August  2019, 39(8): 4747-4770. doi: 10.3934/dcds.2019193

On stratified water waves with critical layers and Coriolis forces

University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received  October 2018 Revised  January 2019 Published  May 2019

We consider nonlinear traveling waves in a two-dimensional fluid subject to the effects of vorticity, stratification, and in-plane Coriolis forces. We first observe that the terms representing the Coriolis forces can be completely eliminated by a change of variables. This does not appear to be well-known, and helps to organize some of the existing literature.

Second we give a rigorous existence result for periodic waves in a two-layer system with a free surface and constant densities and vorticities in each layer, allowing for the presence of critical layers. We augment the problem with four physically-motivated constraints, and phrase our hypotheses directly in terms of the explicit dispersion relation for the problem. This approach smooths the way for further generalizations, some of which we briefly outline at the end of the paper.

Citation: Miles H. Wheeler. On stratified water waves with critical layers and Coriolis forces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4747-4770. doi: 10.3934/dcds.2019193
References:
[1]

A. Aivaliotis, On the symmetry of equatorial travelling water waves with constant vorticity and stagnation points, Nonlinear Anal. Real World Appl., 34 (2017), 159-171. doi: 10.1016/j.nonrwa.2016.08.010. Google Scholar

[2]

A. Akers and S. Walsh, Solitary water waves with discontinuous vorticity, Journal de Mathématiques Pures et Appliquées, 124 (2019), 220-272. doi: 10.1016/j.matpur.2018.06.008. Google Scholar

[3]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003, An introduction. doi: 10.1515/9781400884339. Google Scholar

[4]

R. M. ChenS. Walsh and M. H. Wheeler, Existence and qualitative theory for stratified solitary water waves, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 517-576. doi: 10.1016/j.anihpc.2017.06.003. Google Scholar

[5]

A. Compelli and R. Ivanov, On the dynamics of internal waves interacting with the equatorial undercurrent, J. Nonlinear Math. Phys., 22 (2015), 531-539. doi: 10.1080/14029251.2015.1113052. Google Scholar

[6]

A. Compelli and R. Ivanov, The dynamics of flat surface internal geophysical waves with currents, J. Math. Fluid Mech., 19 (2017), 329-344. doi: 10.1007/s00021-016-0283-4. Google Scholar

[7]

A. ConstantinR. I. Ivanov and C.-I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016), 1417-1447. doi: 10.1007/s00205-016-0990-2. Google Scholar

[8]

A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785. Google Scholar

[9]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873. Google Scholar

[10]

A. Constantin, On equatorial wind waves, Differential Integral Equations, 26 (2013), 237-252. Google Scholar

[11]

A. ConstantinW. Strauss and E. Vǎrvǎrucǎ, Global bifurcation of steady gravity water waves with critical layers, Acta Math., 217 (2016), 195-262. doi: 10.1007/s11511-017-0144-x. Google Scholar

[12]

A. Constantin and W. A. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046. Google Scholar

[13]

A. Constantin and W. A. 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]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[15]

M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'amplitude finie, (French) 1934. 75 pp. Google Scholar

[16]

M. D. Groves, Steady water waves, J. Nonlinear Math. Phys., 11 (2004), 435-460. doi: 10.2991/jnmp.2004.11.4.2. Google Scholar

[17]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, in Annual Review of Fluid Mechanics. Vol. 38, Annual Reviews, Palo Alto, CA, 38 (2006), 395–425. doi: 10.1146/annurev.fluid.38.050304.092129. Google Scholar

[18]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037. doi: 10.1080/03605302.2012.734889. Google Scholar

[19]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487. doi: 10.1090/S0033-569X-2013-01293-8. Google Scholar

[20]

D. Henry, Internal equatorial water waves in the $f$-plane, J. Nonlinear Math. Phys., 22 (2015), 499-506. doi: 10.1080/14029251.2015.1113046. Google Scholar

[21]

D. Henry and A.-V. Matioc, On the existence of equatorial wind waves, Nonlinear Anal., 101 (2014), 113-123. doi: 10.1016/j.na.2014.01.018. Google Scholar

[22]

D. Henry and A.-V. Matioc, On the symmetry of steady equatorial wind waves, Nonlinear Anal. Real World Appl., 18 (2014), 50-56. doi: 10.1016/j.nonrwa.2014.01.009. Google Scholar

[23]

H.-C. Hsu, Exact nonlinear internal equatorial waves in the $f$-plane, J. Math. Fluid Mech., 19 (2017), 367-374. doi: 10.1007/s00021-016-0285-2. Google Scholar

[24]

D. Ionescu-Kruse and C. I. Martin, Periodic equatorial water flows from a Hamiltonian perspective, J. Differential Equations, 262 (2017), 4451-4474. doi: 10.1016/j.jde.2017.01.001. Google Scholar

[25]

D. Ionescu-Kruse and A.-V. Matioc, Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories, Discrete Contin. Dyn. Syst., 34 (2014), 3045-3060. doi: 10.3934/dcds.2014.34.3045. Google Scholar

[26]

V. Kozlov, N. Kuznetsov and E. Lokharu, Steady water waves with vorticity: An analysis of the dispersion equation, J. Fluid Mech., 751 (2014), R3, 13pp. doi: 10.1017/jfm.2014.322. Google Scholar

[27]

V. Kozlov and N. Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents, Arch. Ration. Mech. Anal., 214 (2014), 971-1018. doi: 10.1007/s00205-014-0787-0. Google Scholar

[28]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. Google Scholar

[29]

H. Le, Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind, Discrete Contin. Dyn. Syst., 38 (2018), 3357-3385. doi: 10.3934/dcds.2018144. Google Scholar

[30]

T. Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei, 33 (1924), 141-150. Google Scholar

[31]

C. I. Martin, On periodic geophysical water flows with discontinuous vorticity in the equatorial $f$-plane approximation, Philos. Trans. Roy. Soc. A, 376 (2018), 20170096, 23pp. doi: 10.1098/rsta.2017.0096. Google Scholar

[32]

A.-V. Matioc, Steady internal water waves with a critical layer bounded by the wave surface, J. Nonlinear Math. Phys., 19 (2012), 1250008, 21pp. doi: 10.1142/S1402925112500088. Google Scholar

[33]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66. doi: 10.1002/mma.1670100105. Google Scholar

[34]

J. W. Miles, Solitary waves, in Annual Review of Fluid Mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 43 (1980), 11–43. Google Scholar

[35]

A. I. Nekrasov, On steady waves, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3.Google Scholar

[36]

D. V. Nilsson, Internal gravity-capillary solitary waves in finite depth, Math. Methods Appl. Sci., 40 (2017), 1053-1080. doi: 10.1002/mma.4036. Google Scholar

[37]

R. Quirchmayr, On irrotational flows beneath periodic traveling equatorial waves, J. Math. Fluid Mech., 19 (2017), 283-304. doi: 10.1007/s00021-016-0280-7. Google Scholar

[38]

B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocities, Discrete Contin. Dyn. Syst., 20 (2008), 139-158. doi: 10.3934/dcds.2008.20.139. Google Scholar

[39]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1. Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar

[41]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005. Google Scholar

[42]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583. Google Scholar

[43]

S. WalshO. Bühler and J. Shatah, Steady water waves in the presence of wind, SIAM J. Math. Anal., 45 (2013), 2182-2227. doi: 10.1137/120880124. Google Scholar

[44]

L.-J. Wang, Small-amplitude solitary and generalized solitary traveling waves in a gravity two-layer fluid with vorticity, Nonlinear Anal., 150 (2017), 159-193. doi: 10.1016/j.na.2016.11.012. Google Scholar

[45]

M. H. Wheeler, Simplified models for equatorial waves with vertical structure, Oceanography, 31 (2018), 36-41. doi: 10.5670/oceanog.2018.307. Google Scholar

show all references

References:
[1]

A. Aivaliotis, On the symmetry of equatorial travelling water waves with constant vorticity and stagnation points, Nonlinear Anal. Real World Appl., 34 (2017), 159-171. doi: 10.1016/j.nonrwa.2016.08.010. Google Scholar

[2]

A. Akers and S. Walsh, Solitary water waves with discontinuous vorticity, Journal de Mathématiques Pures et Appliquées, 124 (2019), 220-272. doi: 10.1016/j.matpur.2018.06.008. Google Scholar

[3]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003, An introduction. doi: 10.1515/9781400884339. Google Scholar

[4]

R. M. ChenS. Walsh and M. H. Wheeler, Existence and qualitative theory for stratified solitary water waves, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 517-576. doi: 10.1016/j.anihpc.2017.06.003. Google Scholar

[5]

A. Compelli and R. Ivanov, On the dynamics of internal waves interacting with the equatorial undercurrent, J. Nonlinear Math. Phys., 22 (2015), 531-539. doi: 10.1080/14029251.2015.1113052. Google Scholar

[6]

A. Compelli and R. Ivanov, The dynamics of flat surface internal geophysical waves with currents, J. Math. Fluid Mech., 19 (2017), 329-344. doi: 10.1007/s00021-016-0283-4. Google Scholar

[7]

A. ConstantinR. I. Ivanov and C.-I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016), 1417-1447. doi: 10.1007/s00205-016-0990-2. Google Scholar

[8]

A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785. Google Scholar

[9]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873. Google Scholar

[10]

A. Constantin, On equatorial wind waves, Differential Integral Equations, 26 (2013), 237-252. Google Scholar

[11]

A. ConstantinW. Strauss and E. Vǎrvǎrucǎ, Global bifurcation of steady gravity water waves with critical layers, Acta Math., 217 (2016), 195-262. doi: 10.1007/s11511-017-0144-x. Google Scholar

[12]

A. Constantin and W. A. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046. Google Scholar

[13]

A. Constantin and W. A. 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]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[15]

M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'amplitude finie, (French) 1934. 75 pp. Google Scholar

[16]

M. D. Groves, Steady water waves, J. Nonlinear Math. Phys., 11 (2004), 435-460. doi: 10.2991/jnmp.2004.11.4.2. Google Scholar

[17]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, in Annual Review of Fluid Mechanics. Vol. 38, Annual Reviews, Palo Alto, CA, 38 (2006), 395–425. doi: 10.1146/annurev.fluid.38.050304.092129. Google Scholar

[18]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037. doi: 10.1080/03605302.2012.734889. Google Scholar

[19]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487. doi: 10.1090/S0033-569X-2013-01293-8. Google Scholar

[20]

D. Henry, Internal equatorial water waves in the $f$-plane, J. Nonlinear Math. Phys., 22 (2015), 499-506. doi: 10.1080/14029251.2015.1113046. Google Scholar

[21]

D. Henry and A.-V. Matioc, On the existence of equatorial wind waves, Nonlinear Anal., 101 (2014), 113-123. doi: 10.1016/j.na.2014.01.018. Google Scholar

[22]

D. Henry and A.-V. Matioc, On the symmetry of steady equatorial wind waves, Nonlinear Anal. Real World Appl., 18 (2014), 50-56. doi: 10.1016/j.nonrwa.2014.01.009. Google Scholar

[23]

H.-C. Hsu, Exact nonlinear internal equatorial waves in the $f$-plane, J. Math. Fluid Mech., 19 (2017), 367-374. doi: 10.1007/s00021-016-0285-2. Google Scholar

[24]

D. Ionescu-Kruse and C. I. Martin, Periodic equatorial water flows from a Hamiltonian perspective, J. Differential Equations, 262 (2017), 4451-4474. doi: 10.1016/j.jde.2017.01.001. Google Scholar

[25]

D. Ionescu-Kruse and A.-V. Matioc, Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories, Discrete Contin. Dyn. Syst., 34 (2014), 3045-3060. doi: 10.3934/dcds.2014.34.3045. Google Scholar

[26]

V. Kozlov, N. Kuznetsov and E. Lokharu, Steady water waves with vorticity: An analysis of the dispersion equation, J. Fluid Mech., 751 (2014), R3, 13pp. doi: 10.1017/jfm.2014.322. Google Scholar

[27]

V. Kozlov and N. Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents, Arch. Ration. Mech. Anal., 214 (2014), 971-1018. doi: 10.1007/s00205-014-0787-0. Google Scholar

[28]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. Google Scholar

[29]

H. Le, Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind, Discrete Contin. Dyn. Syst., 38 (2018), 3357-3385. doi: 10.3934/dcds.2018144. Google Scholar

[30]

T. Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei, 33 (1924), 141-150. Google Scholar

[31]

C. I. Martin, On periodic geophysical water flows with discontinuous vorticity in the equatorial $f$-plane approximation, Philos. Trans. Roy. Soc. A, 376 (2018), 20170096, 23pp. doi: 10.1098/rsta.2017.0096. Google Scholar

[32]

A.-V. Matioc, Steady internal water waves with a critical layer bounded by the wave surface, J. Nonlinear Math. Phys., 19 (2012), 1250008, 21pp. doi: 10.1142/S1402925112500088. Google Scholar

[33]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66. doi: 10.1002/mma.1670100105. Google Scholar

[34]

J. W. Miles, Solitary waves, in Annual Review of Fluid Mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 43 (1980), 11–43. Google Scholar

[35]

A. I. Nekrasov, On steady waves, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3.Google Scholar

[36]

D. V. Nilsson, Internal gravity-capillary solitary waves in finite depth, Math. Methods Appl. Sci., 40 (2017), 1053-1080. doi: 10.1002/mma.4036. Google Scholar

[37]

R. Quirchmayr, On irrotational flows beneath periodic traveling equatorial waves, J. Math. Fluid Mech., 19 (2017), 283-304. doi: 10.1007/s00021-016-0280-7. Google Scholar

[38]

B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocities, Discrete Contin. Dyn. Syst., 20 (2008), 139-158. doi: 10.3934/dcds.2008.20.139. Google Scholar

[39]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1. Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar

[41]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005. Google Scholar

[42]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583. Google Scholar

[43]

S. WalshO. Bühler and J. Shatah, Steady water waves in the presence of wind, SIAM J. Math. Anal., 45 (2013), 2182-2227. doi: 10.1137/120880124. Google Scholar

[44]

L.-J. Wang, Small-amplitude solitary and generalized solitary traveling waves in a gravity two-layer fluid with vorticity, Nonlinear Anal., 150 (2017), 159-193. doi: 10.1016/j.na.2016.11.012. Google Scholar

[45]

M. H. Wheeler, Simplified models for equatorial waves with vertical structure, Oceanography, 31 (2018), 36-41. doi: 10.5670/oceanog.2018.307. Google Scholar

Figure 1.  Fluid configurations with multiple layers using the notation (1.1) and (1.2). (a) A configuration with N = 4 layers and a rigid lid. (b) A configuration with N = 2 layers and a free surface. This is the type of configuration which will be considered in Section 3
Figure 2.  Shear flows $\overline U(z)$ corresponding to the stream functions $\overline\Psi_1, \overline\Psi_2$ in (3.9). Both flows have $\omega_2 < 0 < \omega_1$ and $c > 0$. (a) A flow with a critical layer at the marked point in $D_1$ where $\overline U_1 = c$. (b) A flow without a critical layer
[1]

Walter A. Strauss. Vorticity jumps in steady water waves. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101

[2]

David Henry, Bogdan--Vasile Matioc. On the regularity of steady periodic stratified water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1453-1464. doi: 10.3934/cpaa.2012.11.1453

[3]

Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4713-4729. doi: 10.3934/dcds.2019191

[4]

Delia Ionescu-Kruse. Short-wavelength instabilities of edge waves in stratified water. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2053-2066. doi: 10.3934/dcds.2015.35.2053

[5]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397

[6]

Mats Ehrnström. Deep-water waves with vorticity: symmetry and rotational behaviour. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 483-491. doi: 10.3934/dcds.2007.19.483

[7]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109

[8]

Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607

[9]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103

[10]

Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016

[11]

Delia Ionescu-Kruse. Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1475-1496. doi: 10.3934/cpaa.2012.11.1475

[12]

Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114

[13]

Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045

[14]

Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5521-5541. doi: 10.3934/dcds.2019225

[15]

Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1

[16]

R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497

[17]

Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465

[18]

Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1

[19]

Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267

[20]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (54)
  • HTML views (125)
  • Cited by (0)

Other articles
by authors

[Back to Top]