March  2017, 37(3): 1323-1358. doi: 10.3934/dcds.2017055

High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs

Department of Applied Mathematics, University of Washington, Campus Box 352420, Seattle, WA, 98195, USA

* Corresponding author: deconinc@uw.edu

Received  March 2016 Revised  October 2016 Published  December 2016

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.

Citation: Deconinck Bernard, Olga Trichtchenko. High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1323-1358. doi: 10.3934/dcds.2017055
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.  Google Scholar

[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.  Google Scholar

[3]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1981.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[6]

D. J. Benney, Non-linear gravity wave interactions, J. Fluid Mech., 14 (1962), 577-584.  doi: 10.1017/S0022112062001469.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. C. BronskiM. 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.  Google Scholar

[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.  Google Scholar

[11]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1955.  Google Scholar

[12]

W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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

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NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1. 0. 8 of 2014-04-25. Online companion to 41. Google Scholar

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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.  Google Scholar

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B. Deconinck and K. Oliveras, The instability of periodic surface gravity waves, J. Fluid Mech., 675 (2011), 141-167.  doi: 10.1017/S0022112011000073.  Google Scholar

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M. EhrnströM. 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.  Google Scholar

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M. Ehrnström and H. Kalisch, Traveling waves for the Whitham equation, Differential and Integral Equations, 22 (2009), 1193-1210.   Google Scholar

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L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons Classics in Mathematics. Springer, Berlin, english edition, 2007.  Google Scholar

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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.  Google Scholar

[23]

M. GrillakisJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

V. Hur and L. Tao, Wave breaking in a shallow water model, arXiv: 1608.04681, 2016. Google Scholar

[30]

C. K. R. T. JonesR. MarangellP. 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.  Google Scholar

[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.  Google Scholar

[32]

M. A. JohnsonK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[38]

H. P. McKean, Boussinesq's equation on the circle, Comm. Pure Appl. Math., 34 (1981), 599-691.  doi: 10.1002/cpa.3160340502.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[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.  Google Scholar

[43]

A. C. Scott, A nonlinear klein-gordon equation, Amer. J. Phys., 37 (1969), 52-61.  doi: 10.1119/1.1975404.  Google Scholar

[44]

N. SanfordK. KodamaJ. 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.  Google Scholar

[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.  Google Scholar

[46]

G. B. Stokes, On the theory of oscillatory waves, Mathematical and Physical Papers, 1 (1847), 197-229.  doi: 10.1017/CBO9780511702242.013.  Google Scholar

[47]

J. -M. Vanden-Broeck, Gravity-capillary Free-Surface Flows Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511730276.  Google Scholar

[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.  Google Scholar

[49]

G. B. Whitham, Non-linear dispersion of water waves, J. Fluid Mech., 27 (1967), 399-412.  doi: 10.1017/S0022112067000424.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

G. B. Whitham, Linear and Nonlinear Waves Wiley-Interscience, New York, NY, 1974. Pure and Applied Mathematics.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

V. E. ZakharovS. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. J. Benney, Non-linear gravity wave interactions, J. Fluid Mech., 14 (1962), 577-584.  doi: 10.1017/S0022112062001469.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. C. BronskiM. 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.  Google Scholar

[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.  Google Scholar

[11]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1955.  Google Scholar

[12]

W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. Deconinck and K. Oliveras, The instability of periodic surface gravity waves, J. Fluid Mech., 675 (2011), 141-167.  doi: 10.1017/S0022112011000073.  Google Scholar

[19]

M. EhrnströM. 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.  Google Scholar

[20]

M. Ehrnström and H. Kalisch, Traveling waves for the Whitham equation, Differential and Integral Equations, 22 (2009), 1193-1210.   Google Scholar

[21]

L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons Classics in Mathematics. Springer, Berlin, english edition, 2007.  Google Scholar

[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.  Google Scholar

[23]

M. GrillakisJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

V. Hur and L. Tao, Wave breaking in a shallow water model, arXiv: 1608.04681, 2016. Google Scholar

[30]

C. K. R. T. JonesR. MarangellP. 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.  Google Scholar

[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.  Google Scholar

[32]

M. A. JohnsonK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[38]

H. P. McKean, Boussinesq's equation on the circle, Comm. Pure Appl. Math., 34 (1981), 599-691.  doi: 10.1002/cpa.3160340502.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[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.  Google Scholar

[43]

A. C. Scott, A nonlinear klein-gordon equation, Amer. J. Phys., 37 (1969), 52-61.  doi: 10.1119/1.1975404.  Google Scholar

[44]

N. SanfordK. KodamaJ. 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.  Google Scholar

[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.  Google Scholar

[46]

G. B. Stokes, On the theory of oscillatory waves, Mathematical and Physical Papers, 1 (1847), 197-229.  doi: 10.1017/CBO9780511702242.013.  Google Scholar

[47]

J. -M. Vanden-Broeck, Gravity-capillary Free-Surface Flows Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511730276.  Google Scholar

[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.  Google Scholar

[49]

G. B. Whitham, Non-linear dispersion of water waves, J. Fluid Mech., 27 (1967), 399-412.  doi: 10.1017/S0022112067000424.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

G. B. Whitham, Linear and Nonlinear Waves Wiley-Interscience, New York, NY, 1974. Pure and Applied Mathematics.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

V. E. ZakharovS. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  A cartoon of the bifurcation structure of the traveling waves for a third-order ($M=3$) system: solution branches bifurcate away from the trivial zero-amplitude solution at specific values of the traveling wave speed $c$
Figure 2.  Colliding eigenvalues in the complex plane as a parameter is increased. On the left, two eigenvalues are moving towards each other on the positive imaginary axis, accompanied by a complex conjugate pair on the negative imaginary axis. In the middle, the eigenvalues in each pair have collided. On the right, a Hamiltonian Hopf bifurcation occurs: the collided eigenvalues separate, leaving the imaginary axis (implying that the two Krein signatures were different)
Figure 3.  The graphical interpretation of the collision condition (3.12). The solid curve is the graph of the dispersion relation $\omega(k)$. The slope of the dashed line in the first quadrant is the right-hand side in (3.12). The slope of the parallel dotted line is its left-hand side
Figure 4.  The amplitude vs. $c$ bifurcation plots for the traveling-wave solutions of the generalized KdV equation (3.20). (a) The KdV equation, $n=1$, for the cnoidal wave solutions (3.24). (b) The mKdV equation, $n=2$, for the cnoidal wave solutions (3.26). Lastly, (c) shows the bifurcation plot for the snoidal wave solutions (3.27) of mKdV, $n=2$. Note that all bifurcation branches start at $(-1,0)$, as stated above. Further, for all solutions here the speed $c$ and the amplitude $\to \infty$ as $\kappa\to 1$. This is a consequence of enforcing the $2\pi$-periodicity of the solution, which results in non-smooth limit solution
Figure 5.  (a) The imaginary part of $\lambda_n^{(\mu)}\in (-0.7, 0.7)$ as a function of $\mu\in[-1/4, 1/4)$. Different curves correspond to different half-integer values of $n$. (b) The curves $\Omega(k+n)$, for various (integer) values of $n$, illustrating that collisions occur at the origin only
Figure 6.  (a) The profile of a $2\pi$-periodic small-amplitude traveling wave solution of the Whitham equation (2.1) with $c\approx 0.7697166847$, computed using a cosine collocation method with 128 Fourier modes, see [44]. (b) The stability spectrum of this solution, computed using the Fourier-Floquet-Hill method [13] with $128$ modes and 2000 different values of the Floquet parameter $\mu$. The presence of a modulational instability is clear, but no high-frequency instabilities are observed, in agreement with the theory presented. Note that the hallmark bubbles of instability were looked for far outside of the region displayed here
Figure 7.  (a) The dispersion relation for the Whitham equation (curve), together with the line through the origin of slope $\omega(1)/1$, representing the right-hand side of (3.12). (b) The curves $\Omega(k+n)$, for various (integer) values of $n$, illustrating that collisions occur at the origin only
Figure 8.  The graphical interpretation of the collision condition (4.10). The dashed curves are the graphs of the dispersion relations $\omega_1(k)$ and $\omega_2(k)$. The slope of the segment $P_1P_2$ is the right-hand side in (4.10). The collision condition (4.10) seeks points whose abscissas are an integer apart, so that at least one of the segments $P_3P_4$, $P_3P_6$, $P_5P_4$ or $P_5P_6$ is parallel to the segment $P_1P_2$
Figure 9.  (a) The two branches of the dispersion relation for the Sine-Gordon equation. The line segment $P_1 P_2$ has slope $\omega(1)/1$, representing the right-hand side of (4.10). The slope of the parallel line segment $P_3 P_4$ represents the left-hand side of (4.10). (b) The two families of curves $\Omega_1(k+n)$ (red, solid) and $\Omega_2(k+n)$ (black, dashed), for various (integer) values of $n$, illustrating that many collisions occur away from the origin
Figure 10.  (a) A small-amplitude $2\pi$-periodic superluminal solution of the SG equation ($c\approx 1.236084655663$). (b) A blow-up of the numerically computed stability spectrum in a neighborhood of the origin, illustrating the presence of a modulational instability, but the absence of high-frequency instabilities
Figure 11.  The domain for the water wave problem. Here $z=0$ is the equation of the surface for flat water, $z=-h$ is the flat bottom
Figure 12.  (a) The two branches of the dispersion relation for the water wave problem ($g=1$, $h=1$). The line through the origin has slope $\omega_1(1)/1$, representing the right-hand side of (4.10). (b) The two families of curves $\Omega_1(k+n)$ (red, solid) and $\Omega_2(k+n)$ (black, dashed), for various (integer) values of $n$, illustrating that many collisions occur away from the origin. (c) The origin of the high-frequency instability closest to the origin as a function of depth $h$
Figure 13.  (a) A small-amplitude traveling wave solution of the Boussines-Whitham equation (5.1) with $c\approx 1.0498515$. (b) The numerically computed stability spectrum. (c) A blow-up of the stability spectrum in a neighborhood of the origin. (d) A blow-up of the stability spectrum around what appears as a horizontal segment visible in (b) immediately above the longest segment appearing horizontal. More detail is given in the main text
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