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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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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