November  2013, 33(11&12): 4841-4873. doi: 10.3934/dcds.2013.33.4841

Periodic traveling--wave solutions of nonlinear dispersive evolution equations

1. 

Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, 38152

2. 

Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States

Received  February 2012 Revised  November 2012 Published  May 2013

For a general class of nonlinear, dispersive wave equations, existence of periodic, traveling-wave solutions is studied. These traveling waveforms are the analog of the classical cnoidal-wave solutions of the Korteweg-de Vries equation. They are determined to be stable to perturbation of the same period. Their large wavelength limit is shown to be solitary waves.
Citation: Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841
References:
[1]

L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Non-local models for nonlinear, dispersive waves,, Physica D, 40 (1989), 360. doi: 10.1016/0167-2789(89)90050-X.

[2]

J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations,, Applied Analysis (Baton Rouge, 221 (1999), 1. doi: 10.1090/conm/221/03116.

[3]

J. P. Albert, J. L. Bona and D. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves,, Physica D, 24 (1987), 343. doi: 10.1016/0167-2789(87)90084-4.

[4]

C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation - a nonlinear Neumann problem in the plane,, Acta Math., 167 (1991), 107. doi: 10.1007/BF02392447.

[5]

J. Angulo Pava, "Nonlinear Dispersive Equations,", Mathematical Surveys and Monographs, 156 (2009).

[6]

J. Angulo Pava, J. L. Bona and M. Scialom, Stability of cnoidal waves,, Advances in Differenticial Equations, 11 (2006), 1321.

[7]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, J. Fluid Mech., 29 (1967), 559. doi: 10.1017/S002211206700103X.

[8]

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

[9]

T. B. Benjamin, Lectures on nonlinear wave motion,, Lect. Appl. Math., 15 (1974), 3.

[10]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Phil. Trans. Royal Soc. London, 272 (1972), 47. doi: 10.1098/rsta.1972.0032.

[11]

D. P. Bennett, R. W. Brown, S. E. Stansfield, J. D. Stroughair and J. L. Bona, The stability of internal solitary waves in stratified fluids,, Math. Proc. Cambridge Philos. Soc., 94 (1983), 351. doi: 10.1017/S0305004100061193.

[12]

J. L. Bona, On the stability theory of solitary waves,, Proc. Royal Soc. London, 344 (1975), 367. doi: 10.1098/rspa.1975.0106.

[13]

J. L. Bona, Convergence of periodic wave trains in the limit of large wavelength,, Appl. Sci. Res., 37 (1981), 21. doi: 10.1007/BF00382614.

[14]

J. L. Bona, On solitary waves and their role in the evolution of long waves,, in, (1989), 183.

[15]

J. L. Bona and H. Kalisch, Models for internal waves in deep water,, Discrete Cont. Dynamical Sys., 6 (2000), 1. doi: 10.3934/dcds.2000.6.1.

[16]

J. L. Bona, Y. Liu and N. Nguyen, Stability of solitary waves in higher-order Sobolev spaces,, Commun. Math. Sci., 2 (2004), 35.

[17]

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

[18]

J. L. Bona and A. Soyeur, On the stability of solitary-wave solutions of model equations for long waves,, J. Nonlinear Sci., 4 (1994), 449. doi: 10.1007/BF02430641.

[19]

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.

[20]

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

[21]

H. Chen, Existence of periodic traveling-wave solutions of nonlinear, dispersive wave equations,, Nonlinearity, 17 (2004), 2041. doi: 10.1088/0951-7715/17/6/003.

[22]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (2000).

[23]

A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-De Vries equation,, SIAM Review, 14 (1972), 582. doi: 10.1137/1014101.

[24]

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,, Philosophical Magazine, 39 (1895), 422. doi: 10.1080/14786449508620739.

[25]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, Part I,, Ann. Inst. H. Poincaré, 1 (1984), 109.

[26]

L. Molinet, J.-C. Saut and N. Tzvetkov, Ill-posedness Issues for the Benjamin-Ono and related equations,, SIAM J. Math. Anal., 33 (2001), 982. doi: 10.1137/S0036141001385307.

[27]

I. A. Svendsen and J. Buhr Hansen, Laboratory generation of waves of constant form,, in the 14th Coastal Engr. Conf., (1974), 321.

[28]

I. A. Svendsen and J. Buhr Hansen, On the deformation of periodic long waves over a gently sloping bottom,, J. Fluid Mech., 87 (1978), 433. doi: 10.1017/S0022112078001706.

[29]

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

show all references

References:
[1]

L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Non-local models for nonlinear, dispersive waves,, Physica D, 40 (1989), 360. doi: 10.1016/0167-2789(89)90050-X.

[2]

J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations,, Applied Analysis (Baton Rouge, 221 (1999), 1. doi: 10.1090/conm/221/03116.

[3]

J. P. Albert, J. L. Bona and D. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves,, Physica D, 24 (1987), 343. doi: 10.1016/0167-2789(87)90084-4.

[4]

C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation - a nonlinear Neumann problem in the plane,, Acta Math., 167 (1991), 107. doi: 10.1007/BF02392447.

[5]

J. Angulo Pava, "Nonlinear Dispersive Equations,", Mathematical Surveys and Monographs, 156 (2009).

[6]

J. Angulo Pava, J. L. Bona and M. Scialom, Stability of cnoidal waves,, Advances in Differenticial Equations, 11 (2006), 1321.

[7]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, J. Fluid Mech., 29 (1967), 559. doi: 10.1017/S002211206700103X.

[8]

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

[9]

T. B. Benjamin, Lectures on nonlinear wave motion,, Lect. Appl. Math., 15 (1974), 3.

[10]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Phil. Trans. Royal Soc. London, 272 (1972), 47. doi: 10.1098/rsta.1972.0032.

[11]

D. P. Bennett, R. W. Brown, S. E. Stansfield, J. D. Stroughair and J. L. Bona, The stability of internal solitary waves in stratified fluids,, Math. Proc. Cambridge Philos. Soc., 94 (1983), 351. doi: 10.1017/S0305004100061193.

[12]

J. L. Bona, On the stability theory of solitary waves,, Proc. Royal Soc. London, 344 (1975), 367. doi: 10.1098/rspa.1975.0106.

[13]

J. L. Bona, Convergence of periodic wave trains in the limit of large wavelength,, Appl. Sci. Res., 37 (1981), 21. doi: 10.1007/BF00382614.

[14]

J. L. Bona, On solitary waves and their role in the evolution of long waves,, in, (1989), 183.

[15]

J. L. Bona and H. Kalisch, Models for internal waves in deep water,, Discrete Cont. Dynamical Sys., 6 (2000), 1. doi: 10.3934/dcds.2000.6.1.

[16]

J. L. Bona, Y. Liu and N. Nguyen, Stability of solitary waves in higher-order Sobolev spaces,, Commun. Math. Sci., 2 (2004), 35.

[17]

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

[18]

J. L. Bona and A. Soyeur, On the stability of solitary-wave solutions of model equations for long waves,, J. Nonlinear Sci., 4 (1994), 449. doi: 10.1007/BF02430641.

[19]

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.

[20]

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

[21]

H. Chen, Existence of periodic traveling-wave solutions of nonlinear, dispersive wave equations,, Nonlinearity, 17 (2004), 2041. doi: 10.1088/0951-7715/17/6/003.

[22]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (2000).

[23]

A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-De Vries equation,, SIAM Review, 14 (1972), 582. doi: 10.1137/1014101.

[24]

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,, Philosophical Magazine, 39 (1895), 422. doi: 10.1080/14786449508620739.

[25]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, Part I,, Ann. Inst. H. Poincaré, 1 (1984), 109.

[26]

L. Molinet, J.-C. Saut and N. Tzvetkov, Ill-posedness Issues for the Benjamin-Ono and related equations,, SIAM J. Math. Anal., 33 (2001), 982. doi: 10.1137/S0036141001385307.

[27]

I. A. Svendsen and J. Buhr Hansen, Laboratory generation of waves of constant form,, in the 14th Coastal Engr. Conf., (1974), 321.

[28]

I. A. Svendsen and J. Buhr Hansen, On the deformation of periodic long waves over a gently sloping bottom,, J. Fluid Mech., 87 (1978), 433. doi: 10.1017/S0022112078001706.

[29]

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

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