Remarks on solitary waves in equations with nonlocal cubic terms

Johanna Ulvedal Marstrander

doi:10.3934/jcd.2024012

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2024, Volume 11, Issue 3: 274-288. Doi: 10.3934/jcd.2024012

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Remarks on solitary waves in equations with nonlocal cubic terms

Johanna Ulvedal Marstrander

Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, 7491 Trondheim, Norway

Received: July 2023

Revised: January 2024

Early access: February 20, 2024

Published online: February 20, 2024

The author acknowledges the support of the project IMod (Grant No. 325114) from the Research Council of Norway. Part of this work was carried out during the workshop MaGIC 2023.

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Abstract

In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form

$ \begin{equation*} \partial_t u + \partial_x(\Lambda^s u + u\Lambda^r u^2) = 0, \end{equation*} $

where $ \Lambda^s, \Lambda^r $ are Bessel-type Fourier multipliers. The linear operator may be of low fractional order, $ s>0 $, while the operator on the nonlinear part is assumed to act slightly smoother, $ r<s-1 $. The problem is related to the mathematical theory of water waves; we build upon previous works on similar equations, extending them to allow for a nonlocal nonlinearity. Mathematical tools include constrained minimization, Lion's concentration–compactness principle, spectral estimates, and product estimates in fractional Sobolev spaces.

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Mathematics Subject Classification: 76B15, 76B25, 35A15.

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References

[1]	J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, in Applied Analysis(Baton Rouge, LA, 1996), vol. 221 of Contemp. Math., 1999, 1-29. doi: 10.1090/conm/221/03116.
[2]	M. N. Arnesen, Existence of solitary-wave solutions to nonlocal equations, Discrete Contin. Dyn. Syst., 36 (2016), 3483-3510. doi: 10.3934/dcds.2016.36.3483.
[3]	M. N. Arnesen, M. Ehrnstrom and A. G. Stefanov, A maximisation technique for solitary waves: The case of the nonlocally dispersive Whitham equation, preprint, 2023, arXiv: 2303.14036.
[4]	J. L. Bona, P. E. Souganidis and W. A. Strauss, Stability and Instability of Solitary Waves of Korteweg-de Vries Type, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 411 (1987), 395-412. doi: 10.1098/rspa.1987.0073.
[5]	B. Buffoni, Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation, Arch. Rational Mech. Anal., 173 (2004), 25-68. doi: 10.1007/s00205-004-0310-0.
[6]	E. Dinvay and D. Nilsson, Solitary wave solutions of a Whitham-Boussinesq system, Nonlinear Anal. Real World Appl., 60 (2021), Paper No. 103280, 24 pp. doi: 10.1016/j.nonrwa.2020.103280.
[7]	V. Duchêne, D. Nilsson and E. Wahlén, Solitary wave solutions to a class of modified Green-Naghdi systems, J. Math. Fluid Mech., 20 (2018), 1059-1091. doi: 10.1007/s00021-017-0355-0.
[8]	M. Ehrnström, M. D. Groves and D. Nilsson, Existence of Davey–Stewartson type solitary waves for the fully dispersive Kadomtsev–Petviashvilii equation, SIAM J. Math. Anal., 54 (2022), 4954-4986. doi: 10.1137/21M1451518.
[9]	M. Ehrnström, 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.
[10]	M. Ehrnström, K. Nik and C. Walker, A direct construction of a full family of Whitham solitary waves, Proc. Amer. Math. Soc., 151 (2023), 1247-1261. doi: 10.1090/proc/16191.
[11]	L. Grafakos, Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics, Springer New York, New York, 2014. doi: 10.1007/978-1-4939-1194-3.
[12]	M. D. Groves and S.-M. Sun, Fully localised solitary-wave solutions of the three-dimensional gravity–capillary water-wave problem, Arch Rational Mech Anal, 188 (2008), 1-91. doi: 10.1007/s00205-007-0085-1.
[13]	M. D. Groves and E. Wahlén, On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water, J. Math. Fluid Mech., 13 (2011), 593-627. doi: 10.1007/s00021-010-0034-x.
[14]	A. Ionescu and F. Pusateri, Global regularity for 2D water waves with surface tension, Mem. Amer. Math. Soc., 256 (2018), v+124 pp. doi: 10.1090/memo/1227.
[15]	D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.
[16]	F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of burgers and hyperbolic equations Ⅰ: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537. doi: 10.1137/130912001.
[17]	P. L. Lions, The concentration-compactness principle in the calculus of variations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. doi: 10.1016/s0294-1449(16)30428-0.
[18]	O. I. H. Maehlen, Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 4113-4130. doi: 10.3934/dcds.2020174.
[19]	J. U. Marstrander, Solitary waves for dispersive equations with Coifman-Meyer nonlinearities, preprint, 2023, arXiv: 2307.03628.
[20]	M. C. Ørke, Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations, preprint, 2022, arXiv: 2201.13159.
[21]	T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter, 2011. doi: 10.1515/9783110812411.
[22]	A. Stefanov and J. D. Wright, Small amplitude traveling waves in the full-dispersion Whitham equation, J. Dynam. Differential Equations, 32 (2020), 85-99. doi: 10.1007/s10884-018-9713-8.
[23]	M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, 12 (1987), 1133-1173. doi: 10.1080/03605308708820522.