July  2020, 40(7): 4131-4162. doi: 10.3934/dcds.2020175

On the normalized ground states for the Kawahara equation and a fourth order NLS

1460, Jayhawk Blvd., Department of Mathematics, University of Kansas, Lawrence, KS 66049, USA

* Corresponding author: Atanas G. Stefanov

Received  February 2019 Revised  November 2019 Published  April 2020

Fund Project: The first author was partially supported as a graduate research assistant from grant NSF-DMS 1614734. The second author is partially supported by NSF-DMS 1908626

We consider the Kawahara model and two fourth order semi-linear Schrödinger equations in any spatial dimension. We construct the corresponding normalized ground states, which we rigorously show to be spectrally stable.

For the Kawahara model, our results provide a significant extension in parameter space of the current rigorous results. In fact, our results establish (modulo an additional technical assumption, which should be satisfied at least generically), spectral stability for all normalized waves constructed therein - in all dimensions, for all acceptable values of the parameters. This, combined with the results of [5], provides orbital stability, for all normalized waves enjoying the non-degeneracy property. The validity of the non-degeneracy property for generic waves remains an intriguing open question.

At the same time, we verify and clarify recent numerical simulations of the spectral stability of these solitons. For the fourth order NLS models, we improve upon recent results on spectral stability of very special, explicit solutions in the one dimensional case. Our multidimensional results for fourth order anisotropic NLS seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second derivatives vs. pure Laplacian) has implications on the range of existence and stability of the normalized waves.

Citation: Iurii Posukhovskyi, Atanas G. Stefanov. On the normalized ground states for the Kawahara equation and a fourth order NLS. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4131-4162. doi: 10.3934/dcds.2020175
References:
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J. P. Albert, Positivity properties and stability of solitary-wave solutions of model equations for long waves, Comm. Partial Differential Equations, 17 (1992), 1-22.  doi: 10.1080/03605309208820831.  Google Scholar

[2]

T. de Andrade, F. Cristófani and F. Natali, Orbital stability of periodic traveling wave solutions for the Kawahara equation, J. Math. Phys., 58 (2017), 11pp. doi: 10.1063/1.4980016.  Google Scholar

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J. Angulo Pava, On the instability of solitary-wave solutions for fifth-order water wave models, Electron. J. Differential Equations, 2003, 18pp.  Google Scholar

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T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074.  Google Scholar

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D. BonheureJ.-B. CasterasE. M. dos Santos and R. Nascimento, Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation, SIAM J. Math. Anal., 50 (2018), 5027-5071.  doi: 10.1137/17M1154138.  Google Scholar

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T. Bridges and G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal., 33 (2002), 1356-1378.  doi: 10.1137/S0036141099361494.  Google Scholar

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T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar

[8]

W. Craig and M. D. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389.  doi: 10.1016/0165-2125(94)90003-5.  Google Scholar

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W. FengM. Stanislavova and A. Stefanov, On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion, Commun. Pure. Appl. Anal., 17 (2018), 1371-1385.  doi: 10.3934/cpaa.2018067.  Google Scholar

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M. Goldberg, personal communication., Google Scholar

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M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary wavs in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

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M. D. Groves, Solitary-wave solutions to a class of fifth-order model equations, Nonlinearity, 11 (1998), 341-353.  doi: 10.1088/0951-7715/11/2/009.  Google Scholar

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M. HaragusE. Lombardi and A. Scheel, Spectral stability of wave trains in the Kawahara equation, J. Math. Fluid Mech., 8 (2006), 482-509.  doi: 10.1007/s00021-005-0185-3.  Google Scholar

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J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Phys. D, 32 (1988), 253-268.  doi: 10.1016/0167-2789(88)90054-1.  Google Scholar

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A. T. Ill'ichev and A. Y. Semenov, Stability of solitary waves in dispersive media described by a fifth-order evolution equation, Theor. Comput. Fluid Dyn., 3 (1992), 307-326.  doi: 10.1007/BF00417931.  Google Scholar

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T. KapitulaG. Kevrekidis and B. Sandstede, Counting eigenvalues via Krein signature in infinite-dimensional Hamitonial systems, Phys. D, 195 (2004), 263-282.  doi: 10.1016/j.physd.2004.03.018.  Google Scholar

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T. KapitulaP. G. Kevrekidis and B. Sandstede, Addendum: ``Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems", Phys. D, 201 (2005), 199-201.  doi: 10.1016/j.physd.2004.11.015.  Google Scholar

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T. Kapitula and A. Stefanov, A Hamiltonian-Krein (instability) index theory for solitary waves to KdV-like eigenvalue problems, Stud. Appl. Math., 132 (2014), 183-211.  doi: 10.1111/sapm.12031.  Google Scholar

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V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: KdV-type equations, Phys. Lett. A, 210 (1996), 77-84.  doi: 10.1016/0375-9601(95)00752-0.  Google Scholar

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V. I. Karpman, Stabilization of soliton instabilities by higher order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996). doi: 10.1103/physreve.53.r1336.  Google Scholar

[23]

V. I. Karpman and A. Shagalov, Solitons and their stability in high dispersive systems. I. Fourth order nonlinear Schrödinger-type equations with power-law nonlinearities, Phys. Lett. A, 228 (1997), 59-65.  doi: 10.1016/S0375-9601(97)00063-7.  Google Scholar

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V. I. Karpman and A. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.  Google Scholar

[25]

V. I. Karpman, Lyapunov approach to the soliton stability in highly dispersive systems. I. Fourth order nonlinear Schrödinger equations, Phys. Lett. A, 215 (1996), 254-256.  doi: 10.1016/0375-9601(96)00231-9.  Google Scholar

[26]

R. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264.  doi: 10.1143/JPSJ.33.260.  Google Scholar

[27]

G. Kevrekidis, A. Stefanov, Y. Tsolias and J. Maraver, Quartic generalizations of the nonlinear Schrödinger model in two-dimensions: Theoretical analysis and numerical computations, work in progress. Google Scholar

[28]

S. Kichenassamy and P. J. Olver, Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, SIAM J. Math. Anal., 23 (1992), 1141-1166.  doi: 10.1137/0523064.  Google Scholar

[29]

S. P. Levandosky, A stability analysis of fifth-order water-wave models, Phys. D, 125 (1999), 222-240.  doi: 10.1016/S0167-2789(98)00245-0.  Google Scholar

[30]

S. Levandosky, Stability of solitary waves of a fifth-order water wave model, Phys. D, 227 (2007), 162-172.  doi: 10.1016/j.physd.2007.01.006.  Google Scholar

[31]

Z. Lin and C. Zeng, Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs, preprint, arXiv: 1703.04016. Google Scholar

[32]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[33]

F. Natali and A. Pastor, The fourth order dispersive nonlinear Schrödinger equation: Orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14 (2015), 1326-1347.  doi: 10.1137/151004884.  Google Scholar

[34]

D. E. Pelinovsky, Spectral stability on nonlinear waves in KdV-type evolution equations, in Nonlinear Physical Systems, Mech. Eng. Solid Mech. Ser., Wiley, Hoboken, NJ, 2014,377–400. doi: 10.1002/9781118577608.ch17.  Google Scholar

[35]

Z. Wang, Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation, Discrete Contin. Dyn. Syst., 37 (2017), 4091-4108.  doi: 10.3934/dcds.2017174.  Google Scholar

show all references

References:
[1]

J. P. Albert, Positivity properties and stability of solitary-wave solutions of model equations for long waves, Comm. Partial Differential Equations, 17 (1992), 1-22.  doi: 10.1080/03605309208820831.  Google Scholar

[2]

T. de Andrade, F. Cristófani and F. Natali, Orbital stability of periodic traveling wave solutions for the Kawahara equation, J. Math. Phys., 58 (2017), 11pp. doi: 10.1063/1.4980016.  Google Scholar

[3]

J. Angulo Pava, On the instability of solitary-wave solutions for fifth-order water wave models, Electron. J. Differential Equations, 2003, 18pp.  Google Scholar

[4]

T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074.  Google Scholar

[5]

D. BonheureJ.-B. CasterasE. M. dos Santos and R. Nascimento, Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation, SIAM J. Math. Anal., 50 (2018), 5027-5071.  doi: 10.1137/17M1154138.  Google Scholar

[6]

T. Bridges and G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal., 33 (2002), 1356-1378.  doi: 10.1137/S0036141099361494.  Google Scholar

[7]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar

[8]

W. Craig and M. D. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389.  doi: 10.1016/0165-2125(94)90003-5.  Google Scholar

[9]

W. FengM. Stanislavova and A. Stefanov, On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion, Commun. Pure. Appl. Anal., 17 (2018), 1371-1385.  doi: 10.3934/cpaa.2018067.  Google Scholar

[10]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $ \bf R$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[11]

M. Goldberg, personal communication., Google Scholar

[12]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary wavs in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[13]

M. D. Groves, Solitary-wave solutions to a class of fifth-order model equations, Nonlinearity, 11 (1998), 341-353.  doi: 10.1088/0951-7715/11/2/009.  Google Scholar

[14]

M. HaragusE. Lombardi and A. Scheel, Spectral stability of wave trains in the Kawahara equation, J. Math. Fluid Mech., 8 (2006), 482-509.  doi: 10.1007/s00021-005-0185-3.  Google Scholar

[15]

J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Phys. D, 32 (1988), 253-268.  doi: 10.1016/0167-2789(88)90054-1.  Google Scholar

[16]

A. T. Ill'ichev and A. Y. Semenov, Stability of solitary waves in dispersive media described by a fifth-order evolution equation, Theor. Comput. Fluid Dyn., 3 (1992), 307-326.  doi: 10.1007/BF00417931.  Google Scholar

[17]

T. KapitulaG. Kevrekidis and B. Sandstede, Counting eigenvalues via Krein signature in infinite-dimensional Hamitonial systems, Phys. D, 195 (2004), 263-282.  doi: 10.1016/j.physd.2004.03.018.  Google Scholar

[18]

T. KapitulaP. G. Kevrekidis and B. Sandstede, Addendum: ``Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems", Phys. D, 201 (2005), 199-201.  doi: 10.1016/j.physd.2004.11.015.  Google Scholar

[19]

T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences, 185, Springer, New York, 2013. doi: 10.1007/978-1-4614-6995-7.  Google Scholar

[20]

T. Kapitula and A. Stefanov, A Hamiltonian-Krein (instability) index theory for solitary waves to KdV-like eigenvalue problems, Stud. Appl. Math., 132 (2014), 183-211.  doi: 10.1111/sapm.12031.  Google Scholar

[21]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: KdV-type equations, Phys. Lett. A, 210 (1996), 77-84.  doi: 10.1016/0375-9601(95)00752-0.  Google Scholar

[22]

V. I. Karpman, Stabilization of soliton instabilities by higher order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996). doi: 10.1103/physreve.53.r1336.  Google Scholar

[23]

V. I. Karpman and A. Shagalov, Solitons and their stability in high dispersive systems. I. Fourth order nonlinear Schrödinger-type equations with power-law nonlinearities, Phys. Lett. A, 228 (1997), 59-65.  doi: 10.1016/S0375-9601(97)00063-7.  Google Scholar

[24]

V. I. Karpman and A. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.  Google Scholar

[25]

V. I. Karpman, Lyapunov approach to the soliton stability in highly dispersive systems. I. Fourth order nonlinear Schrödinger equations, Phys. Lett. A, 215 (1996), 254-256.  doi: 10.1016/0375-9601(96)00231-9.  Google Scholar

[26]

R. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264.  doi: 10.1143/JPSJ.33.260.  Google Scholar

[27]

G. Kevrekidis, A. Stefanov, Y. Tsolias and J. Maraver, Quartic generalizations of the nonlinear Schrödinger model in two-dimensions: Theoretical analysis and numerical computations, work in progress. Google Scholar

[28]

S. Kichenassamy and P. J. Olver, Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, SIAM J. Math. Anal., 23 (1992), 1141-1166.  doi: 10.1137/0523064.  Google Scholar

[29]

S. P. Levandosky, A stability analysis of fifth-order water-wave models, Phys. D, 125 (1999), 222-240.  doi: 10.1016/S0167-2789(98)00245-0.  Google Scholar

[30]

S. Levandosky, Stability of solitary waves of a fifth-order water wave model, Phys. D, 227 (2007), 162-172.  doi: 10.1016/j.physd.2007.01.006.  Google Scholar

[31]

Z. Lin and C. Zeng, Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs, preprint, arXiv: 1703.04016. Google Scholar

[32]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[33]

F. Natali and A. Pastor, The fourth order dispersive nonlinear Schrödinger equation: Orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14 (2015), 1326-1347.  doi: 10.1137/151004884.  Google Scholar

[34]

D. E. Pelinovsky, Spectral stability on nonlinear waves in KdV-type evolution equations, in Nonlinear Physical Systems, Mech. Eng. Solid Mech. Ser., Wiley, Hoboken, NJ, 2014,377–400. doi: 10.1002/9781118577608.ch17.  Google Scholar

[35]

Z. Wang, Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation, Discrete Contin. Dyn. Syst., 37 (2017), 4091-4108.  doi: 10.3934/dcds.2017174.  Google Scholar

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