Solitary wave collisions for the Whitham equation. (English) Zbl 1513.76047
Summary: In this article, we numerically study overtaking collisions of two solitary waves for the Whitham equation. We find regimes in which solitary wave interactions maintain two well separated crests at any given time and regimes where the number of local maxima varies according to the laws \(2\rightarrow 1\rightarrow 2\rightarrow 1\rightarrow 2\) or \(2\rightarrow 1\rightarrow 2\). It shows that the geometric Lax-categorization of the Korteweg-de Vries equation (KdV) for two-soliton collisions still holds for the Whitham equation. However, differently from the KdV and the Euler equations, we show that an algebraic Lax-categorization for the Whitham equation based on the ratio of the amplitude of the initial solitary waves is not possible.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76B20 | Ship waves |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 76B55 | Internal waves for incompressible inviscid fluids |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Software:
MatlabReferences:
| [1] | Carter, JD, Bidirectional Whitham equations as models of waves on shallow water, Wave Motion, 82, 51-61 (2018) · Zbl 1524.35519 · doi:10.1016/j.wavemoti.2018.07.004 |
| [2] | Carter JD, Kalisch H, Kharif C, Abid M (2021) The cubic vortical Whitham equation. arXiv:2110.02072v1 [physics.flu-dyn] |
| [3] | Craig, W.; Guynne, P.; Hammack, J.; Henderson, D.; Sulem, C., Solitary water wave interactions, Phys Fluids, 18 (2006) · Zbl 1185.76463 · doi:10.1063/1.2205916 |
| [4] | Deconinck, B.; Trichtchenko, O., High-frequency instabilities of small-amplitude Hamiltonian PDEs, DCDS, 37, 3, 1323-1358 (2015) · Zbl 1365.37057 |
| [5] | Ehrnström, M.; Kalisch, H., Traveling waves for the Whitham equation, Differ Integral Equ, 22, 1193-1210 (2009) · Zbl 1240.35449 |
| [6] | Ehrnström, M.; Wahlén, E., On Whitham’s conjecture of a highest cusped wave for a nonlocal dispersive equation, Ann I H Poincare-An, 36, 769-799 (2019) · Zbl 1423.35059 |
| [7] | Flamarion, MV; Ribeiro-Jr, R., Solitary water wave interactions for the forced Korteweg-de Vries equation, Comput Appl Math, 40, 312 (2021) · Zbl 1499.76023 · doi:10.1007/s40314-021-01700-6 |
| [8] | Flamarion, MV; Milewski, PA; Nachbin, A., Rotational waves generated by current-topography interaction, Stud Appl Math, 142, 433-464 (2019) · Zbl 1418.35303 · doi:10.1111/sapm.12253 |
| [9] | Hur, VM; Pandey, AK, Modulational instability in a full-dispersion shallow water model, Stud Appl Math, 142, 3-47 (2019) · Zbl 1420.35241 · doi:10.1111/sapm.12231 |
| [10] | Joseph, A., Investigating seafloors and oceans (2016), New York: Elsevier, New York |
| [11] | Kalisch H, Alejo MA, Corcho AJ, Pilod D (2022) Breather solutions to the Cubic Whitham equation. arXiv:2201.12074v2 [nlin.PS] |
| [12] | Klein, C.; Linares, F.; Pilod, D.; Saut, JC, On Whitham and related equations, Stud Appl Math, 140, 133-177 (2018) · Zbl 1393.35208 · doi:10.1111/sapm.12194 |
| [13] | Lax, PD, Integrals of nonlinear equations of evolution and solitary waves, Commun Pure Appl Math, 21, 467-490 (1968) · Zbl 0162.41103 · doi:10.1002/cpa.3160210503 |
| [14] | Mirie, RM; Su, CH, Collisions between two solitary waves. Part 2. A numerical study., J Fluid Mech, 115, 475-492 (1982) · Zbl 0487.76029 · doi:10.1017/S002211208200086X |
| [15] | Moldabayev, D.; Kalisch, H.; Dutykh, D., The Whitham equation as a model for surface water waves, Phys. D, 309, 99-107 (2015) · Zbl 1364.76032 · doi:10.1016/j.physd.2015.07.010 |
| [16] | Sanford, N.; Kodama, K.; Carter, JD; Kalisch, H., Stability of traveling wave solutions to the Whitham equation, Phys Lett A, 378, 2100-2107 (2014) · Zbl 1331.35310 · doi:10.1016/j.physleta.2014.04.067 |
| [17] | Trefethen, LN, Spectral methods in MATLAB (2001), Philadelphia: SIAM, Philadelphia · Zbl 0953.68643 |
| [18] | Trillo, S.; Klein, M.; Clauss, GF; Onorato, M., Observation of dispersive shock waves developing from initial depressions in shallow water, Phys. D, 333, 276-284 (2016) · doi:10.1016/j.physd.2016.01.007 |
| [19] | Weidman, PD; Maxworthy, T., Experiments on strong interaction between solitary waves, J Fluid Mech, 85, 417-431 (1978) · doi:10.1017/S0022112078000713 |
| [20] | Whitham, GB, Variational methods and applications to water waves, Proc R Soc Lond A, 229, 6-25 (1967) · Zbl 0163.21104 |
| [21] | Whitham, GB, Linear and nonlinear waves (1974), New York: Wiley, New York · Zbl 0373.76001 |
| [22] | Zabusky, M.; Kruskal, N., Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys Rev Lett, 15, 240-243 (1965) · Zbl 1201.35174 · doi:10.1103/PhysRevLett.15.240 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
