Exact solution for nonlinear deep-water waves

Shuwen Song; Yuxin Wang; Jian Song; Shuwen Song; Yuxin Wang; Jian Song

doi:10.3934/math.2026523

AIMS Mathematics

2026, Volume 11, Issue 5: 12705-12717. doi: 10.3934/math.2026523

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Exact solution for nonlinear deep-water waves

College of Sciences Inner Mongolia University of Technology, Hohhot, China

Received: 01 February 2026 Revised: 30 March 2026 Accepted: 14 April 2026 Published: 08 May 2026
MSC : 53Q30, 76B15

In this paper, we developed a generalized analytical framework for nonlinear deep-water waves by incorporating time-varying background flows $ (s_{1}(t), s_{2}(t)) $, effectively extending the classical Gerstner wave theory to unsteady environments. Within the framework of nonlinear partial differential equations, exact solutions for particle trajectories, velocity fields, and vorticity were rigorously derived using a time-varying Lagrangian mapping, while volume conservation throughout the evolutionary process was ensured via Jacobian analysis. A self-consistent pressure function was constructed to satisfy the nonlinear dynamic boundary conditions at the free surface. The core contribution of this study lies in the derivation of a modulated dispersion relation, which reveals how the coupling mechanism between wave dynamics and background acceleration induces an effective gravity shift, thereby explicitly regulating the phase velocity. By integrating these time-varying background flow terms, this work provides a robust theoretical foundation for characterizing transient wave-current interactions and nonlinear evolutionary processes in complex, non-steady ocean systems.
- Gerstner wave,
- exact solution,
- unsteady environments,
- dispersion relation,
- Lagrangian coordinates
Citation: Shuwen Song, Yuxin Wang, Jian Song. Exact solution for nonlinear deep-water waves[J]. AIMS Mathematics, 2026, 11(5): 12705-12717. doi: 10.3934/math.2026523

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Abstract

In this paper, we developed a generalized analytical framework for nonlinear deep-water waves by incorporating time-varying background flows $ (s_{1}(t), s_{2}(t)) $, effectively extending the classical Gerstner wave theory to unsteady environments. Within the framework of nonlinear partial differential equations, exact solutions for particle trajectories, velocity fields, and vorticity were rigorously derived using a time-varying Lagrangian mapping, while volume conservation throughout the evolutionary process was ensured via Jacobian analysis. A self-consistent pressure function was constructed to satisfy the nonlinear dynamic boundary conditions at the free surface. The core contribution of this study lies in the derivation of a modulated dispersion relation, which reveals how the coupling mechanism between wave dynamics and background acceleration induces an effective gravity shift, thereby explicitly regulating the phase velocity. By integrating these time-varying background flow terms, this work provides a robust theoretical foundation for characterizing transient wave-current interactions and nonlinear evolutionary processes in complex, non-steady ocean systems.

References

[1]	H. Lamb, Hydrodynamics, 6 Eds., Cambridge: Cambridge University Press, 1993.
[2]	C. C. Mei, M. A. Stiassnie, D. K. P. Yue, Theory and applications of ocean surface waves, 3 Eds., World Scientific, 2018. https://doi.org/10.1142/10212
[3]	A. E. Gill, Atmosphere-ocean dynamics, Academic Press, 1982.
[4]	J. J. Stoker, Water waves: The mathematical theory with applications, John Wiley & Sons, 1992. https://doi.org/10.1002/9781118033159
[5]	A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, Philadelphia: SIAM, 2011. https://doi.org/10.1137/1.9781611971873
[6]	A. A. Abrashkin, E. N. Pelinovsky, Gerstner waves and their generalizations in hydrodynamics and geophysics, Phys. Usp., 65 (2022), 453–467. https://doi.org/10.3367/UFNe.2021.05.038980 doi: 10.3367/UFNe.2021.05.038980
[7]	J. E. H. Weber, An interfacial Gerstner-type trapped wave, Wave Motion, 77 (2018), 186–194. https://doi.org/10.1016/j.wavemoti.2017.12.002 doi: 10.1016/j.wavemoti.2017.12.002
[8]	J. E. H. Weber, A Lagrangian study of internal Gerstner- and Stokes-type gravity waves, Wave Motion, 88 (2019), 257–264. https://doi.org/10.1016/j.wavemoti.2019.06.002 doi: 10.1016/j.wavemoti.2019.06.002
[9]	A. Constantin, W. Strauss, Exact periodic traveling water waves with vorticity, C. R. Math., 335 (2002), 797–800. https://doi.org/10.1016/S1631-073X(02)02565-7 doi: 10.1016/S1631-073X(02)02565-7
[10]	A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity, Commun. Pure Appl. Math., 57 (2004), 481–527. https://doi.org/10.1002/cpa.3046 doi: 10.1002/cpa.3046
[11]	A. Constantin, J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171–181. https://doi.org/10.1017/S0022112003006773 doi: 10.1017/S0022112003006773
[12]	A. Constantin, J. Escher, Symmetry of steady deep-water waves with vorticity, Eur. J. Appl. Math., 15 (2004), 755–768. https://doi.org/10.1017/S0956792504005777 doi: 10.1017/S0956792504005777
[13]	Q. Lin, On the regularity of the solution for incompressible 3D Navier-Stokes equation with periodic boundary conditions, Russ. J. Math. Phys., 31 (2024), 255–275. https://doi.org/10.1134/S1061920824020092 doi: 10.1134/S1061920824020092
[14]	S. V. Filatov, V. M. Parfenyev, S. S. Vergeles, M. Y. Brazhnikov, A. A. Levchenko, V. V. Lebedev, Nonlinear generation of vorticity by surface waves, Phys. Rev. Lett., 116 (2016), 054501. https://doi.org/10.1103/PhysRevLett.116.054501 doi: 10.1103/PhysRevLett.116.054501
[15]	C. I. Martin, Geophysical water flows with constant vorticity and centripetal terms, Ann. Mat. Pura Appl., 200 (2021), 101–116. https://doi.org/10.1007/s10231-020-00985-4 doi: 10.1007/s10231-020-00985-4
[16]	M. Kluczek, Nonlinear water wave models with vorticity, PhD Thesis, University College Cork, 2019.
[17]	D. Henry, T. Lyons, Pollard waves with underlying currents, Proc. Amer. Math. Soc., 149 (2021), 1175–1188. https://doi.org/10.1090/PROC/15309 doi: 10.1090/PROC/15309
[18]	R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511624056
[19]	G. D. Crapper, Introduction to water waves, Ellis Horwood, 1984.
[20]	L. M. Milne-Thomson, Theoretical hydrodynamics, 5 Eds., Dover Publications, 2011.
[21]	J. S. Russell, Report on waves, In: Report of the 14th Meeting of the British Association for the Advancement of Science, York, London, 1844,311.
[22]	D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422–443. https://doi.org/10.1080/14786449508620739 doi: 10.1080/14786449508620739
[23]	F. Gerstner, Theorie der Wellen, Ann. Phys., 32 (1809), 412–415. https://doi.org/10.1002/andp.18090320808
[24]	A. Constantin, On the deep water wave motion, J. Phys. A Math. Gen., 34 (2001), 1405. https://doi.org/10.1088/0305-4470/34/7/313 doi: 10.1088/0305-4470/34/7/313
[25]	P. F. Han, T. Bao, Dynamical behavior of multiwave interaction solutions for the (3+1)-dimensional Kadomtsev-Petviashvili-Bogoyavlensky-Konopelchenko equation, Nonlinear Dyn., 111 (2023), 4753–4768. https://doi.org/10.1007/s11071-022-08097-9 doi: 10.1007/s11071-022-08097-9
[26]	P. F. Han, K. Zhu, W. X. Ma, Novel superposition solutions for an extended (3+1)-dimensional generalized shallow water wave equation in fluid mechanics and plasma physics, Qual. Theory Dyn. Syst., 24 (2025), 150. https://doi.org/10.1007/s12346-025-01310-1 doi: 10.1007/s12346-025-01310-1
[27]	Z. P. Yang, W. P. Zhong, M. Belić, Dark localized waves in shallow waters: analysis within an extended Boussinesq system, Chin. Phys. Lett., 41 (2024), 044201. https://doi.org/10.1088/0256-307X/41/4/044201 doi: 10.1088/0256-307X/41/4/044201
[28]	M. H. Rafiq, M. N. Rafiq, J. Lin, Nonlinear localized waves and positon solutions for nonlocal Boussinesq equations in shallow water applications, Phys. Lett. A, 573 (2026), 131341. https://doi.org/10.1016/j.physleta.2026.131341 doi: 10.1016/j.physleta.2026.131341
[29]	P. F. Han, K. Zhu, F. Zhang, W. X. Ma, Y. Zhang, Inverse scattering transform for the fourth-order nonlinear Schrödinger equation with fully asymmetric non-zero boundary conditions, Chinese J. Phys., 96 (2025), 577–602. https://doi.org/10.1016/j.cjph.2025.05.021 doi: 10.1016/j.cjph.2025.05.021
[30]	P. F. Han, R. S. Ye, Y. Zhang, Inverse scattering transform for the coupled Lakshmanan-Porsezian-Daniel equations with non-zero boundary conditions in optical fiber communications, Math. Comput. Simul., 232 (2025), 483–503. https://doi.org/10.1016/j.matcom.2025.01.008 doi: 10.1016/j.matcom.2025.01.008
[31]	A. Bennett, Lagrangian fluid dynamics, Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511734939

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