2014, 4(3): 315-363. doi: 10.3934/mcrf.2014.4.315

Existence theory of capillary-gravity waves on water of finite depth

1. 

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061

Received  March 2013 Revised  January 2014 Published  April 2014

This review article discusses the recent developments on the existence of two-dimensional and three-dimensional capillary-gravity waves on water of finite-depth. The Korteweg-de Vries (KdV) equation and Kadomtsev-Petviashvili (KP) equation are derived formally from the exact governing equations and the solitary-wave solutions and other solution are obtained for these model equations. Recent results on the existence of solutions of the exact governing equations near the solutions of these model equations are presented and various two- and three-dimensional solutions of the exact equations are provided. The ideas and methods to obtain the existence results are briefly discussed.
Citation: Shu-Ming Sun. Existence theory of capillary-gravity waves on water of finite depth. Mathematical Control & Related Fields, 2014, 4 (3) : 315-363. doi: 10.3934/mcrf.2014.4.315
References:
[1]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform,, SIAM Studies in Applied Mathematics, (1981).

[2]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, 2nd edition, (2003).

[3]

J. Albert and J. Angulo Pava, Existence and stability of ground-state solutions of a Schrödinger-KdV system,, Proc. Roy. Soc. Edinburgh A, 133 (2003), 987. doi: 10.1017/S030821050000278X.

[4]

J. C. Alexander, R. L. Pego and R. L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation,, Phys. Lett. A, 226 (1997), 187. doi: 10.1016/S0375-9601(96)00921-8.

[5]

C. J. Amick and K. Kirchgässner, A theory of solitary water-waves in the presence of surface tension,, Arch. Rat. Mech. Anal., 105 (1989), 1. doi: 10.1007/BF00251596.

[6]

K. Appert and J. Vaclavik, Dynamics of coupled solitons,, Phys. Fluids, 20 (1977), 1845. doi: 10.1063/1.861802.

[7]

J. T. Beale, The existence of solitary water waves,, Comm. Pure Appl. Math., 30 (1977), 373. doi: 10.1002/cpa.3160300402.

[8]

J. T. Beale, Exact solitary water waves with capillary ripples at infinity,, Comm. Pure Appl. Math., 44 (1991), 211. doi: 10.1002/cpa.3160440204.

[9]

T. B. Benjamin, The stability of solitary waves,, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153. doi: 10.1098/rspa.1972.0074.

[10]

J. L. Bona, On the stability theory of solitary waves,, Proc. Roy. Soc. London Ser. A, 344 (1975), 363. doi: 10.1098/rspa.1975.0106.

[11]

J. Bona and H. Q. Chen, Solitary waves in nonlinear dispersive systems,, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 313. doi: 10.3934/dcdsb.2002.2.313.

[12]

J. L. Bona, P. E. Souganidis and W. A. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type,, Proc. Roy. Soc. London Ser. A, 411 (1987), 395. doi: 10.1098/rspa.1987.0073.

[13]

J. V. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire,, C. R. Acad. Sci. Paris, 72 (1871), 755.

[14]

J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal,, C. R. Acad. Sci. Paris, 73 (1871), 256.

[15]

J. V. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond,, J. Math. Pures Appl., 17 (1872), 55.

[16]

J. V. Boussinesq, Essai sur la théorie des eaux courantes,, Mém. prés. div. Sav. Acad. Sci. Inst. Fr., 23 (1877), 1.

[17]

H. Brezis and L. Nirenberg, Remarks on finding critical points,, Commun. Pure. Appl. Math., 44 (1991), 939. doi: 10.1002/cpa.3160440808.

[18]

T. J. Bridges, Spatial Hamiltonian structure, energy flux and the water-wave problem,, Proc. Roy. Soc. London Ser. A, 439 (1992), 297. doi: 10.1098/rspa.1992.0151.

[19]

T. J. Bridges, Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference,, J. Nonlinear Sci., 4 (1994), 221. doi: 10.1007/BF02430633.

[20]

B. Buffoni, Existence and conditional energetic stability capillary-gravity solitary water waves by minimisation,, Arch. Rational Mech. Anal., 173 (2004), 25. doi: 10.1007/s00205-004-0310-0.

[21]

B. Buffoni, M. D. Groves, S. M. Sun and E. Wahlen, Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves,, J. Diff. eqns., 254 (2013), 1006. doi: 10.1016/j.jde.2012.10.007.

[22]

B. Buffoni, M. D. Groves and J. F. Toland, A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers,, Phil. Trans. Roy. Soc. Lond. A, 354 (1996), 575. doi: 10.1098/rsta.1996.0020.

[23]

J. G. Byatt-Smith, On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter,, Eur. J. Appl. Math., 2 (1991), 133. doi: 10.1017/S0956792500000449.

[24]

A. R. Champneys, J.-M. Vanden-Broeck and G. J. Lord, Do true elevation gravity-capillary solitary waves exist? A numerical investigation,, J. Fluid Mech., 454 (2002), 403. doi: 10.1017/S0022112001007200.

[25]

S. Deng and S. M. Sun, Three-dimensional gravity-capillary waves on water-small surface tension case,, Phys. D, 238 (2009), 1735. doi: 10.1016/j.physd.2009.05.012.

[26]

S. Deng and S. M. Sun, Exact theory of three-dimensional water waves at the critical speed,, SIAM J. Math. Anal., 42 (2010), 2721. doi: 10.1137/09077922X.

[27]

W. Craig and D. P. Nicholls, Travelling two and three dimensional capillary gravity water waves,, SIAM J. Math. Anal., 32 (2000), 323. doi: 10.1137/S0036141099354181.

[28]

A. de Bouard and J. C. Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 211. doi: 10.1016/S0294-1449(97)80145-X.

[29]

F. Dias and G. Iooss, Water-waves as a spatial dynamical system,, in Handbook of Mathematical Fluid Dynamics, (2003), 443. doi: 10.1016/S1874-5792(03)80012-5.

[30]

K. O. Friedrichs and D. H. Hyers, The existence of solitary waves,, Comm. Pure Appl. Math., 7 (1954), 517. doi: 10.1002/cpa.3160070305.

[31]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3,, Ann. of Math., 175 (2012), 691. doi: 10.4007/annals.2012.175.2.6.

[32]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9.

[33]

R. Grimshaw and N. Joshi, Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation,, SIAM J. Appl. Math., 55 (1995), 124. doi: 10.1137/S0036139993243825.

[34]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II,, J. Funct. Anal., 94 (1990), 308. doi: 10.1016/0022-1236(90)90016-E.

[35]

M. D. Groves, Solitary-wave solutions to a class of fifth-order model equations,, Nonlinearity, 11 (1998), 341. doi: 10.1088/0951-7715/11/2/009.

[36]

M. D. Groves, An existence theory for three-dimensional periodic travelling gravity-capillary water waves with bounded transverse profiles,, Phys. D, 152/153 (2001), 395. doi: 10.1016/S0167-2789(01)00182-8.

[37]

M. D. Groves, M. Haragus and S. M. Sun, Transverse instability of gravity-capillary line solitary water waves,, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 421. doi: 10.1016/S0764-4442(01)02080-8.

[38]

M. D. Groves, M. Haragus-Courcelle and S. M. Sun, A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves,, Phil. Trans. Royal Soc. Lond. A., 360 (2002), 2189. doi: 10.1098/rsta.2002.1066.

[39]

M. D. Groves and A. Mielke, A spatial dynamics approach to three-dimensional gravity-capillary steady water waves,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 83. doi: 10.1017/S0308210500000809.

[40]

M. Groves and S. M. Sun, Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem,, Arch. Rational Mech. Anal., 188 (2008), 1. doi: 10.1007/s00205-007-0085-1.

[41]

M. Haragus and A. Scheel, Finite-wavelength stability of capillary-gravity solitary waves,, Comm. Math. Phys., 225 (2002), 487. doi: 10.1007/s002200100590.

[42]

J. K. Hunter and J.-M. Vanden-Broeck, Solitary and periodic gravity-capillary waves of finite amplitude,, J. Fluid Mech., 134 (1983), 205. doi: 10.1017/S0022112083003316.

[43]

G. Iooss, Gravity and capillary-gravity periodic travelling waves for two superposed fluid layers, one being of infinite depth,, J. Math. Fluid Mech., 1 (1999), 24. doi: 10.1007/s000210050003.

[44]

G. Iooss and K. Kirchgässner, Bifurcation d'ondes solitaires en présence d'une faible tension superficielle,, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 265.

[45]

G. Iooss and K. Kirchgässner, Water waves for small surface tension: An approach via normal form,, Proc. Royal Soc. Edinburgh Sect., 122 (1992), 267. doi: 10.1017/S0308210500021119.

[46]

G. Iooss and M. C. Pérouème, Perturbed homoclinic solutions in reversible 1:1 resonance vector fields,, J. Diff. Equ., 102 (1993), 62. doi: 10.1006/jdeq.1993.1022.

[47]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997). doi: 10.1017/CBO9780511624056.

[48]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media,, Sov. Phys. Dokl., 15 (1970), 539.

[49]

S. Kichenassamy, Existence of solitary waves for water-wave models,, Nonlinearity, 10 (1997), 133. doi: 10.1088/0951-7715/10/1/009.

[50]

K. Kirchgässner, Nonlinearly resonant surface waves and homoclinic bifurcation,, Advances in Applied Mechanics, 26 (1988), 135. doi: 10.1016/S0065-2156(08)70288-6.

[51]

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. doi: 10.1080/14786449508620739.

[52]

M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth,, Stud. Appl. Math., 85 (1991), 129.

[53]

M. A. Lavrentiev, On the theory of long waves (1943); A contribution to the theory of long waves, (1947),, Amer. Math. Soc. Transl., 102 (1954), 3.

[54]

S. Levandosky, Stability and instability of fourth-order solitary waves,, Dyn. Diff. Eqns., 10 (1999), 151. doi: 10.1023/A:1022644629950.

[55]

T. Levi-Civita, D'etermination rigoureuse des ondes permanentes d'ampleur finie,, Math. Ann., 93 (1925), 264. doi: 10.1007/BF01449965.

[56]

P. L. Lions, The concentration-compactness principle in the calculus of variations - the locally compact case, part 1,, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 1 (1984), 109.

[57]

P. L. Lions, The concentration-compactness principle in the calculus of variations - the locally compact case, part 2,, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 1 (1984), 223.

[58]

E. Lombardi, Homoclinic orbits to exponentially small periodic orbits for a class of reversible systems. Application to water waves,, Arch. Rat. Mech. Anal., 137 (1997), 227. doi: 10.1007/s002050050029.

[59]

J. C. Luke, A variational principle for a fluid with a free surface,, J. Fluid Mech., 27 (1967), 395. doi: 10.1017/S0022112067000412.

[60]

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

[61]

A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds,, Berlin, (1991).

[62]

A. Mielke, On the energetic stability of solitary water waves,, Phil. Trans. Royal Soc. Lond. A., 360 (2002), 2337. doi: 10.1098/rsta.2002.1067.

[63]

K. Nishikawa, H. Hojo, K. Mima and H. Ikezi, Coupled nonlinear electron-plasma and ion-acoustic waves,, Phys. Rev. Lett., 33 (1974), 148. doi: 10.1103/PhysRevLett.33.148.

[64]

R. L. Pego and J. R. Quintero, Two-dimensional solitary waves for a Benny-Luke equation,, Physica D, 132 (1999), 476. doi: 10.1016/S0167-2789(99)00058-5.

[65]

R. L. Pego and S. M. Sun, On the transverse linear instability of solitary water waves with large surface tension,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 733. doi: 10.1017/S0308210500003450.

[66]

R. L. Pego and S. M. Sun, Asymptotic linear stability of solitary water waves,, preprint., ().

[67]

R. L. Pego and M. I. Weinstein, Eigenvalues and instabilities of solitary waves,, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47. doi: 10.1098/rsta.1992.0055.

[68]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305. doi: 10.1007/BF02101705.

[69]

Y. Pomeau, A. Ramani and B. Grammaticos, Structural instability of the Korteweg-de Vries solitons under a singular perturbation,, Physica D, 31 (1988), 127. doi: 10.1016/0167-2789(88)90018-8.

[70]

L. Rayleigh, On waves,, Lecture Notes in Applied Mechanics, 5 (2002), 95. doi: 10.1007/978-3-540-45626-1_8.

[71]

J. Reeder and M. Shinbrot, Three-dimensional, nonlinear wave interaction in water of constant depth,, Nonlinear Anal. TMA, 5 (1981), 303. doi: 10.1016/0362-546X(81)90035-3.

[72]

F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves,, Invent. Math., 184 (2011), 257. doi: 10.1007/s00222-010-0290-7.

[73]

R. Sachs, On the existence of small amplitude waves with strong surface tension,, J. Differential Eqns., 90 (1991), 31. doi: 10.1016/0022-0396(91)90159-7.

[74]

J. Scott Russell, Report on waves,, in Report of the 14th Meeting of the British Association for the Advancement of Science, (1844), 311.

[75]

J. J. Stoker, Water Waves: The Mathematical Theory with Applications,, Interscience, (1957).

[76]

S. M. Sun, Existence of a generalized solitary wave solution for water with positive Bond number smaller than 1/3,, J. Math. Anal. Appl., 156 (1991), 471. doi: 10.1016/0022-247X(91)90410-2.

[77]

S. M. Sun, Non-existence of truly solitary waves in water with small surface tension,, R. Soc. Lond. Proc. Ser. A, 455 (1999), 2191. doi: 10.1098/rspa.1999.0399.

[78]

S. M. Sun and M. C. Shen, Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave,, J. Math. Anal. Appl., 172 (1993), 533. doi: 10.1006/jmaa.1993.1042.

[79]

M. Tajiri and Y. Murakami, The periodic soliton resonance: Solutions to the Kadomtsev-Petviashvili equation with positive dispersion,, Phys. Lett. A, 143 (1990), 217. doi: 10.1016/0375-9601(90)90742-7.

[80]

E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I,, 2nd edition, (1962).

[81]

J.-M. Vanden-Broeck, Elevation solitary waves with surface tension,, Phys. Fluids A, 11 (1991), 2659.

[82]

X. P. Wang, M. J. Ablowitz and H. Segur, Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation,, Physica D, 78 (1994), 241. doi: 10.1016/0167-2789(94)90118-X.

[83]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Comm. Pure Appl. Math., 39 (1986), 51. doi: 10.1002/cpa.3160390103.

[84]

M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, Commun. Part. Diff. Eqns., 12 (1987), 1133. doi: 10.1080/03605308708820522.

[85]

S. Wu, Almost global wellposedness of the 2-D full water wave problem,, Invent. Math., 177 (2009), 45. doi: 10.1007/s00222-009-0176-8.

[86]

S. Wu, Global wellposedness of the 3-D full water wave problem,, Invent. Math., 184 (2011), 125. doi: 10.1007/s00222-010-0288-1.

[87]

T. S. Yang and T. R. Akylas, On asymmetric gravity-capillary solitary waves,, J. Fluid Mech., 330 (1997), 215. doi: 10.1017/S0022112096003643.

[88]

V. E. Zakharov, Instability and nonlinear oscillations of solitons,, JETP Lett., 22 (1975), 172.

show all references

References:
[1]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform,, SIAM Studies in Applied Mathematics, (1981).

[2]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, 2nd edition, (2003).

[3]

J. Albert and J. Angulo Pava, Existence and stability of ground-state solutions of a Schrödinger-KdV system,, Proc. Roy. Soc. Edinburgh A, 133 (2003), 987. doi: 10.1017/S030821050000278X.

[4]

J. C. Alexander, R. L. Pego and R. L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation,, Phys. Lett. A, 226 (1997), 187. doi: 10.1016/S0375-9601(96)00921-8.

[5]

C. J. Amick and K. Kirchgässner, A theory of solitary water-waves in the presence of surface tension,, Arch. Rat. Mech. Anal., 105 (1989), 1. doi: 10.1007/BF00251596.

[6]

K. Appert and J. Vaclavik, Dynamics of coupled solitons,, Phys. Fluids, 20 (1977), 1845. doi: 10.1063/1.861802.

[7]

J. T. Beale, The existence of solitary water waves,, Comm. Pure Appl. Math., 30 (1977), 373. doi: 10.1002/cpa.3160300402.

[8]

J. T. Beale, Exact solitary water waves with capillary ripples at infinity,, Comm. Pure Appl. Math., 44 (1991), 211. doi: 10.1002/cpa.3160440204.

[9]

T. B. Benjamin, The stability of solitary waves,, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153. doi: 10.1098/rspa.1972.0074.

[10]

J. L. Bona, On the stability theory of solitary waves,, Proc. Roy. Soc. London Ser. A, 344 (1975), 363. doi: 10.1098/rspa.1975.0106.

[11]

J. Bona and H. Q. Chen, Solitary waves in nonlinear dispersive systems,, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 313. doi: 10.3934/dcdsb.2002.2.313.

[12]

J. L. Bona, P. E. Souganidis and W. A. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type,, Proc. Roy. Soc. London Ser. A, 411 (1987), 395. doi: 10.1098/rspa.1987.0073.

[13]

J. V. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire,, C. R. Acad. Sci. Paris, 72 (1871), 755.

[14]

J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal,, C. R. Acad. Sci. Paris, 73 (1871), 256.

[15]

J. V. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond,, J. Math. Pures Appl., 17 (1872), 55.

[16]

J. V. Boussinesq, Essai sur la théorie des eaux courantes,, Mém. prés. div. Sav. Acad. Sci. Inst. Fr., 23 (1877), 1.

[17]

H. Brezis and L. Nirenberg, Remarks on finding critical points,, Commun. Pure. Appl. Math., 44 (1991), 939. doi: 10.1002/cpa.3160440808.

[18]

T. J. Bridges, Spatial Hamiltonian structure, energy flux and the water-wave problem,, Proc. Roy. Soc. London Ser. A, 439 (1992), 297. doi: 10.1098/rspa.1992.0151.

[19]

T. J. Bridges, Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference,, J. Nonlinear Sci., 4 (1994), 221. doi: 10.1007/BF02430633.

[20]

B. Buffoni, Existence and conditional energetic stability capillary-gravity solitary water waves by minimisation,, Arch. Rational Mech. Anal., 173 (2004), 25. doi: 10.1007/s00205-004-0310-0.

[21]

B. Buffoni, M. D. Groves, S. M. Sun and E. Wahlen, Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves,, J. Diff. eqns., 254 (2013), 1006. doi: 10.1016/j.jde.2012.10.007.

[22]

B. Buffoni, M. D. Groves and J. F. Toland, A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers,, Phil. Trans. Roy. Soc. Lond. A, 354 (1996), 575. doi: 10.1098/rsta.1996.0020.

[23]

J. G. Byatt-Smith, On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter,, Eur. J. Appl. Math., 2 (1991), 133. doi: 10.1017/S0956792500000449.

[24]

A. R. Champneys, J.-M. Vanden-Broeck and G. J. Lord, Do true elevation gravity-capillary solitary waves exist? A numerical investigation,, J. Fluid Mech., 454 (2002), 403. doi: 10.1017/S0022112001007200.

[25]

S. Deng and S. M. Sun, Three-dimensional gravity-capillary waves on water-small surface tension case,, Phys. D, 238 (2009), 1735. doi: 10.1016/j.physd.2009.05.012.

[26]

S. Deng and S. M. Sun, Exact theory of three-dimensional water waves at the critical speed,, SIAM J. Math. Anal., 42 (2010), 2721. doi: 10.1137/09077922X.

[27]

W. Craig and D. P. Nicholls, Travelling two and three dimensional capillary gravity water waves,, SIAM J. Math. Anal., 32 (2000), 323. doi: 10.1137/S0036141099354181.

[28]

A. de Bouard and J. C. Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 211. doi: 10.1016/S0294-1449(97)80145-X.

[29]

F. Dias and G. Iooss, Water-waves as a spatial dynamical system,, in Handbook of Mathematical Fluid Dynamics, (2003), 443. doi: 10.1016/S1874-5792(03)80012-5.

[30]

K. O. Friedrichs and D. H. Hyers, The existence of solitary waves,, Comm. Pure Appl. Math., 7 (1954), 517. doi: 10.1002/cpa.3160070305.

[31]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3,, Ann. of Math., 175 (2012), 691. doi: 10.4007/annals.2012.175.2.6.

[32]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9.

[33]

R. Grimshaw and N. Joshi, Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation,, SIAM J. Appl. Math., 55 (1995), 124. doi: 10.1137/S0036139993243825.

[34]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II,, J. Funct. Anal., 94 (1990), 308. doi: 10.1016/0022-1236(90)90016-E.

[35]

M. D. Groves, Solitary-wave solutions to a class of fifth-order model equations,, Nonlinearity, 11 (1998), 341. doi: 10.1088/0951-7715/11/2/009.

[36]

M. D. Groves, An existence theory for three-dimensional periodic travelling gravity-capillary water waves with bounded transverse profiles,, Phys. D, 152/153 (2001), 395. doi: 10.1016/S0167-2789(01)00182-8.

[37]

M. D. Groves, M. Haragus and S. M. Sun, Transverse instability of gravity-capillary line solitary water waves,, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 421. doi: 10.1016/S0764-4442(01)02080-8.

[38]

M. D. Groves, M. Haragus-Courcelle and S. M. Sun, A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves,, Phil. Trans. Royal Soc. Lond. A., 360 (2002), 2189. doi: 10.1098/rsta.2002.1066.

[39]

M. D. Groves and A. Mielke, A spatial dynamics approach to three-dimensional gravity-capillary steady water waves,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 83. doi: 10.1017/S0308210500000809.

[40]

M. Groves and S. M. Sun, Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem,, Arch. Rational Mech. Anal., 188 (2008), 1. doi: 10.1007/s00205-007-0085-1.

[41]

M. Haragus and A. Scheel, Finite-wavelength stability of capillary-gravity solitary waves,, Comm. Math. Phys., 225 (2002), 487. doi: 10.1007/s002200100590.

[42]

J. K. Hunter and J.-M. Vanden-Broeck, Solitary and periodic gravity-capillary waves of finite amplitude,, J. Fluid Mech., 134 (1983), 205. doi: 10.1017/S0022112083003316.

[43]

G. Iooss, Gravity and capillary-gravity periodic travelling waves for two superposed fluid layers, one being of infinite depth,, J. Math. Fluid Mech., 1 (1999), 24. doi: 10.1007/s000210050003.

[44]

G. Iooss and K. Kirchgässner, Bifurcation d'ondes solitaires en présence d'une faible tension superficielle,, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 265.

[45]

G. Iooss and K. Kirchgässner, Water waves for small surface tension: An approach via normal form,, Proc. Royal Soc. Edinburgh Sect., 122 (1992), 267. doi: 10.1017/S0308210500021119.

[46]

G. Iooss and M. C. Pérouème, Perturbed homoclinic solutions in reversible 1:1 resonance vector fields,, J. Diff. Equ., 102 (1993), 62. doi: 10.1006/jdeq.1993.1022.

[47]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997). doi: 10.1017/CBO9780511624056.

[48]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media,, Sov. Phys. Dokl., 15 (1970), 539.

[49]

S. Kichenassamy, Existence of solitary waves for water-wave models,, Nonlinearity, 10 (1997), 133. doi: 10.1088/0951-7715/10/1/009.

[50]

K. Kirchgässner, Nonlinearly resonant surface waves and homoclinic bifurcation,, Advances in Applied Mechanics, 26 (1988), 135. doi: 10.1016/S0065-2156(08)70288-6.

[51]

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. doi: 10.1080/14786449508620739.

[52]

M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth,, Stud. Appl. Math., 85 (1991), 129.

[53]

M. A. Lavrentiev, On the theory of long waves (1943); A contribution to the theory of long waves, (1947),, Amer. Math. Soc. Transl., 102 (1954), 3.

[54]

S. Levandosky, Stability and instability of fourth-order solitary waves,, Dyn. Diff. Eqns., 10 (1999), 151. doi: 10.1023/A:1022644629950.

[55]

T. Levi-Civita, D'etermination rigoureuse des ondes permanentes d'ampleur finie,, Math. Ann., 93 (1925), 264. doi: 10.1007/BF01449965.

[56]

P. L. Lions, The concentration-compactness principle in the calculus of variations - the locally compact case, part 1,, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 1 (1984), 109.

[57]

P. L. Lions, The concentration-compactness principle in the calculus of variations - the locally compact case, part 2,, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 1 (1984), 223.

[58]

E. Lombardi, Homoclinic orbits to exponentially small periodic orbits for a class of reversible systems. Application to water waves,, Arch. Rat. Mech. Anal., 137 (1997), 227. doi: 10.1007/s002050050029.

[59]

J. C. Luke, A variational principle for a fluid with a free surface,, J. Fluid Mech., 27 (1967), 395. doi: 10.1017/S0022112067000412.

[60]

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

[61]

A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds,, Berlin, (1991).

[62]

A. Mielke, On the energetic stability of solitary water waves,, Phil. Trans. Royal Soc. Lond. A., 360 (2002), 2337. doi: 10.1098/rsta.2002.1067.

[63]

K. Nishikawa, H. Hojo, K. Mima and H. Ikezi, Coupled nonlinear electron-plasma and ion-acoustic waves,, Phys. Rev. Lett., 33 (1974), 148. doi: 10.1103/PhysRevLett.33.148.

[64]

R. L. Pego and J. R. Quintero, Two-dimensional solitary waves for a Benny-Luke equation,, Physica D, 132 (1999), 476. doi: 10.1016/S0167-2789(99)00058-5.

[65]

R. L. Pego and S. M. Sun, On the transverse linear instability of solitary water waves with large surface tension,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 733. doi: 10.1017/S0308210500003450.

[66]

R. L. Pego and S. M. Sun, Asymptotic linear stability of solitary water waves,, preprint., ().

[67]

R. L. Pego and M. I. Weinstein, Eigenvalues and instabilities of solitary waves,, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47. doi: 10.1098/rsta.1992.0055.

[68]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305. doi: 10.1007/BF02101705.

[69]

Y. Pomeau, A. Ramani and B. Grammaticos, Structural instability of the Korteweg-de Vries solitons under a singular perturbation,, Physica D, 31 (1988), 127. doi: 10.1016/0167-2789(88)90018-8.

[70]

L. Rayleigh, On waves,, Lecture Notes in Applied Mechanics, 5 (2002), 95. doi: 10.1007/978-3-540-45626-1_8.

[71]

J. Reeder and M. Shinbrot, Three-dimensional, nonlinear wave interaction in water of constant depth,, Nonlinear Anal. TMA, 5 (1981), 303. doi: 10.1016/0362-546X(81)90035-3.

[72]

F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves,, Invent. Math., 184 (2011), 257. doi: 10.1007/s00222-010-0290-7.

[73]

R. Sachs, On the existence of small amplitude waves with strong surface tension,, J. Differential Eqns., 90 (1991), 31. doi: 10.1016/0022-0396(91)90159-7.

[74]

J. Scott Russell, Report on waves,, in Report of the 14th Meeting of the British Association for the Advancement of Science, (1844), 311.

[75]

J. J. Stoker, Water Waves: The Mathematical Theory with Applications,, Interscience, (1957).

[76]

S. M. Sun, Existence of a generalized solitary wave solution for water with positive Bond number smaller than 1/3,, J. Math. Anal. Appl., 156 (1991), 471. doi: 10.1016/0022-247X(91)90410-2.

[77]

S. M. Sun, Non-existence of truly solitary waves in water with small surface tension,, R. Soc. Lond. Proc. Ser. A, 455 (1999), 2191. doi: 10.1098/rspa.1999.0399.

[78]

S. M. Sun and M. C. Shen, Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave,, J. Math. Anal. Appl., 172 (1993), 533. doi: 10.1006/jmaa.1993.1042.

[79]

M. Tajiri and Y. Murakami, The periodic soliton resonance: Solutions to the Kadomtsev-Petviashvili equation with positive dispersion,, Phys. Lett. A, 143 (1990), 217. doi: 10.1016/0375-9601(90)90742-7.

[80]

E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I,, 2nd edition, (1962).

[81]

J.-M. Vanden-Broeck, Elevation solitary waves with surface tension,, Phys. Fluids A, 11 (1991), 2659.

[82]

X. P. Wang, M. J. Ablowitz and H. Segur, Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation,, Physica D, 78 (1994), 241. doi: 10.1016/0167-2789(94)90118-X.

[83]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Comm. Pure Appl. Math., 39 (1986), 51. doi: 10.1002/cpa.3160390103.

[84]

M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, Commun. Part. Diff. Eqns., 12 (1987), 1133. doi: 10.1080/03605308708820522.

[85]

S. Wu, Almost global wellposedness of the 2-D full water wave problem,, Invent. Math., 177 (2009), 45. doi: 10.1007/s00222-009-0176-8.

[86]

S. Wu, Global wellposedness of the 3-D full water wave problem,, Invent. Math., 184 (2011), 125. doi: 10.1007/s00222-010-0288-1.

[87]

T. S. Yang and T. R. Akylas, On asymmetric gravity-capillary solitary waves,, J. Fluid Mech., 330 (1997), 215. doi: 10.1017/S0022112096003643.

[88]

V. E. Zakharov, Instability and nonlinear oscillations of solitons,, JETP Lett., 22 (1975), 172.

[1]

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

[2]

Frédéric Rousset, Nikolay Tzvetkov. On the transverse instability of one dimensional capillary-gravity waves. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 859-872. doi: 10.3934/dcdsb.2010.13.859

[3]

Mark Jones. The bifurcation of interfacial capillary-gravity waves under O(2) symmetry. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1183-1204. doi: 10.3934/cpaa.2011.10.1183

[4]

Kristoffer Varholm. Solitary gravity-capillary water waves with point vortices. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3927-3959. doi: 10.3934/dcds.2016.36.3927

[5]

Kenta Ohi, Tatsuo Iguchi. A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1205-1240. doi: 10.3934/dcds.2009.23.1205

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030

[13]

Samuel Walsh. Steady stratified periodic gravity waves with surface tension II: Global bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3287-3315. doi: 10.3934/dcds.2014.34.3287

[14]

Samuel Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3241-3285. doi: 10.3934/dcds.2014.34.3241

[15]

Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997

[16]

Mats Ehrnström, Gabriele Villari. Recent progress on particle trajectories in steady water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 539-559. doi: 10.3934/dcdsb.2009.12.539

[17]

David M. Ambrose, Jerry L. Bona, David P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1113-1137. doi: 10.3934/dcdsb.2012.17.1113

[18]

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

[19]

Gerhard Tulzer. On the symmetry of steady periodic water waves with stagnation points. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1577-1586. doi: 10.3934/cpaa.2012.11.1577

[20]

David Henry, Hung-Chu Hsu. Instability of equatorial water waves in the $f-$plane. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 909-916. doi: 10.3934/dcds.2015.35.909

2017 Impact Factor: 0.542

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]