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 and Continuous Dynamical Systems, 2020, 40 (7) : 4131-4162. doi: 10.3934/dcds.2020175
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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

M. Goldberg, personal communication.,

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

M. Goldberg, personal communication.,

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

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