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Modified energy functionals and the NLS approximation
High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs
Department of Applied Mathematics, University of Washington, Campus Box 352420, Seattle, WA, 98195, USA |
Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition for the presence of high-frequency (i.e., not modulational) instabilities of small-amplitude periodic solutions of Hamiltonian partial differential equations is presented, entirely in terms of the Hamiltonian of the linearized problem. With the exception of a Krein signature calculation, the theory is completely phrased in terms of the dispersion relation of the linear problem. The general theory changes as the Poisson structure of the Hamiltonian partial differential equation is changed. Two important cases of such Poisson structures are worked out in full generality. An example not fitting these two important cases is presented as well, using a candidate Boussinesq-Whitham equation.
References:
[1] |
V. I. Arnol'd and S. P. Novikov, editors,
Dynamical Systems. Ⅳ volume 4 of Encyclopaedia of Mathematical Sciences Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-662-06793-2. |
[2] |
V. I. Arnol'd,
Mathematical Methods of Classical Mechanics volume 60 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[3] |
M. J. Ablowitz and H. Segur,
Solitons and the Inverse Scattering Transform Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1981. |
[4] |
N. Bottman and B. Deconinck,
KdV cnoidal waves are spectrally stable, Discrete Contin. Dyn. Syst., 25 (2009), 1163-1180.
doi: 10.3934/dcds.2009.25.1163. |
[5] |
N. Bottman, B. Deconinck and M. Nivala, Elliptic solutions of the defocusing nls equation are stable J. Phys. A, 44 (2011), 285201, 24pp.
doi: 10.1088/1751-8113/44/28/285201. |
[6] |
D. J. Benney,
Non-linear gravity wave interactions, J. Fluid Mech., 14 (1962), 577-584.
doi: 10.1017/S0022112062001469. |
[7] |
T. B. Benjamin,
Instability of periodic wavetrains in nonlinear dispersive systems, Proc. Roy. Soc. (London) Ser. A, 299 (1967), 59-76.
doi: 10.1098/rspa.1967.0123. |
[8] |
J. C. Bronski and M. A. Johnson,
The modulational instability for a generalized Korteweg-de Vries equation, Arch. Ration. Mech. Anal., 197 (2010), 357-400.
doi: 10.1007/s00205-009-0270-5. |
[9] |
J. C. Bronski, M. A. Johnson and T. Kapitula,
An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1141-1173.
doi: 10.1017/S0308210510001216. |
[10] |
T. J. Bridges and A. Mielke,
A proof of the Benjamin-Feir instability, Arch. Rational Mech. Anal., 133 (1995), 145-198.
doi: 10.1007/BF00376815. |
[11] |
E. A. Coddington and N. Levinson,
Theory of Ordinary Differential Equations McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1955. |
[12] |
W. Craig and C. Sulem,
Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.
doi: 10.1006/jcph.1993.1164. |
[13] |
B. Deconinck and J. N. Kutz,
Computing spectra of linear operators using the Floquet-Fourier-Hill method, Journal of Computational Physics, 219 (2006), 296-321.
doi: 10.1016/j.jcp.2006.03.020. |
[14] |
B. Deconinck and T. Kapitula,
The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. A, 374 (2010), 4018-4022.
doi: 10.1016/j.physleta.2010.08.007. |
[15] |
B. Deconinck and T. Kapitula, On the orbital (in)stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations, Submitted for Publication, pages 1–24, 2013. Google Scholar |
[16] |
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1. 0. 8 of 2014-04-25. Online companion to 41. Google Scholar |
[17] |
B. Deconinck and M. Nivala,
The stability analysis of the periodic traveling wave solutions of the mKdV equation, Stud. Appl. Math., 126 (2011), 17-48.
doi: 10.1111/j.1467-9590.2010.00496.x. |
[18] |
B. Deconinck and K. Oliveras,
The instability of periodic surface gravity waves, J. Fluid Mech., 675 (2011), 141-167.
doi: 10.1017/S0022112011000073. |
[19] |
M. Ehrnströ, M. D. Groves and E. Wahlén,
On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 2903-2936.
doi: 10.1088/0951-7715/25/10/2903. |
[20] |
M. Ehrnström and H. Kalisch,
Traveling waves for the Whitham equation, Differential and Integral Equations, 22 (2009), 1193-1210.
|
[21] |
L. D. Faddeev and L. A. Takhtajan,
Hamiltonian Methods in the Theory of Solitons Classics in Mathematics. Springer, Berlin, english edition, 2007. |
[22] |
C. S. Gardner,
The Korteweg-deVries equation and generalizations. Ⅳ the Korteweg-deVries equation as a Hamiltonian system, J. Math. Phys., 12 (1971), 1548-1551.
doi: 10.1063/1.1665772. |
[23] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[24] |
NJ. L. Hammack and D. M. Henderson, Resonant interactions among surface water waves, In
Annual review of fluid mechanics, Annual Reviews, Palo Alto, CA, 25 (1993), 55–97. |
[25] |
V. Hur and M. Johnson,
Modulational instability in the Whitham equation of water waves, Stud. Appl. Math., 134 (2015), 120-143.
doi: 10.1111/sapm.12061. |
[26] |
M. Hǎrǎguş and T. Kapitula,
On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, Phys. D, 237 (2008), 2649-2671.
doi: 10.1016/j.physd.2008.03.050. |
[27] |
V. Hur and A. K. Pandey, Modulational instability in a full-dispersion shallow water model, Phys. D, 325 (2016), 98–112, arXiv: 1608.04685. Google Scholar |
[28] |
P. D. Hislop and I. M. Sigal,
Introduction to Spectral Theory volume 113 of Applied Mathematical Sciences Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-0741-2. |
[29] |
V. Hur and L. Tao, Wave breaking in a shallow water model, arXiv: 1608.04681, 2016. Google Scholar |
[30] |
C. K. R. T. Jones, R. Marangell, P. D. Miller and R. G. Plaza,
On the stability analysis of periodic sine-Gordon traveling waves, Phys. D, 251 (2013), 63-74.
doi: 10.1016/j.physd.2013.02.003. |
[31] |
C. K. R. T. Jones, R. Marangell, P. D. Miller and R. G. Plaza, Spectral and modulational stability of periodic wavetrains for the nonlinear klein-gordon equation, J. Differential Equations, 257 (2014), 4632–4703, arXiv: 1312.1132 [math. AP].
doi: 10.1016/j.jde.2014.09.004. |
[32] |
M. A. Johnson, K. Zumbrun and J. C. Bronski,
On the modulation equations and stability of periodic generalized Korteweg-de Vries waves via Bloch decompositions, Phys. D, 239 (2010), 2057-2065.
doi: 10.1016/j.physd.2010.07.012. |
[33] |
R. Kollar and P. D. Miller,
Graphical krein signature theory and evans-krein functions, SIAM Rev., 56 (2014), 73-123.
doi: 10.1137/120891423. |
[34] |
T. Kapitula and K. Promislow,
Spectral and Dynamical Stability of Nonlinear Waves volume 185 of Applied Mathematical Sciences Springer, New York, 2013.
doi: 10.1007/978-1-4614-6995-7. |
[35] |
M. G. Kreǐn,
A generalization of some investigations of A. M. Lyapunov on linear differential equations with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.), 73 (1950), 445-448.
|
[36] |
M. G. Kreǐn,
On the application of an algebraic proposition in the theory of matrices of monodromy, Uspehi Matem. Nauk (N.S.), 6 (1951), 171-177.
|
[37] |
R. S. MacKay, Stability of equilibria of Hamiltonian systems, In Nonlinear phenomena and
chaos (Malvern, 1985), Malvern Phys. Ser. , pages 254–270. Hilger, Bristol, 1986. |
[38] |
H. P. McKean,
Boussinesq's equation on the circle, Comm. Pure Appl. Math., 34 (1981), 599-691.
doi: 10.1002/cpa.3160340502. |
[39] |
J. D. Meiss,
Differential Dynamical Systems volume 14 of Mathematical Modeling and Computation Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.
doi: 10.1137/1.9780898718232. |
[40] |
R. S. MacKay and P. G. Saffman,
Stability of water waves, Proc. Roy. Soc. London Ser. A, 406 (1986), 115-125.
doi: 10.1098/rspa.1986.0068. |
[41] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, editors,
NIST Handbook of Mathematical Functions Cambridge University Press, New York, NY, 2010. Print companion to [16]. |
[42] |
O. M. Phillips,
On the dynamics of unsteady gravity waves of finite amplitude. Ⅰ. The elementary interactions, J. Fluid Mech., 9 (1960), 193-217.
doi: 10.1017/S0022112060001043. |
[43] |
A. C. Scott,
A nonlinear klein-gordon equation, Amer. J. Phys., 37 (1969), 52-61.
doi: 10.1119/1.1975404. |
[44] |
N. Sanford, K. Kodama, J. D. Carter and H. Kalisch,
Stability of traveling wave solutions to the Whitham equation, Physics Letters A, 378 (2014), 2100-2107.
doi: 10.1016/j.physleta.2014.04.067. |
[45] |
I. Stakgold,
Boundary Value Problems of Mathematical Physics. Vol. Ⅰ,Ⅱ, volume 29 of Classics in Applied Mathematics Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
doi: 10.1137/1.9780898719475. |
[46] |
G. B. Stokes,
On the theory of oscillatory waves, Mathematical and Physical Papers, 1 (1847), 197-229.
doi: 10.1017/CBO9780511702242.013. |
[47] |
J. -M. Vanden-Broeck,
Gravity-capillary Free-Surface Flows Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511730276. |
[48] |
J. -C. van der Meer,
The Hamiltonian Hopf Bifurcation volume 1160 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985.
doi: 10.1007/BFb0080357. |
[49] |
G. B. Whitham,
Non-linear dispersion of water waves, J. Fluid Mech., 27 (1967), 399-412.
doi: 10.1017/S0022112067000424. |
[50] |
G. B. Whitham,
Variational methods and applications to water waves, Hyperbolic Equations and Waves, (1970), 153-172.
doi: 10.1007/978-3-642-87025-5_16. |
[51] |
G. B. Whitham, Variational methods and applications to water waves, In Hyperbolic equations and waves (Rencontres, Battelle Res. Inst. , Seattle, Wash. , 1968), pages 153–172. Springer, Berlin, 1970. |
[52] |
G. B. Whitham,
Linear and Nonlinear Waves Wiley-Interscience, New York, NY, 1974. Pure and Applied Mathematics. |
[53] |
V. E. Zakharov,
Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.
doi: 10.1007/BF00913182. |
[54] |
V. E. Zakharov and L. D. Faddeev,
Korteweg -de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl., (2016), 277-284.
doi: 10.1142/9789814340960_0023. |
[55] |
V. E. Zakharov, S. L. Musher and A. M. Rubenchik,
Hamiltonian approach to the description of nonlinear plasma phenomena, Phys. Rep., 129 (1985), 285-366.
doi: 10.1016/0370-1573(85)90040-7. |
[56] |
V. E. Zakharov and L. A. Ostrovsky,
Modulation instability: The beginning, Phys. D, 238 (2009), 540-548.
doi: 10.1016/j.physd.2008.12.002. |
show all references
References:
[1] |
V. I. Arnol'd and S. P. Novikov, editors,
Dynamical Systems. Ⅳ volume 4 of Encyclopaedia of Mathematical Sciences Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-662-06793-2. |
[2] |
V. I. Arnol'd,
Mathematical Methods of Classical Mechanics volume 60 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[3] |
M. J. Ablowitz and H. Segur,
Solitons and the Inverse Scattering Transform Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1981. |
[4] |
N. Bottman and B. Deconinck,
KdV cnoidal waves are spectrally stable, Discrete Contin. Dyn. Syst., 25 (2009), 1163-1180.
doi: 10.3934/dcds.2009.25.1163. |
[5] |
N. Bottman, B. Deconinck and M. Nivala, Elliptic solutions of the defocusing nls equation are stable J. Phys. A, 44 (2011), 285201, 24pp.
doi: 10.1088/1751-8113/44/28/285201. |
[6] |
D. J. Benney,
Non-linear gravity wave interactions, J. Fluid Mech., 14 (1962), 577-584.
doi: 10.1017/S0022112062001469. |
[7] |
T. B. Benjamin,
Instability of periodic wavetrains in nonlinear dispersive systems, Proc. Roy. Soc. (London) Ser. A, 299 (1967), 59-76.
doi: 10.1098/rspa.1967.0123. |
[8] |
J. C. Bronski and M. A. Johnson,
The modulational instability for a generalized Korteweg-de Vries equation, Arch. Ration. Mech. Anal., 197 (2010), 357-400.
doi: 10.1007/s00205-009-0270-5. |
[9] |
J. C. Bronski, M. A. Johnson and T. Kapitula,
An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1141-1173.
doi: 10.1017/S0308210510001216. |
[10] |
T. J. Bridges and A. Mielke,
A proof of the Benjamin-Feir instability, Arch. Rational Mech. Anal., 133 (1995), 145-198.
doi: 10.1007/BF00376815. |
[11] |
E. A. Coddington and N. Levinson,
Theory of Ordinary Differential Equations McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1955. |
[12] |
W. Craig and C. Sulem,
Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.
doi: 10.1006/jcph.1993.1164. |
[13] |
B. Deconinck and J. N. Kutz,
Computing spectra of linear operators using the Floquet-Fourier-Hill method, Journal of Computational Physics, 219 (2006), 296-321.
doi: 10.1016/j.jcp.2006.03.020. |
[14] |
B. Deconinck and T. Kapitula,
The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. A, 374 (2010), 4018-4022.
doi: 10.1016/j.physleta.2010.08.007. |
[15] |
B. Deconinck and T. Kapitula, On the orbital (in)stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations, Submitted for Publication, pages 1–24, 2013. Google Scholar |
[16] |
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1. 0. 8 of 2014-04-25. Online companion to 41. Google Scholar |
[17] |
B. Deconinck and M. Nivala,
The stability analysis of the periodic traveling wave solutions of the mKdV equation, Stud. Appl. Math., 126 (2011), 17-48.
doi: 10.1111/j.1467-9590.2010.00496.x. |
[18] |
B. Deconinck and K. Oliveras,
The instability of periodic surface gravity waves, J. Fluid Mech., 675 (2011), 141-167.
doi: 10.1017/S0022112011000073. |
[19] |
M. Ehrnströ, M. D. Groves and E. Wahlén,
On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 2903-2936.
doi: 10.1088/0951-7715/25/10/2903. |
[20] |
M. Ehrnström and H. Kalisch,
Traveling waves for the Whitham equation, Differential and Integral Equations, 22 (2009), 1193-1210.
|
[21] |
L. D. Faddeev and L. A. Takhtajan,
Hamiltonian Methods in the Theory of Solitons Classics in Mathematics. Springer, Berlin, english edition, 2007. |
[22] |
C. S. Gardner,
The Korteweg-deVries equation and generalizations. Ⅳ the Korteweg-deVries equation as a Hamiltonian system, J. Math. Phys., 12 (1971), 1548-1551.
doi: 10.1063/1.1665772. |
[23] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[24] |
NJ. L. Hammack and D. M. Henderson, Resonant interactions among surface water waves, In
Annual review of fluid mechanics, Annual Reviews, Palo Alto, CA, 25 (1993), 55–97. |
[25] |
V. Hur and M. Johnson,
Modulational instability in the Whitham equation of water waves, Stud. Appl. Math., 134 (2015), 120-143.
doi: 10.1111/sapm.12061. |
[26] |
M. Hǎrǎguş and T. Kapitula,
On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, Phys. D, 237 (2008), 2649-2671.
doi: 10.1016/j.physd.2008.03.050. |
[27] |
V. Hur and A. K. Pandey, Modulational instability in a full-dispersion shallow water model, Phys. D, 325 (2016), 98–112, arXiv: 1608.04685. Google Scholar |
[28] |
P. D. Hislop and I. M. Sigal,
Introduction to Spectral Theory volume 113 of Applied Mathematical Sciences Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-0741-2. |
[29] |
V. Hur and L. Tao, Wave breaking in a shallow water model, arXiv: 1608.04681, 2016. Google Scholar |
[30] |
C. K. R. T. Jones, R. Marangell, P. D. Miller and R. G. Plaza,
On the stability analysis of periodic sine-Gordon traveling waves, Phys. D, 251 (2013), 63-74.
doi: 10.1016/j.physd.2013.02.003. |
[31] |
C. K. R. T. Jones, R. Marangell, P. D. Miller and R. G. Plaza, Spectral and modulational stability of periodic wavetrains for the nonlinear klein-gordon equation, J. Differential Equations, 257 (2014), 4632–4703, arXiv: 1312.1132 [math. AP].
doi: 10.1016/j.jde.2014.09.004. |
[32] |
M. A. Johnson, K. Zumbrun and J. C. Bronski,
On the modulation equations and stability of periodic generalized Korteweg-de Vries waves via Bloch decompositions, Phys. D, 239 (2010), 2057-2065.
doi: 10.1016/j.physd.2010.07.012. |
[33] |
R. Kollar and P. D. Miller,
Graphical krein signature theory and evans-krein functions, SIAM Rev., 56 (2014), 73-123.
doi: 10.1137/120891423. |
[34] |
T. Kapitula and K. Promislow,
Spectral and Dynamical Stability of Nonlinear Waves volume 185 of Applied Mathematical Sciences Springer, New York, 2013.
doi: 10.1007/978-1-4614-6995-7. |
[35] |
M. G. Kreǐn,
A generalization of some investigations of A. M. Lyapunov on linear differential equations with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.), 73 (1950), 445-448.
|
[36] |
M. G. Kreǐn,
On the application of an algebraic proposition in the theory of matrices of monodromy, Uspehi Matem. Nauk (N.S.), 6 (1951), 171-177.
|
[37] |
R. S. MacKay, Stability of equilibria of Hamiltonian systems, In Nonlinear phenomena and
chaos (Malvern, 1985), Malvern Phys. Ser. , pages 254–270. Hilger, Bristol, 1986. |
[38] |
H. P. McKean,
Boussinesq's equation on the circle, Comm. Pure Appl. Math., 34 (1981), 599-691.
doi: 10.1002/cpa.3160340502. |
[39] |
J. D. Meiss,
Differential Dynamical Systems volume 14 of Mathematical Modeling and Computation Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.
doi: 10.1137/1.9780898718232. |
[40] |
R. S. MacKay and P. G. Saffman,
Stability of water waves, Proc. Roy. Soc. London Ser. A, 406 (1986), 115-125.
doi: 10.1098/rspa.1986.0068. |
[41] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, editors,
NIST Handbook of Mathematical Functions Cambridge University Press, New York, NY, 2010. Print companion to [16]. |
[42] |
O. M. Phillips,
On the dynamics of unsteady gravity waves of finite amplitude. Ⅰ. The elementary interactions, J. Fluid Mech., 9 (1960), 193-217.
doi: 10.1017/S0022112060001043. |
[43] |
A. C. Scott,
A nonlinear klein-gordon equation, Amer. J. Phys., 37 (1969), 52-61.
doi: 10.1119/1.1975404. |
[44] |
N. Sanford, K. Kodama, J. D. Carter and H. Kalisch,
Stability of traveling wave solutions to the Whitham equation, Physics Letters A, 378 (2014), 2100-2107.
doi: 10.1016/j.physleta.2014.04.067. |
[45] |
I. Stakgold,
Boundary Value Problems of Mathematical Physics. Vol. Ⅰ,Ⅱ, volume 29 of Classics in Applied Mathematics Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
doi: 10.1137/1.9780898719475. |
[46] |
G. B. Stokes,
On the theory of oscillatory waves, Mathematical and Physical Papers, 1 (1847), 197-229.
doi: 10.1017/CBO9780511702242.013. |
[47] |
J. -M. Vanden-Broeck,
Gravity-capillary Free-Surface Flows Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511730276. |
[48] |
J. -C. van der Meer,
The Hamiltonian Hopf Bifurcation volume 1160 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985.
doi: 10.1007/BFb0080357. |
[49] |
G. B. Whitham,
Non-linear dispersion of water waves, J. Fluid Mech., 27 (1967), 399-412.
doi: 10.1017/S0022112067000424. |
[50] |
G. B. Whitham,
Variational methods and applications to water waves, Hyperbolic Equations and Waves, (1970), 153-172.
doi: 10.1007/978-3-642-87025-5_16. |
[51] |
G. B. Whitham, Variational methods and applications to water waves, In Hyperbolic equations and waves (Rencontres, Battelle Res. Inst. , Seattle, Wash. , 1968), pages 153–172. Springer, Berlin, 1970. |
[52] |
G. B. Whitham,
Linear and Nonlinear Waves Wiley-Interscience, New York, NY, 1974. Pure and Applied Mathematics. |
[53] |
V. E. Zakharov,
Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.
doi: 10.1007/BF00913182. |
[54] |
V. E. Zakharov and L. D. Faddeev,
Korteweg -de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl., (2016), 277-284.
doi: 10.1142/9789814340960_0023. |
[55] |
V. E. Zakharov, S. L. Musher and A. M. Rubenchik,
Hamiltonian approach to the description of nonlinear plasma phenomena, Phys. Rep., 129 (1985), 285-366.
doi: 10.1016/0370-1573(85)90040-7. |
[56] |
V. E. Zakharov and L. A. Ostrovsky,
Modulation instability: The beginning, Phys. D, 238 (2009), 540-548.
doi: 10.1016/j.physd.2008.12.002. |













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