[29]	J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 10 (2004), 485-499.
[30]	J. Lenells, Stability for the periodic Camassa-Holm equation, Math. Scand., 97 (2005), 188-200. doi: 10.7146/math.scand.a-14971.
[31]	J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differ. Equ., 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007.
[32]	J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82. doi: 10.1016/j.jmaa.2004.11.038.
[33]	Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.
[34]	T. Lyons, The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209-218. doi: 10.1007/s00021-016-0249-6.
[35]	R. Quirchmayr, A new highly nonlinear shallow water wave equation, J. Evol. Equations, 16 (2016), 539-567. doi: 10.1007/s00028-015-0312-4.
[36]	H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff and K. Socha, Stabilizing the Benjamin-Feir instability, J. Fluid Mech., 539 (2005), 229-271. doi: 10.1017/S002211200500563X.
[37]	G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics 140 AMS, Providence, RI, 2012.
[38]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.
[39]	E. Varvaruca and G. S. Weiss, The Stokes conjecture for waves with vorticity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 861-885. doi: 10.1016/j.anihpc.2012.05.001.

show all references

A. Geyer acknowledges the support of the Austrian Science Fund (FWF) project J3452 "Dynamical Systems Methods in Hydrodynamics". R. Quirchmayr acknowledges the support of the Austrian Science Fund (FWF), Grant W1245.

References:

[1]	T. B. Benjamin and J. E. Feir, The disintegration of wavetrains in deep water, J. Fluid Mech., 27 (1967), 417-430.
[2]	J. L. Bona, P. E. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 411 (1987), 395-412. doi: 10.1098/rspa.1987.0073.
[3]	G. Brüll, M. Ehrnström, A. Geyer and L. Pei, Symmetric solutions of evolutionary partial differential equations, Nonlinearity, 30 (2017), 3932-3950.
[4]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[5]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.
[6]	A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. : Oceans, 117 (2012), C05029. doi: 10.1029/2012JC007879.
[7]	A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. doi: 10.1175/JPO-D-13-0174.1.
[8]	A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, 81 SIAM Philadelphia, 2011.
[9]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.
[10]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.
[11]	A. Constantin and W. Strauss, Stability of peakons, Commun. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.
[12]	A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x.
[13]	A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.
[14]	B. Deconinck and T. Kapitula, The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. Sect. A Gen. At. Solid State Phys., 374 (2010), 4018-4022. doi: 10.1016/j.physleta.2010.08.007.
[15]	A. Degasperis and M. Procesi, Asymptotic integrability, In A. Degasperis and G. Gaeta, editors, Symmetry and Perturbation Theory, pages 23–37, World Scientific, Singapore, 1999.
[16]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006.
[17]	N. Duruk Mutlubas and A. Geyer, Orbital stability of solitary waves of moderate amplitude in shallow water, J. Differ. Equations, 255 (2013), 254-263. doi: 10.1016/j.jde.2013.04.010.
[18]	M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not., 2009 (2009), 4578-4596.
[19]	A. Gasull and A. Geyer, Traveling surface waves of moderate amplitude in shallow water, Nonlinear Anal. Theory, Methods Appl., 102 (2014), 105-119. doi: 10.1016/j.na.2014.02.005.
[20]	A. Geyer, Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude, J. Nonlinear Math. Phys., 22 (2015), 545-551. doi: 10.1080/14029251.2015.1129492.
[21]	A. Geyer and V. Mañosa, Singular solutions for a class of traveling wave equations, Nonlinear Anal. Real World Appl., 31 (2016), 57-76. doi: 10.1016/j.nonrwa.2016.01.009.
[22]	J. Guggenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983. doi: 10.1007/978-1-4612-1140-2.
[23]	D. Henry, Equatorially trapped nonlinear water waves in a $β$-plane approximation with centripetal forces, J. Fluid Mech. , 804 (2016), R1, 11 pp.
[24]	E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975.
[25]	V. M. Hur, Analyticity of rotational flows beneath solitary water waves, Int. Math. Res. Not. IMRN, 2012 (2012), 2550-2570.
[26]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.
[27]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739.
[28]	J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2.
[29]	J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 10 (2004), 485-499.
[30]	J. Lenells, Stability for the periodic Camassa-Holm equation, Math. Scand., 97 (2005), 188-200. doi: 10.7146/math.scand.a-14971.
[31]	J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differ. Equ., 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007.
[32]	J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82. doi: 10.1016/j.jmaa.2004.11.038.
[33]	Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.
[34]	T. Lyons, The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209-218. doi: 10.1007/s00021-016-0249-6.
[35]	R. Quirchmayr, A new highly nonlinear shallow water wave equation, J. Evol. Equations, 16 (2016), 539-567. doi: 10.1007/s00028-015-0312-4.
[36]	H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff and K. Socha, Stabilizing the Benjamin-Feir instability, J. Fluid Mech., 539 (2005), 229-271. doi: 10.1017/S002211200500563X.
[37]	G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics 140 AMS, Providence, RI, 2012.
[38]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.
[39]	E. Varvaruca and G. S. Weiss, The Stokes conjecture for waves with vorticity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 861-885. doi: 10.1016/j.anihpc.2012.05.001.

Figure 1. A selection of some traveling wave solutions of (1). The waves on the left side from top to bottom are of the following types: smooth periodic, peaked periodic, cusped periodic, periodic with peaked crests and cusped troughs, periodic with peaked crests and troughs, composite, composite with plateaus. Right side top to bottom: smooth solitary, peaked solitary, cusped solitary, wavefront, compactly supported anticusp, multi-crest with decay, multi-peak with decay.

Download full-size image

Download as PowerPoint slide

Figure 2. (a) the graphs of the functions $\alpha_2$ [bold], $\alpha_1$ [plain] and $\bar u$ [dashed]; (b) the graphs of $K_{\alpha_2}$ [bold], $K_{\alpha_1}$ [plain] and $K_0$ [dashed], cf. (24) and (29).

Download full-size image

Download as PowerPoint slide

Figure 3. The graph of $\varphi$ for (a) $c>\bar c$ and (b) $c = \bar c$.

Download full-size image

Download as PowerPoint slide

Figure 4. The graph of $\varphi$ for increasing values of $c \in (-\infty, \bar c)$.

Download full-size image

Download as PowerPoint slide

Figure 5. In (5a) we sketch a phase portrait representing scenario Ⅲ in Table 1, which yields a smooth solitary and smooth periodic traveling waves as illustrated in (5b).

Download full-size image

Download as PowerPoint slide

Figure 6. Sketches of phase portraits for scenario Ⅰ, i.e. $K <K_0(c)$.

Download full-size image

Download as PowerPoint slide

Figure 7. Sketches of phase portraits for scenario Ⅱ, i.e. $K = K_0(c)$

Download full-size image

Download as PowerPoint slide

Figure 8. Sketches of the phase portraits of scenario Ⅲ: (8a)-(8d) yield smooth periodic waves, (8a) and (8d) yield smooth solitary waves, (8b) yields peaked solitary waves, (8c) yields peaked periodic and-provided that $c\in (c_0,c_1)$-both periodic and solitary cusped traveling waves.

Download full-size image

Download as PowerPoint slide

Figure 9. Sketches of peaked (9a) and cusped (9b) traveling waves.

Download full-size image

Download as PowerPoint slide

Figure 10. Sketches of phase portraits in scenario Ⅳ, i.e. $K = K_{\alpha_1}(c)$.

Download full-size image

Download as PowerPoint slide

Figure 11. Some examples of composite waves constructed from orbits corresponding to scenario Ⅳ: smooth (11a) and peaked (11b) traveling wave solutions with plateaus at height $\alpha_1$, smooth (11c) and peaked (11d) multi-crested solutions with decay, and smooth (11e) and non-smooth (11f) compactons.

Download full-size image

Download as PowerPoint slide

Figure 12. Sketches of phase portraits in scenario Ⅴ, i.e. $K\in(K_{\alpha_1}(c),K_{\alpha_2}(c))$.

Download full-size image

Download as PowerPoint slide

Figure 13. Traveling waves constructed from orbits corresponding to the phase portrait in Scenario Ⅴ, Fig. 12a: Fig. 13a shows solitary and periodic anti-cusped waves. Fig. 13b shows composite waves: a steep wave front and a periodic composition.

Download full-size image

Download as PowerPoint slide

Figure 14. Traveling waves constructed from orbits corresponding to the phase portrait in Scenario Ⅴ, Fig. 12b: Fig. 14a shows solitary and periodic anti-peaked waves. Fig. 14b shows composite waves: a wave front and a periodic composition.

Download full-size image

Download as PowerPoint slide

Figure 15. Sketches of phase portraits in scenario Ⅵ, i.e. $K = K_{\alpha_2}(c)$.

Download full-size image

Download as PowerPoint slide

Figure 16. Sketches of phase portraits in scenario Ⅶ, i.e. $K>K_{\alpha_2}(c)$.

Download full-size image

Download as PowerPoint slide

Figure 17. In Fig. 17a we see sketches of an anti-cusped and an anti-peaked solitary wave taking the constant value $\alpha_2$ outside a bounded interval-they correspond to the dark blue lines in Fig. 15a and 15b, respectively. The first image in Fig. 17b shows a periodic wave with peaked crests and cusped troughs. In the second sketch in Fig. 17b we see a periodic wave with peaked crests and troughs. These waves correspond to the dark blue lines in Fig. 16a and 16b, respectively. In Fig. 17c we see an example of a composite wave constructed from orbits of different energy levels in Fig. 16b.

Download full-size image

Download as PowerPoint slide

Figure 18. Sketches of phase portraits in scenarios Ⅰ and Ⅱ.

Download full-size image

Download as PowerPoint slide

Figure 19. Sketches of phase portraits in scenarios Ⅲ and Ⅳ.

Download full-size image

Download as PowerPoint slide

Figure 20. Sketches of the phase portrait corresponding to scenario Ⅴ, i.e. $K>K_{\alpha}$.

Download full-size image

Download as PowerPoint slide

Figure 21. Sketches of Cantor waves with peak elements (a) and cusp elements (b).

Download full-size image

Download as PowerPoint slide

Table 1. A list of all possible scenarios for the ordering of fixed points on the horizontal axis. Here s stands for saddle, c for center and n means that the Jacobi matrix at the fixed point is nilpotent.

scenario	parameter	order relation	fixed points and type
Ⅰ	$K < K_0(c)$	-	-
Ⅱ	$K = K_0(c)$	$\bar u <\alpha_1 <\alpha_2 $	$ (\bar u,0)$ n
Ⅲ	$K_0(c) <K <K_{\alpha_1}(c)$	$u_1 < u_2<\alpha_1 <\alpha_2$	$(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm c$
Ⅳ	$K = K_{\alpha_1}(c)$	$u_1 < u_2=\alpha_1 <\alpha_2$	$(u_1,0)$ $\mathrm s$, $(\alpha_1,0)$ $\mathrm n$
Ⅴ	$K_{\alpha_1}(c) < K < K_{\alpha_2}(c)$	$u_1 <\alpha_1 < u_2 <\alpha_2$	$(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm s$
Ⅵ	$K = K_{\alpha_2}(c)$	$u_1 <\alpha_1 < u_2 = \alpha_2$	$(u_1,0)$ $\mathrm s$, $(\alpha_2,0)$ $\mathrm n$
Ⅶ	$K > K_{\alpha_2}(c)$	$u_1 <\alpha_1 <\alpha_2 <u_2$	$(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm c$

Download as excel

scenario	parameter	order relation	fixed points and type
Ⅰ	$K < K_0(c)$	-	-
Ⅱ	$K = K_0(c)$	$\bar u <\alpha_1 <\alpha_2 $	$ (\bar u,0)$ n
Ⅲ	$K_0(c) <K <K_{\alpha_1}(c)$	$u_1 < u_2<\alpha_1 <\alpha_2$	$(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm c$
Ⅳ	$K = K_{\alpha_1}(c)$	$u_1 < u_2=\alpha_1 <\alpha_2$	$(u_1,0)$ $\mathrm s$, $(\alpha_1,0)$ $\mathrm n$
Ⅴ	$K_{\alpha_1}(c) < K < K_{\alpha_2}(c)$	$u_1 <\alpha_1 < u_2 <\alpha_2$	$(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm s$
Ⅵ	$K = K_{\alpha_2}(c)$	$u_1 <\alpha_1 < u_2 = \alpha_2$	$(u_1,0)$ $\mathrm s$, $(\alpha_2,0)$ $\mathrm n$
Ⅶ	$K > K_{\alpha_2}(c)$	$u_1 <\alpha_1 <\alpha_2 <u_2$	$(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm c$

[1]	Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103
[2]	Anwar Ja'afar Mohamad Jawad, Mohammad Mirzazadeh, Anjan Biswas. Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1155-1164. doi: 10.3934/dcdss.2015.8.1155
[3]	Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239
[4]	David Henry, Hung-Chu Hsu. Instability of equatorial water waves in the $f-$plane. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 909-916. doi: 10.3934/dcds.2015.35.909
[5]	Xingxing Liu. Stability in the energy space of the sum of $ N $ solitary waves for an equation modelling shallow water waves of moderate amplitude. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022105
[6]	Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 115-125. doi: 10.3934/dcdsb.2007.8.115
[7]	Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115
[8]	Mouhamadou Aliou M. T. Baldé, Diaraf Seck. Coupling the shallow water equation with a long term dynamics of sand dunes. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1521-1551. doi: 10.3934/dcdss.2016061
[9]	Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089
[10]	Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
[11]	David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[12]	Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397
[13]	Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5521-5541. doi: 10.3934/dcds.2019225
[14]	Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[15]	Shuichi Kawashima, Peicheng Zhu. Traveling waves for models of phase transitions of solids driven by configurational forces. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 309-323. doi: 10.3934/dcdsb.2011.15.309
[16]	Chiun-Chuan Chen, Hung-Yu Chien, Chih-Chiang Huang. A variational approach to three-phase traveling waves for a gradient system. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4737-4765. doi: 10.3934/dcds.2021055
[17]	Delia Ionescu-Kruse. Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 531-543. doi: 10.3934/dcds.2007.19.531
[18]	Joseph Thirouin. Classification of traveling waves for a quadratic Szegő equation. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3099-3122. doi: 10.3934/dcds.2019128
[19]	Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[20]	Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Anna Geyer Ronald Quirchmayr
on Google Scholar
Anna Geyer Ronald Quirchmayr

2021 Impact Factor: 1.588

Tools

Article outline

Figures and Tables

2021 Impact Factor: 1.588

Tools

Figures and Tables

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Anna Geyer Ronald Quirchmayr
on Google Scholar
Anna Geyer Ronald Quirchmayr

[Back to Top]