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

Periodic traveling--wave solutions of nonlinear dispersive evolution equations

Hongqiu Chen 1, and  Jerry L. Bona 2,

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.
Keywords: generalized cnoidal waves, stability theory., solitary waves, Period traveling waves, generalized Korteweg--de Vries equations, non-local dispersion relations.
Mathematics Subject Classification: Primary: 35C07, 35C08, 35Q35, 35Q51, 35Q53, 35S10, 76B15, 76B25; Secondary: 35C10, 35Q86, 86A0.
Citation: Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems, 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), 360-392. doi: 10.1016/0167-2789(89)90050-X.  Google Scholar

[2]

J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, Applied Analysis (Baton Rouge, LA 1996), Contemp. Math. 221, American Math. Soc., Providence, RI (1999), 1-29. doi: 10.1090/conm/221/03116.  Google Scholar

[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), 343-366. doi: 10.1016/0167-2789(87)90084-4.  Google Scholar

[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), 107-126. doi: 10.1007/BF02392447.  Google Scholar

[5]

J. Angulo Pava, "Nonlinear Dispersive Equations," Mathematical Surveys and Monographs, 156, American Math. Soc.: Providence, RI 2009.  Google Scholar

[6]

J. Angulo Pava, J. L. Bona and M. Scialom, Stability of cnoidal waves, Advances in Differenticial Equations, 11 (2006), 1321-1374.  Google Scholar

[7]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X.  Google Scholar

[8]

T. B. Benjamin, The stability of solitary waves, Proc. Royal Soc. London Ser. A, 328 (1972), 153-183. doi: 10.1098/rspa.1972.0074.  Google Scholar

[9]

T. B. Benjamin, Lectures on nonlinear wave motion, Lect. Appl. Math., 15, (ed. A. Newell) American Math. Soc., Providence, RI (1974), 3-47.  Google Scholar

[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, Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[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), 351-379. doi: 10.1017/S0305004100061193.  Google Scholar

[12]

J. L. Bona, On the stability theory of solitary waves, Proc. Royal Soc. London, Ser. A, 344 (1975), 367-374. doi: 10.1098/rspa.1975.0106.  Google Scholar

[13]

J. L. Bona, Convergence of periodic wave trains in the limit of large wavelength, Appl. Sci. Res., 37 (1981), 21-30. doi: 10.1007/BF00382614.  Google Scholar

[14]

J. L. Bona, On solitary waves and their role in the evolution of long waves, in "Applications of Nonlinear Analysis in the Physical Sciences" (eds. H. Amann, N. Bazley and K. Kirchgässner), Pitman: London (1989), 183-205. Google Scholar

[15]

J. L. Bona and H. Kalisch, Models for internal waves in deep water, Discrete Cont. Dynamical Sys., 6 (2000), 1-20. doi: 10.3934/dcds.2000.6.1.  Google Scholar

[16]

J. L. Bona, Y. Liu and N. Nguyen, Stability of solitary waves in higher-order Sobolev spaces, Commun. Math. Sci., 2 (2004), 35-52.  Google Scholar

[17]

J. L. Bona, P. E. Souganidis and W. A. Strauss, Stability and instability of solitary waves of KdV-type, Proc. Royal Soc. London, Ser. A, 411 (1987), 395-412. doi: 10.1098/rspa.1987.0073.  Google Scholar

[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), 449-470. doi: 10.1007/BF02430641.  Google Scholar

[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), 755-759. Google Scholar

[20]

J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal, C. R. Acad. Sci. Paris, 73 (1871), 256-260. Google Scholar

[21]

H. Chen, Existence of periodic traveling-wave solutions of nonlinear, dispersive wave equations, Nonlinearity, 17 (2004), 2041-2056. doi: 10.1088/0951-7715/17/6/003.  Google Scholar

[22]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Math. Soc.: Providence, 2000.  Google Scholar

[23]

A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-De Vries equation, SIAM Review, 14 (1972), 582-643. doi: 10.1137/1014101.  Google Scholar

[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), 422-443. doi: 10.1080/14786449508620739.  Google Scholar

[25]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, Part I, Ann. Inst. H. Poincaré, Analysis Nonlinear, 1 (1984), 109-145.  Google Scholar

[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), 982-988. doi: 10.1137/S0036141001385307.  Google Scholar

[27]

I. A. Svendsen and J. Buhr Hansen, Laboratory generation of waves of constant form, in the 14th Coastal Engr. Conf., Copenhagen (1974) Chapt. 17, pp. 321-339. Google Scholar

[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), 433-448. doi: 10.1017/S0022112078001706.  Google Scholar

[29]

M. I. Weinstein, Liapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.  Google Scholar

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), 360-392. doi: 10.1016/0167-2789(89)90050-X.  Google Scholar

[2]

J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, Applied Analysis (Baton Rouge, LA 1996), Contemp. Math. 221, American Math. Soc., Providence, RI (1999), 1-29. doi: 10.1090/conm/221/03116.  Google Scholar

[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), 343-366. doi: 10.1016/0167-2789(87)90084-4.  Google Scholar

[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), 107-126. doi: 10.1007/BF02392447.  Google Scholar

[5]

J. Angulo Pava, "Nonlinear Dispersive Equations," Mathematical Surveys and Monographs, 156, American Math. Soc.: Providence, RI 2009.  Google Scholar

[6]

J. Angulo Pava, J. L. Bona and M. Scialom, Stability of cnoidal waves, Advances in Differenticial Equations, 11 (2006), 1321-1374.  Google Scholar

[7]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X.  Google Scholar

[8]

T. B. Benjamin, The stability of solitary waves, Proc. Royal Soc. London Ser. A, 328 (1972), 153-183. doi: 10.1098/rspa.1972.0074.  Google Scholar

[9]

T. B. Benjamin, Lectures on nonlinear wave motion, Lect. Appl. Math., 15, (ed. A. Newell) American Math. Soc., Providence, RI (1974), 3-47.  Google Scholar

[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, Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[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), 351-379. doi: 10.1017/S0305004100061193.  Google Scholar

[12]

J. L. Bona, On the stability theory of solitary waves, Proc. Royal Soc. London, Ser. A, 344 (1975), 367-374. doi: 10.1098/rspa.1975.0106.  Google Scholar

[13]

J. L. Bona, Convergence of periodic wave trains in the limit of large wavelength, Appl. Sci. Res., 37 (1981), 21-30. doi: 10.1007/BF00382614.  Google Scholar

[14]

J. L. Bona, On solitary waves and their role in the evolution of long waves, in "Applications of Nonlinear Analysis in the Physical Sciences" (eds. H. Amann, N. Bazley and K. Kirchgässner), Pitman: London (1989), 183-205. Google Scholar

[15]

J. L. Bona and H. Kalisch, Models for internal waves in deep water, Discrete Cont. Dynamical Sys., 6 (2000), 1-20. doi: 10.3934/dcds.2000.6.1.  Google Scholar

[16]

J. L. Bona, Y. Liu and N. Nguyen, Stability of solitary waves in higher-order Sobolev spaces, Commun. Math. Sci., 2 (2004), 35-52.  Google Scholar

[17]

J. L. Bona, P. E. Souganidis and W. A. Strauss, Stability and instability of solitary waves of KdV-type, Proc. Royal Soc. London, Ser. A, 411 (1987), 395-412. doi: 10.1098/rspa.1987.0073.  Google Scholar

[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), 449-470. doi: 10.1007/BF02430641.  Google Scholar

[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), 755-759. Google Scholar

[20]

J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal, C. R. Acad. Sci. Paris, 73 (1871), 256-260. Google Scholar

[21]

H. Chen, Existence of periodic traveling-wave solutions of nonlinear, dispersive wave equations, Nonlinearity, 17 (2004), 2041-2056. doi: 10.1088/0951-7715/17/6/003.  Google Scholar

[22]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Math. Soc.: Providence, 2000.  Google Scholar

[23]

A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-De Vries equation, SIAM Review, 14 (1972), 582-643. doi: 10.1137/1014101.  Google Scholar

[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), 422-443. doi: 10.1080/14786449508620739.  Google Scholar

[25]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, Part I, Ann. Inst. H. Poincaré, Analysis Nonlinear, 1 (1984), 109-145.  Google Scholar

[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), 982-988. doi: 10.1137/S0036141001385307.  Google Scholar

[27]

I. A. Svendsen and J. Buhr Hansen, Laboratory generation of waves of constant form, in the 14th Coastal Engr. Conf., Copenhagen (1974) Chapt. 17, pp. 321-339. Google Scholar

[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), 433-448. doi: 10.1017/S0022112078001706.  Google Scholar

[29]

M. I. Weinstein, Liapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.  Google Scholar

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