Traveling wave solutions of a highly nonlinear shallow water equation

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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 & Continuous Dynamical Systems - A, 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. Google Scholar
[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. Google Scholar
[3]	G. Brüll, M. Ehrnström, A. Geyer and L. Pei, Symmetric solutions of evolutionary partial differential equations, Nonlinearity, 30 (2017), 3932-3950. Google Scholar
[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. Google Scholar
[5]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar
[6]	A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. : Oceans, 117 (2012), C05029. doi: 10.1029/2012JC007879. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. Google Scholar
[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. Google Scholar
[18]	M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not., 2009 (2009), 4578-4596. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	D. Henry, Equatorially trapped nonlinear water waves in a $β$-plane approximation with centripetal forces, J. Fluid Mech. , 804 (2016), R1, 11 pp. Google Scholar
[24]	E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975. Google Scholar
[25]	V. M. Hur, Analyticity of rotational flows beneath solitary water waves, Int. Math. Res. Not. IMRN, 2012 (2012), 2550-2570. Google Scholar
[26]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 10 (2004), 485-499. Google Scholar
[30]	J. Lenells, Stability for the periodic Camassa-Holm equation, Math. Scand., 97 (2005), 188-200. doi: 10.7146/math.scand.a-14971. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics 140 AMS, Providence, RI, 2012. Google Scholar
[38]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar
[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. Google Scholar

show all references

References:

[1]	T. B. Benjamin and J. E. Feir, The disintegration of wavetrains in deep water, J. Fluid Mech., 27 (1967), 417-430. Google Scholar
[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. Google Scholar
[3]	G. Brüll, M. Ehrnström, A. Geyer and L. Pei, Symmetric solutions of evolutionary partial differential equations, Nonlinearity, 30 (2017), 3932-3950. Google Scholar
[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. Google Scholar
[5]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar
[6]	A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. : Oceans, 117 (2012), C05029. doi: 10.1029/2012JC007879. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. Google Scholar
[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. Google Scholar
[18]	M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not., 2009 (2009), 4578-4596. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	D. Henry, Equatorially trapped nonlinear water waves in a $β$-plane approximation with centripetal forces, J. Fluid Mech. , 804 (2016), R1, 11 pp. Google Scholar
[24]	E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975. Google Scholar
[25]	V. M. Hur, Analyticity of rotational flows beneath solitary water waves, Int. Math. Res. Not. IMRN, 2012 (2012), 2550-2570. Google Scholar
[26]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 10 (2004), 485-499. Google Scholar
[30]	J. Lenells, Stability for the periodic Camassa-Holm equation, Math. Scand., 97 (2005), 188-200. doi: 10.7146/math.scand.a-14971. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics 140 AMS, Providence, RI, 2012. Google Scholar
[38]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar
[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. Google Scholar

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.

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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).

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

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

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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).

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

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

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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.

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

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

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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.

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

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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.

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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.

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

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

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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.

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

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

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

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

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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	$K > K_{\alpha_2}(c)$	$u_1 <\alpha_1 <\alpha_2 <u_2$	$(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm c$

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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	$K > K_{\alpha_2}(c)$	$u_1 <\alpha_1 <\alpha_2 <u_2$	$(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm c$

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2019 Impact Factor: 1.338

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2019 Impact Factor: 1.338

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2019 Impact Factor: 1.338

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Anna Geyer Ronald Quirchmayr
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