Nonlinear non-autonomous Boussinesq equations. (English) Zbl 07907916
Summary: We study solitary wave solutions for a nonlinear and non-autonomous Boussinesq system with initial conditions. Since the variable coefficients introduce distortions and modulations of the solution amplitudes, we implement a multiple-scale approach combining various modes in order to capture the coupling between the nonlinear evolution and the effect of the variable coefficient. The differential system is mapped into a solvable system of nonlinear and non-autonomous ODE which is integrable by recursion procedures. We show that even in the limiting autonomous case, the multiple-scale approach gives a new possibly integrable dispersionless coupled envelope system, which deserves further study. We validate our theoretical results with numerical simulations, and we study their stability.
MSC:
| 35Q51 | Soliton equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35G50 | Systems of nonlinear higher-order PDEs |
| 34E13 | Multiple scale methods for ordinary differential equations |
| 93C70 | Time-scale analysis and singular perturbations in control/observation systems |
References:
| [1] | J. L. Bona, M. Chen, J. -C. Saut; Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory, J. Nonlin. Sci., 12 (2002) 283-318. · Zbl 1022.35044 |
| [2] | J. L. Bona, M. Chen, J. -C. Saut; Global Existence of Smooth Solutions and Stability of Solitary Waves for a Generalized Boussinesq Equation, Comm. Math. Phys., 118 (1988) 15-29. · Zbl 0654.35018 |
| [3] | R. Camassa, D. D. Holm, C. D. Levermore; Long-time effects of bottom topography in shallow water, Physica D 98 (1996) 258-286. · Zbl 0899.76081 |
| [4] | C.-H. Chang, C.-J. Tang, C. Lin; Vortex generation and flow pattern development after a solitary wave passing over a bottom cavity, Computers and Fluids, 53 (2012) 79-92. · Zbl 1271.76053 |
| [5] | C. -L. Chen, X. -Y. Tang, S. -Y. Lou; Solutions of a (2 + 1)-dimensional dispersive long wave equation, Phys. Rev. E 66, 3 (2002) 036605. |
| [6] | S.-J. Chen, X. Lüa, X.-F. Tang; Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients, Comm. Nonlin. Sci. Numer Simulat, 95 (2021) 105628. · Zbl 1456.35072 |
| [7] | S. -K. Chua; On Weighted Sobolev Spaces, Can. J. Math., 48, 3 (1996) 527-541. · Zbl 0858.46026 |
| [8] | D. Clamond, D. Dutykh; Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth, J. Fluid Mech., 844 (2018) 491-518. · Zbl 1460.76644 |
| [9] | P. A. Clarkson, E. Dowie; Rational solutions of the Boussinesq equation and applications to rogue waves, Trans. Math. Appl., 1 (2017) 1-26. · Zbl 1403.37073 |
| [10] | A. Compelli, R. Ivanov, M. Todorov; Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom, Phil. Trans. Roy. Soc. A 376, 2111 (2018) 20170091. · Zbl 1404.76038 |
| [11] | W. Craig, P. Guyenne, D. P. Nichols, C. Sulem; Hamiltonian long-wave expansions for water waves over a rough bottom, Proc. Roy. Soc. A, 461 (2005) 839-873. · Zbl 1145.76325 |
| [12] | C. Dai, J. Zhu, J. Zhang; New exact solutions to the mKdV equation with variable coefficients, Chaos, Solitons and Fractals 27 (2006), 881-886. · Zbl 1088.35513 |
| [13] | F. Dias, D. Dutykh; Dynamics of tsunami waves in Extreme man-made and natural hazards in dynamics of structures (Springer, Dordrecht 2007), 201-224. |
| [14] | D. Dutykh, D. Clamond; Modified shallow water equations for significantly varying seabeds, Appl. Math. Modelling 40 (2016), 9767-9787. · Zbl 1443.76022 |
| [15] | D. Dutykh, O. Goubet; Derivation of dissipative Boussinesq equations using the Dirichlet-to-Neumann operator approach, Math. Comp. Sim., 127 (2016), 80-93. · Zbl 1520.35118 |
| [16] | C. A. J. Fletcher; Computational Techniques for Fluid Dynamics 2: Specific Techniques for Different Flow Categories, 2nd Ed, Springer Series in Computational Physics, 1991. · Zbl 0717.76001 |
| [17] | A. S. Fokas, A. Nachbin; Water waves over a variable bottom: a non-local formulation and conformal mappings, J. Fluid Mech. 695 (2012), 288-309. · Zbl 1250.76035 |
| [18] | M. F. Gobbi, J. T. Kirby, G. E. Wei; A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh) 4 , J. Fluid Mech., 405 (2000), 181-210. · Zbl 0964.76014 |
| [19] | I. M. Gorban; A Numerical Study of Solitary Wave Interactions with a Bottom Step, Con-tinuous and Distributed Systems II: Theory and Applications (2015), 369-387. · Zbl 1335.76023 |
| [20] | J. H. Herterich, F. Dias; Extreme long waves over a varying bathymetry numerical, J. Fluid Mech., 878 (2019), 481-501. · Zbl 1430.76074 |
| [21] | R. Hirota, J. Satsuma; Nonlinear Evolution Equations Generated from the Bäcklund Trans-formation for the Boussinesq Equation, Prog. Th. Phys. 57, 3 (1977), 797-807. · Zbl 1098.81547 |
| [22] | P. A. E. M. Janssen; Nonlinear Four-Wave Interactions and Freak Waves, J. Phys. Oceanog-raphy 33 (2003) 863-884. |
| [23] | O. V. Kaptsova, D. O. Kaptsov; Exact solution of Boussinesq equations for propagation of nonlinear waves, Eur. Phys. J. Plus (2020) 135-723. |
| [24] | D. J. Kaup; A Higher-Order Water-Wave Equation and the Method for Solving It, Prog. Theor. Phys., 54, 2 (1975) 396-408. · Zbl 1079.37514 |
| [25] | B. A. Kupershmidt; Mathematics of dispersive water waves, Commun. Math. Phys., 99, 1 (1985) 51-73. · Zbl 1093.37511 |
| [26] | Y. S. Kivshar, B. A. Malomed; Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys., 61, 4 (1989) 763. |
| [27] | N. Kolkovska, V. M. Vassilev; Solitary Waves to Boussinesq Equation with Linear Restoring Force. AIP Conf. Proc., 2164 (2019) 110005. |
| [28] | Y. Li, W. -X. Ma, J. E. Zhang; Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett., 275 (2000), 60-66. · Zbl 1115.35329 |
| [29] | W.-X. Ma; Integrable couplings of vector AKNS soliton equations, J. Math. Phys., 46 (2005) 033507. · Zbl 1067.37096 |
| [30] | L. Molinet, T. Raafat, I. Zaiter; The classical Boussinesq system revisited, arXiv preprint arXiv:2001.11870 (2020) |
| [31] | D. E. Mitsotakis; Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves, Math. Comp. Sim., 80 (2009), 860-873. · Zbl 1256.35089 |
| [32] | B. T. Nadiga, L. G. Margolin, P. K. Smolarkiewicz; Different approximations of shallow fluid flow over an obstacle, Phys. Fluids, 8, 8 (1996), 2066-2077. · Zbl 1082.76509 |
| [33] | O. Nakoulima et al; Solitary wave dynamics in shallow water over periodic topography, Chaos, 15 (2005) 037107. · Zbl 1144.37391 |
| [34] | K. E. Parnell et al.; Ship-induced solitary Riemann waves of depression in Venice Lagoon, Phys. Let. A, 379 (2015), 555-559. |
| [35] | S. Pierini, M. Ghil, M. D. Chekroun; Exploring the pullback attractors of a low-order quasi-geostrophic ocean model: The deterministic case, J. Climate 29, 11 (2016), 4185-4202. |
| [36] | S. Ponce de León A. R. Osborne; Role of Nonlinear Four-Wave Interactions Source Term on the Spectral Shape, J. Mar. Sci. Eng., 8 (2020) 251. |
| [37] | A. A. Saakyan; Convergence of Double Fourier Series after a Change of Variable, Math. Notes, 74, 2 (2003) 255-265; For basic reference to the Lipschitz condition and Fourier series convergence see: R. A. Adams, and J. J. F. Fournier, Sobolev Spaces (Academic Press, 2003). · Zbl 1052.42012 |
| [38] | H. H. Sohrab; Basic Real Analysis, vol. 231 (Birkhäuser 2003) p. 142. · Zbl 1035.26001 |
| [39] | X. -Y. Tang, S. -Y. Lou, Y. Zhang; Localized excitations in (2 + 1)-dimensional systems, Phys. Rev. E 66, 4 (2002) 046601. |
| [40] | B. V. Turesson; Nonlinear Potential Theory and Weighted Sobolev Spaces (Springer 2000) section 1.2.1. · Zbl 0949.31006 |
| [41] | R. M. Vargas-Magaña, P. Panayotaros; A non-local Dirichlet-to-Neumann operator: A Whitham-Boussinesq long-wave model for variable topography, Wave Motion 65 (2016) 156-174. · Zbl 1467.76019 |
| [42] | V. M. Vassilev, P. A. Djondjorov, M. T. Hadzhilazova, I. M. Mladenov; Traveling Wave Solutions of the One-Dimensional Boussinesq Paradigm Equation, AIP Conf. Proc. 1561 (2013) 327-332. |
| [43] | J. Yu; Revisiting terrain-following Boussinesq equations on a highly variable periodic bed, J. Ocean Eng. Marine Eng., 5 (2019) 403-412. |
| [44] | J. Yu; Waveform of gravity and capillary-gravity waves over bathymetry, Phys. Rev. Fluids, 4 (2019) 014806. |
| [45] | V. E. Zakharov, E. A. Kuznetsov; Multi-scale expansions in the theory of systems integrable by the inverse scattering transform, Physica 18 D (1986) 455-463. · Zbl 0611.35079 |
| [46] | J. E. Zhang, Y. Li; Bidirectional solitons on water, Phys. Rev. E, 67 (2003) 016306. |
| [47] | D. Zhao, Y.-J. Zhang, W.-W. Lou, H.-G. Luo; AKNS hierarchy, Darboux transformation and conservation laws of the 1D nonautonomous nonlinear Schrödinger equations, J. Math. Phys., 52 (2011) 043502. · Zbl 1316.35271 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
