-
Previous Article
On the asymptotic behavior of variational inequalities set in cylinders
- DCDS Home
- This Issue
-
Next Article
Singularity formation and blowup of complex-valued solutions of the modified KdV equation
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 |
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. |
[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. |
[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. |
[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. |
[5] |
J. Angulo Pava, "Nonlinear Dispersive Equations," Mathematical Surveys and Monographs, 156, American Math. Soc.: Providence, RI 2009. |
[6] |
J. Angulo Pava, J. L. Bona and M. Scialom, Stability of cnoidal waves, Advances in Differenticial Equations, 11 (2006), 1321-1374. |
[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. |
[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. |
[9] |
T. B. Benjamin, Lectures on nonlinear wave motion, Lect. Appl. Math., 15, (ed. A. Newell) American Math. Soc., Providence, RI (1974), 3-47. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Math. Soc.: Providence, 2000. |
[23] |
A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-De Vries equation, SIAM Review, 14 (1972), 582-643.
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), 422-443.
doi: 10.1080/14786449508620739. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
J. Angulo Pava, "Nonlinear Dispersive Equations," Mathematical Surveys and Monographs, 156, American Math. Soc.: Providence, RI 2009. |
[6] |
J. Angulo Pava, J. L. Bona and M. Scialom, Stability of cnoidal waves, Advances in Differenticial Equations, 11 (2006), 1321-1374. |
[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. |
[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. |
[9] |
T. B. Benjamin, Lectures on nonlinear wave motion, Lect. Appl. Math., 15, (ed. A. Newell) American Math. Soc., Providence, RI (1974), 3-47. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Math. Soc.: Providence, 2000. |
[23] |
A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-De Vries equation, SIAM Review, 14 (1972), 582-643.
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), 422-443.
doi: 10.1080/14786449508620739. |
[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. |
[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. |
[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. |
[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. |
[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. |
[1] |
Belkacem Said-Houari. Long-time behavior of solutions of the generalized Korteweg--de Vries equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 245-252. doi: 10.3934/dcdsb.2016.21.245 |
[2] |
Philippe Gravejat. Asymptotics of the solitary waves for the generalized Kadomtsev-Petviashvili equations. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 835-882. doi: 10.3934/dcds.2008.21.835 |
[3] |
Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331 |
[4] |
Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397 |
[5] |
Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255 |
[6] |
Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126 |
[7] |
Amin Esfahani, Steve Levandosky. Solitary waves of the rotation-generalized Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 663-700. doi: 10.3934/dcds.2013.33.663 |
[8] |
Jundong Wang, Lijun Zhang, Elena Shchepakina, Vladimir Sobolev. Solitary waves of singularly perturbed generalized KdV equation with high order nonlinearity. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022124 |
[9] |
Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control and Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353 |
[10] |
Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 389-418. doi: 10.3934/cpaa.2014.13.389 |
[11] |
Massimiliano Gubinelli. Rough solutions for the periodic Korteweg--de~Vries equation. Communications on Pure and Applied Analysis, 2012, 11 (2) : 709-733. doi: 10.3934/cpaa.2012.11.709 |
[12] |
Nate Bottman, Bernard Deconinck. KdV cnoidal waves are spectrally stable. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1163-1180. doi: 10.3934/dcds.2009.25.1163 |
[13] |
Zengji Du, Xiaojie Lin, Yulin Ren. Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1987-2003. doi: 10.3934/cpaa.2021118 |
[14] |
Yang Yang, Yun-Rui Yang, Xin-Jun Jiao. Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence. Electronic Research Archive, 2020, 28 (1) : 1-13. doi: 10.3934/era.2020001 |
[15] |
John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations and Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001 |
[16] |
Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789 |
[17] |
Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109 |
[18] |
Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i |
[19] |
Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029 |
[20] |
H. Kalisch. Stability of solitary waves for a nonlinearly dispersive equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 709-717. doi: 10.3934/dcds.2004.10.709 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]