On the universality of the incompressible Euler equation on compact manifolds
Wave breaking of periodic solutions to the Fornberg-Whitham equation
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,
Delft University of Technology, Delft Institute of Applied Mathematics, Faculty of EEMCS, Mekelweg 4,2628 CD Delft, The Netherlands
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
Figure(21) / Table(1)
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
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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.
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).
Figure 3. The graph of $\varphi$ for (a) $c>\bar c$ and (b) $c = \bar c$.
Figure 4. The graph of $\varphi$ for increasing values of $c \in (-\infty, \bar c)$.
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).
Figure 6. Sketches of phase portraits for scenario Ⅰ, i.e. $K <K_0(c)$.
Figure 7. Sketches of phase portraits for scenario Ⅱ, i.e. $K = K_0(c)$
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.
Figure 9. Sketches of peaked (9a) and cusped (9b) traveling waves.
Figure 10. Sketches of phase portraits in scenario Ⅳ, i.e. $K = K_{\alpha_1}(c)$.
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.
Figure 12. Sketches of phase portraits in scenario Ⅴ, i.e. $K\in(K_{\alpha_1}(c),K_{\alpha_2}(c))$.
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.
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.
Figure 15. Sketches of phase portraits in scenario Ⅵ, i.e. $K = K_{\alpha_2}(c)$.
Figure 16. Sketches of phase portraits in scenario Ⅶ, i.e. $K>K_{\alpha_2}(c)$.
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.
Figure 18. Sketches of phase portraits in scenarios Ⅰ and Ⅱ.
Figure 19. Sketches of phase portraits in scenarios Ⅲ and Ⅳ.
Figure 20. Sketches of the phase portrait corresponding to scenario Ⅴ, i.e. $K>K_{\alpha}$.
Figure 21. Sketches of Cantor waves with peak elements (a) and cusp elements (b).
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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