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Elliptic and hyperelliptic functions describing the particle motion beneath smallamplitude water waves with constant vorticity
On the formation of singularities for surface water waves
1.  Department of Mathematics, University of Illinois at UrbanaChampaign, Urbana, IL 61801, United States 
References:
[1] 
C. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form,, Acta Math., 148 (1982), 193. 
[2] 
A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. GómezSerrano, Splash singularity for water waves,, preprint, (2011). 
[3] 
Adrian Constantin and Joachim Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. 
[4] 
H. Dong, D. Du and D. Li, Finite time singularities and global wellposedness for fractal Burgers equations,, Indiana Univ. Math. J., 58 (2009), 807. 
[5] 
Vera Mikyoung Hur, Gain of regularity for water waves with surface tension: a model equation,, preprint, (2011). 
[6] 
Michael Selwyn LonguetHiggins, On the forming of sharp corners at a free surface,, Proc. R. Soc. Lond. A, 371 (1980), 453. 
[7] 
Michael Selwyn LonguetHiggins, On the overturning of gravity waves,, Proc. R. Soc. Lond. A, 376 (1981), 377. 
[8] 
P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves,", Translations of Mathematical Monographs, (1994). 
[9] 
D. Howell Peregrine, Breaking waves on beaches,, Ann. Rev. Fluid Mech., 15 (1983), 149. 
[10] 
J.B. Song and M. L. Banner, On determining the onset and strength of breaking for deep water waves. Part I: Unforced irrotational wave groups,, Journal of Physical Oceanography, 32 (2002), 2541. 
[11] 
R. Thom, "Structural Stability and Morphogenesis,", Benjamin, (1975). 
[12] 
Milton Van Dyke, "An Album of Fluid Motion,", Parabolic Press, (1982). 
[13] 
Gerald Bereford Whitham, Variational methods and applications to water waves,, Hyperbolic equations and waves (Rencontre, (1968), 153. 
[14] 
Gerald Beresford Whitham, "Linear and Nonlinear Waves,", Reprint of the 1974 original. Pure and Applied Mathematics, (1974). 
[15] 
Sijue Wu, Wellposedness in Sobolev spaces of the full water wave problem in 2D,, Invent. Math., 130 (1997), 39. 
[16] 
Sijue Wu, Wellposedness in Sobolev spaces of the full water wave problem in 3D,, J. Amer. Math. Soc., 12 (1999), 445. 
[17] 
Sijue Wu, Almost global wellposedness of the 2D full water wave problem,, Invent. Math., 177 (2009), 45. 
[18] 
E. C. Zeeman, Breaking of waves,, Warwick Symposium on Differential Equations and Dynamical Systems, 206 (1971), 2. 
[19] 
Antoni Zygmund, "Trigonometric Series,", Volume 2, (1968). 
show all references
References:
[1] 
C. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form,, Acta Math., 148 (1982), 193. 
[2] 
A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. GómezSerrano, Splash singularity for water waves,, preprint, (2011). 
[3] 
Adrian Constantin and Joachim Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. 
[4] 
H. Dong, D. Du and D. Li, Finite time singularities and global wellposedness for fractal Burgers equations,, Indiana Univ. Math. J., 58 (2009), 807. 
[5] 
Vera Mikyoung Hur, Gain of regularity for water waves with surface tension: a model equation,, preprint, (2011). 
[6] 
Michael Selwyn LonguetHiggins, On the forming of sharp corners at a free surface,, Proc. R. Soc. Lond. A, 371 (1980), 453. 
[7] 
Michael Selwyn LonguetHiggins, On the overturning of gravity waves,, Proc. R. Soc. Lond. A, 376 (1981), 377. 
[8] 
P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves,", Translations of Mathematical Monographs, (1994). 
[9] 
D. Howell Peregrine, Breaking waves on beaches,, Ann. Rev. Fluid Mech., 15 (1983), 149. 
[10] 
J.B. Song and M. L. Banner, On determining the onset and strength of breaking for deep water waves. Part I: Unforced irrotational wave groups,, Journal of Physical Oceanography, 32 (2002), 2541. 
[11] 
R. Thom, "Structural Stability and Morphogenesis,", Benjamin, (1975). 
[12] 
Milton Van Dyke, "An Album of Fluid Motion,", Parabolic Press, (1982). 
[13] 
Gerald Bereford Whitham, Variational methods and applications to water waves,, Hyperbolic equations and waves (Rencontre, (1968), 153. 
[14] 
Gerald Beresford Whitham, "Linear and Nonlinear Waves,", Reprint of the 1974 original. Pure and Applied Mathematics, (1974). 
[15] 
Sijue Wu, Wellposedness in Sobolev spaces of the full water wave problem in 2D,, Invent. Math., 130 (1997), 39. 
[16] 
Sijue Wu, Wellposedness in Sobolev spaces of the full water wave problem in 3D,, J. Amer. Math. Soc., 12 (1999), 445. 
[17] 
Sijue Wu, Almost global wellposedness of the 2D full water wave problem,, Invent. Math., 177 (2009), 45. 
[18] 
E. C. Zeeman, Breaking of waves,, Warwick Symposium on Differential Equations and Dynamical Systems, 206 (1971), 2. 
[19] 
Antoni Zygmund, "Trigonometric Series,", Volume 2, (1968). 
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