July  2012, 11(4): 1497-1522. doi: 10.3934/cpaa.2012.11.1497

A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques

1. 

School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom

Received  June 2011 Revised  September 2011 Published  January 2012

The methods of analysis based on asymptotic expansions, with a small parameter, are briefly outlined. These techniques are then applied to three examples in the theory of water waves, the aim being to demonstrate the effectiveness of this approach. Throughout, we relate this procedure to more rigorous methods.
The classical problem of water waves is formulated, and then various parameter choices are incorporated. The first problem examines the properties of small-amplitude, periodic waves over constant vorticity and, invoking the ideas of breakdown, scaling and matching, the possibility of stagnation is investigated. In the second case, a Camassa-Holm equation is derived; this is found to be relevant only for the horizontal velocity component in the flow at a specific depth. However, this special, integrable equation requires the retention of a number of terms of different asymptotic order--which is the least satisfactory way to use these methods. The final example, which shows how some new aspects of a familiar problem can be obtained by these methods, develops an asymptotic description of edge waves. This leads to a single equation that captures all aspects of the run-up pattern at a shoreline.
Citation: 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
References:
[1]

T. B. Benjamin, J. L. Bona and J. J. Mahoney, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. R. Soc. Lond. A, 227 (1972), 47.   Google Scholar

[2]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity,, Comm. Pure Appl. Math., 64 (2011), 975.   Google Scholar

[3]

A. W. Bush, "Perturbation Methods for Engineers and Scientists,", CRC Press, (1992).   Google Scholar

[4]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.   Google Scholar

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. R. Soc. Lond., A457 (2001), 953.   Google Scholar

[6]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A Maths. Gen., 34 (2001), 9723.   Google Scholar

[7]

A. Constantin, M. Ehrnstrom and E. Wahlen, Symmetry of steady periodic waves with vorticity,, Duke Math. J., 140 (2007), 591.   Google Scholar

[8]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity,, Eur. J. Appl. Math., 15 (2004), 755.   Google Scholar

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559.   Google Scholar

[10]

A. Constantin, V. S. Gerdokov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Probl., 22 (2006), 2197.   Google Scholar

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 192 (2009), 165.   Google Scholar

[12]

A. Constantin and W. Strauss, Exact periodic travelling water waves with vorticity,, C. R. Math. Acad. Sci. Paris, 335 (2002), 797.   Google Scholar

[13]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481.   Google Scholar

[14]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation,, Phil. Trans. Roy. Soc., A365 (2007), 2227.   Google Scholar

[15]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Rat. Mech. Anal., 202 (2011), 133.   Google Scholar

[16]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation,, Arch. Rat. Mech. Anal., 199 (2011), 33.   Google Scholar

[17]

E. T. Copson, "Asymptotic Expansions,", Cambridge University Press, (1967).   Google Scholar

[18]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie,, J. Math. Pures Appl., 13 (1934), 217.   Google Scholar

[19]

M. Ehrnstrom and G. Villari, Linear water waves with vorticity: rotational features and particle paths,, J. Differential Equations, 244 (2008), 1888.   Google Scholar

[20]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their B?cklund transformation and hereditary symmetries,, Physica D, 4 (1981), 821.   Google Scholar

[21]

W. B. Ford, "Divergent Series, Summability and Asymptotics,", Chelsea, (1960).   Google Scholar

[22]

N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows,, J. Fluid Mech., 42 (1970), 401.   Google Scholar

[23]

F. Gerstner, Theorie der Wellen. Abh. d. k. böhm. Ges. d. Wiss,, (1802) [Also see Ann. Phys. Lpz. 2, (1802).   Google Scholar

[24]

G. H. Hardy, "Divergent Series,", Clarendon, (1949).   Google Scholar

[25]

E. J. Hinch, "Perturbation Methods,", Cambridge University Press, (1991).   Google Scholar

[26]

M. H. Holmes, "Introduction to Perturbation Methods,", Springer-Verlag, (1995).   Google Scholar

[27]

R. S. Johnson, On the development of a solitary wave over an uneven bottom,, Proc. Camb. Phil. Soc., 73 (1973), 183.   Google Scholar

[28]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,", CUP, (1997).   Google Scholar

[29]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 457 (2002), 63.   Google Scholar

[30]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dyn. Res., 33 (2003), 97.   Google Scholar

[31]

R. S. Johnson, "Singular Perturbation Theory,", Springer, (2004).   Google Scholar

[32]

R. S. Johnson, Some contributions to the theory of edge waves,, J. Fluid Mech., 524 (2005), 81.   Google Scholar

[33]

R. S. Johnson, Edge waves: theories past and present,, Phil. Trans. R. Soc. Lond., A365 (2007), 2359.   Google Scholar

[34]

R. S. Johnson, Water waves near a shoreline in a flow with vorticity: two classical examples,, J. Nonlinear Math. Phys., 15 (2008), 133.   Google Scholar

[35]

R. S. Johnson, Periodic waves over constant vorticity: some asymptotic results generated by parameter expansions,, Wave Motion, 46 (2009), 339.   Google Scholar

[36]

J. Kevorkian and J. D. Cole, "Perturbation Methods in Applied Mathematics,", Springer-Verlag, (1981).   Google Scholar

[37]

J. Kevorkian and J.D. Cole, "Multiple Scale and Singular Perturbation Methods,", Springer-Verlag, (1996).   Google Scholar

[38]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves,, Eur. J. Mech. B Fluids, 27 (2008), 96.   Google Scholar

[39]

J. Ko and W. Strauss, Effect of vorticity on steady water waves,, J. Fluid Mech., 608 (2008), 197.   Google Scholar

[40]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on anew type of long stationary waves,, Philos. Mag., 39 (1895), 422.   Google Scholar

[41]

H. Lamb, "Hydrodynamics,", (6th edition), (1932).   Google Scholar

[42]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave,, Phys. Fluids, 25 (1982), 586.   Google Scholar

[43]

J. D. Murray, "Asymptotic Analysis,", Clarendon, (1974).   Google Scholar

[44]

D. H. Peregrine, Calculations of the development of an undular bore,, J. Fluid Mech., 25 (1966), 321.   Google Scholar

[45]

W. J. M. Rankine, On the exact form of waves near the surface of deep water,, Phil. Trans. R. Soc. Lond., 153 (1863), 127.   Google Scholar

[46]

D. R. Smith, "Singular-perturbation Theory: An Introduction with Applications,", Cambridge University Press, (1985).   Google Scholar

[47]

G. G. Stokes, Report on recent researches in hydrodynamics,, Rep. 16th Brit. Assoc. Adv. Sci. (1846) 1-20. [See also Papers, (1846), 1.   Google Scholar

[48]

M. D. van Dyke, "Perturbation Methods in Fluid Mechanics,", Academic Press, (1964).   Google Scholar

[49]

E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468.   Google Scholar

[50]

C.-S. Yih, Note on edge waves in a stratified fluid,, J. Fluid Mech., 24 (1966), 765.   Google Scholar

show all references

References:
[1]

T. B. Benjamin, J. L. Bona and J. J. Mahoney, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. R. Soc. Lond. A, 227 (1972), 47.   Google Scholar

[2]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity,, Comm. Pure Appl. Math., 64 (2011), 975.   Google Scholar

[3]

A. W. Bush, "Perturbation Methods for Engineers and Scientists,", CRC Press, (1992).   Google Scholar

[4]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.   Google Scholar

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. R. Soc. Lond., A457 (2001), 953.   Google Scholar

[6]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A Maths. Gen., 34 (2001), 9723.   Google Scholar

[7]

A. Constantin, M. Ehrnstrom and E. Wahlen, Symmetry of steady periodic waves with vorticity,, Duke Math. J., 140 (2007), 591.   Google Scholar

[8]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity,, Eur. J. Appl. Math., 15 (2004), 755.   Google Scholar

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559.   Google Scholar

[10]

A. Constantin, V. S. Gerdokov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Probl., 22 (2006), 2197.   Google Scholar

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 192 (2009), 165.   Google Scholar

[12]

A. Constantin and W. Strauss, Exact periodic travelling water waves with vorticity,, C. R. Math. Acad. Sci. Paris, 335 (2002), 797.   Google Scholar

[13]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481.   Google Scholar

[14]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation,, Phil. Trans. Roy. Soc., A365 (2007), 2227.   Google Scholar

[15]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Rat. Mech. Anal., 202 (2011), 133.   Google Scholar

[16]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation,, Arch. Rat. Mech. Anal., 199 (2011), 33.   Google Scholar

[17]

E. T. Copson, "Asymptotic Expansions,", Cambridge University Press, (1967).   Google Scholar

[18]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie,, J. Math. Pures Appl., 13 (1934), 217.   Google Scholar

[19]

M. Ehrnstrom and G. Villari, Linear water waves with vorticity: rotational features and particle paths,, J. Differential Equations, 244 (2008), 1888.   Google Scholar

[20]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their B?cklund transformation and hereditary symmetries,, Physica D, 4 (1981), 821.   Google Scholar

[21]

W. B. Ford, "Divergent Series, Summability and Asymptotics,", Chelsea, (1960).   Google Scholar

[22]

N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows,, J. Fluid Mech., 42 (1970), 401.   Google Scholar

[23]

F. Gerstner, Theorie der Wellen. Abh. d. k. böhm. Ges. d. Wiss,, (1802) [Also see Ann. Phys. Lpz. 2, (1802).   Google Scholar

[24]

G. H. Hardy, "Divergent Series,", Clarendon, (1949).   Google Scholar

[25]

E. J. Hinch, "Perturbation Methods,", Cambridge University Press, (1991).   Google Scholar

[26]

M. H. Holmes, "Introduction to Perturbation Methods,", Springer-Verlag, (1995).   Google Scholar

[27]

R. S. Johnson, On the development of a solitary wave over an uneven bottom,, Proc. Camb. Phil. Soc., 73 (1973), 183.   Google Scholar

[28]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,", CUP, (1997).   Google Scholar

[29]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 457 (2002), 63.   Google Scholar

[30]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dyn. Res., 33 (2003), 97.   Google Scholar

[31]

R. S. Johnson, "Singular Perturbation Theory,", Springer, (2004).   Google Scholar

[32]

R. S. Johnson, Some contributions to the theory of edge waves,, J. Fluid Mech., 524 (2005), 81.   Google Scholar

[33]

R. S. Johnson, Edge waves: theories past and present,, Phil. Trans. R. Soc. Lond., A365 (2007), 2359.   Google Scholar

[34]

R. S. Johnson, Water waves near a shoreline in a flow with vorticity: two classical examples,, J. Nonlinear Math. Phys., 15 (2008), 133.   Google Scholar

[35]

R. S. Johnson, Periodic waves over constant vorticity: some asymptotic results generated by parameter expansions,, Wave Motion, 46 (2009), 339.   Google Scholar

[36]

J. Kevorkian and J. D. Cole, "Perturbation Methods in Applied Mathematics,", Springer-Verlag, (1981).   Google Scholar

[37]

J. Kevorkian and J.D. Cole, "Multiple Scale and Singular Perturbation Methods,", Springer-Verlag, (1996).   Google Scholar

[38]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves,, Eur. J. Mech. B Fluids, 27 (2008), 96.   Google Scholar

[39]

J. Ko and W. Strauss, Effect of vorticity on steady water waves,, J. Fluid Mech., 608 (2008), 197.   Google Scholar

[40]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on anew type of long stationary waves,, Philos. Mag., 39 (1895), 422.   Google Scholar

[41]

H. Lamb, "Hydrodynamics,", (6th edition), (1932).   Google Scholar

[42]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave,, Phys. Fluids, 25 (1982), 586.   Google Scholar

[43]

J. D. Murray, "Asymptotic Analysis,", Clarendon, (1974).   Google Scholar

[44]

D. H. Peregrine, Calculations of the development of an undular bore,, J. Fluid Mech., 25 (1966), 321.   Google Scholar

[45]

W. J. M. Rankine, On the exact form of waves near the surface of deep water,, Phil. Trans. R. Soc. Lond., 153 (1863), 127.   Google Scholar

[46]

D. R. Smith, "Singular-perturbation Theory: An Introduction with Applications,", Cambridge University Press, (1985).   Google Scholar

[47]

G. G. Stokes, Report on recent researches in hydrodynamics,, Rep. 16th Brit. Assoc. Adv. Sci. (1846) 1-20. [See also Papers, (1846), 1.   Google Scholar

[48]

M. D. van Dyke, "Perturbation Methods in Fluid Mechanics,", Academic Press, (1964).   Google Scholar

[49]

E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468.   Google Scholar

[50]

C.-S. Yih, Note on edge waves in a stratified fluid,, J. Fluid Mech., 24 (1966), 765.   Google Scholar

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