American Institute of Mathematical Sciences

Advanced
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 - 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.  Google Scholar

[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.  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.  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.  doi: 10.1007/BF02392447.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

T. B. Benjamin, Lectures on nonlinear wave motion,, Lect. Appl. Math., 15 (1974), 3.   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, 272 (1972), 47.  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.  doi: 10.1017/S0305004100061193.  Google Scholar

[12]

J. L. Bona, On the stability theory of solitary waves,, Proc. Royal Soc. London, 344 (1975), 367.  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.  doi: 10.1007/BF00382614.  Google Scholar

[14]

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

[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.  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.   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, 411 (1987), 395.  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.  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.   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.   Google Scholar

[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.  Google Scholar

[22]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (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.  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.  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, 1 (1984), 109.   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.  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., (1974), 321.   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.  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.  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.  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, 221 (1999), 1.  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.  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.  doi: 10.1007/BF02392447.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

T. B. Benjamin, Lectures on nonlinear wave motion,, Lect. Appl. Math., 15 (1974), 3.   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, 272 (1972), 47.  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.  doi: 10.1017/S0305004100061193.  Google Scholar

[12]

J. L. Bona, On the stability theory of solitary waves,, Proc. Royal Soc. London, 344 (1975), 367.  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.  doi: 10.1007/BF00382614.  Google Scholar

[14]

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

[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.  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.   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, 411 (1987), 395.  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.  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.   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.   Google Scholar

[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.  Google Scholar

[22]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (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.  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.  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, 1 (1984), 109.   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.  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., (1974), 321.   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.  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.  doi: 10.1002/cpa.3160390103.  Google Scholar

[1]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[2]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[3]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[4]

Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020364
[5]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
[6]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[7]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[8]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021008
[9]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[10]

Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021002
[11]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075
[12]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466
[13]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[14]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
[15]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[16]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
[17]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346
[18]

Jie Shen, Nan Zheng. Efficient and accurate sav schemes for the generalized Zakharov systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 645-666. doi: 10.3934/dcdsb.2020262
[19]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117
[20]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences