×

Plenty of wave solutions to the ill-posed Boussinesq dynamic wave equation under shallow water beneath gravity. (English) Zbl 1485.35115

Summary: This paper applies two computational techniques for constructing novel solitary wave solutions of the ill-posed Boussinesq dynamic wave (IPB) equation. Jacques Hadamard has formulated this model for studying the dynamic behavior of waves in shallow water under gravity. Extended simple equation (ESE) method and novel Riccati expansion (NRE) method have been applied to the investigated model’s converted nonlinear ordinary differential equation through the wave transformation. As a result of this research, many solitary wave solutions have been obtained and represented in different figures in two-dimensional, three-dimensional, and density plots. The explanation of the methods used shows their dynamics and effectiveness in dealing with certain nonlinear evolution equations.

MSC:

35C08 Soliton solutions
35R25 Ill-posed problems for PDEs
35Q35 PDEs in connection with fluid mechanics
76B25 Solitary waves for incompressible inviscid fluids
49M05 Numerical methods based on necessary conditions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D, The euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137, 1-81 (1998) · Zbl 0951.37020 · doi:10.1006/aima.1998.1721
[2] V, Bohmian trajectories and the path integral paradigm: Complexified lagrangian mechanics, Int. J. Bifurcat. Chaos, 19, 2335-2346 (2009) · Zbl 1176.81003 · doi:10.1142/S0218127409024104
[3] M, and breath solutions of the water wave propagation with surface tension via four recent computational schemes, Ain Shams Eng. J., 12, 3031-3041 (2021) · doi:10.1016/j.asej.2020.10.029
[4] M, On semi analytical and numerical simulations for a mathematical biological model; the time-fractional nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation, Chaos Soliton. Fract., 144, 110676 (2021) · Zbl 1498.35506 · doi:10.1016/j.chaos.2021.110676
[5] M. M. Khater, A. Mousa, M. El-Shorbagy, R. A. Attia, Analytical and semi-analytical solutions for Phi-four equation through three recent schemes, <i>Results Phys.</i>, 2021, 103954. doi: <a href=“http://dx.doi.org/10.1016/j.rinp.2021.103954.” target=“_blank”>10.1016/j.rinp.2021.103954.</a>
[6] M, Abundant stable computational solutions of Atangana-Baleanu fractional nonlinear HIV-1 infection of CD4+ T-cells of immunodeficiency syndrome, Results Phys., 144, 103890 (2021) · doi:10.1016/j.rinp.2021.103890
[7] M, Some optical soliton solutions to the perturbed nonlinear Schrödinger equation by modified Khater method, AIP Adv., 11, 025130 (2021) · doi:10.1063/5.0038671
[8] M, Optical soliton structure of the sub-10-fs-pulse propagation model, J. Opt., 50, 109-119 (2021) · doi:10.1007/s12596-020-00667-7
[9] Y, Diverse novel analytical and semi-analytical wave solutions of the generalized (2+1)-dimensional shallow water waves model, AIP Adv., 11, 015223 (2021) · doi:10.1063/5.0036261
[10] R, Computational and numerical simulations for the deoxyribonucleic acid (DNA) model, Discrete Cont. Dyn-S., 14, 3459-3478 (2021) · Zbl 1471.92238 · doi:10.3934/dcdss.2021018
[11] M, Analytical and semi-analytical solutions for time-fractional Cahn-Allen equation, Math. Method. Appl. Sci., 44, 2682-2691 (2021) · Zbl 1470.35114 · doi:10.1002/mma.6951
[12] A, Analytical optical pulses and bifurcation analysis for the traveling optical pulses of the hyperbolic nonlinear Schrödinger equation, Opt. Quant. Electron., 53, 1-19 (2021) · doi:10.1007/s11082-021-02787-1
[13] H, On the conformable nonlinear Schrödinger equation with second order spatiotemporal and group velocity dispersion coefficients, Chinese J. Phys., 72, 403-414 (2021) · doi:10.1016/j.cjph.2021.01.012
[14] F, Dynamics of solitons to the ill-posed Boussinesq equation, Eur. Phys. J. Plus, 132, 1-9 (2017) · doi:10.1140/epjp/i2017-11430-0
[15] S, New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science, Int. J. Opt. Control (IJOCTA), 7, 240-247 (2017) · doi:10.11121/ijocta.01.2017.00495
[16] P, A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: Filtering and regularization techniques, Appl. Math. Comput., 101, 159-207 (1999) · Zbl 0937.76050
[17] S, Auxiliary equation method for ill-posed Boussinesq equation, Phys. Scripta, 94, 085213 (2019) · doi:10.1088/1402-4896/ab1951
[18] C, Dynamical analysis of the nonlinear complex fractional emerging telecommunication model with higher-order dispersive cubic-quintic, Alex. Eng. J., 59, 1425-1433 (2020) · doi:10.1016/j.aej.2020.03.046
[19] M, Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms, Chaos Soliton. Fract., 136, 109824 (2020) · Zbl 1489.92102 · doi:10.1016/j.chaos.2020.109824
[20] C, On new computational and numerical solutions of the modified Zakharov-Kuznetsov equation arising in electrical engineering, Alex. Eng. J., 59, 1099-1105 (2020) · doi:10.1016/j.aej.2019.12.043
[21] M, Analytical and semi-analytical ample solutions of the higher-order nonlinear Schrödinger equation with the non-kerr nonlinear term, Results Phys., 16, 103000 (2020) · doi:10.1016/j.rinp.2020.103000
[22] D, Analytical and numerical simulations for the kinetics of phase separation in iron (Fe-Cr-X (X = Mo, Cu)) based on ternary alloys, Physica A, 537, 122634 (2020) · Zbl 07571789 · doi:10.1016/j.physa.2019.122634
[23] A, Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system, Chaos Soliton. Fract., 131, 109473 (2020) · Zbl 1495.35156 · doi:10.1016/j.chaos.2019.109473
[24] A, Computational solutions of the HIV-1 infection of CD \(4^+\) T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy, Chaos Soliton. Fract., 139, 110092 (2020) · doi:10.1016/j.chaos.2020.110092
[25] B, Symmetry reductions and exact solutions to the ill-posed Boussinesq equation, Int. J. Nonlin. Mech., 72, 80-83 (2015) · doi:10.1016/j.ijnonlinmec.2015.03.004
[26] F, Response solution to ill-posed Boussinesq equation with quasi-periodic forcing of Liouvillean frequency, J. Nonlinear Sci., 30, 657-710 (2021) · Zbl 1437.35017 · doi:10.1007/s00332-019-09587-8
[27] E, Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation, Open Phys., 14, 37-43 (2016) · doi:10.1515/phys-2016-0007
[28] N, Sharp local well-posedness for the “good” Boussinesq equation, J. Differ. Equations, 254, 2393-2433 (2013) · Zbl 1266.35006 · doi:10.1016/j.jde.2012.12.008
[29] R, Open boundary conditions for the primitive and Boussinesq equations, J. Atmos. Sci., 60, 2647-2660 (2003) · doi:10.1175/1520-0469(2003)060<2647:obcftp>2.0.co;2
[30] H, Stable manifolds to bounded solutions in possibly ill-posed PDEs, J. Differ. Equations, 268, 4830-4899 (2020) · Zbl 1448.35564 · doi:10.1016/j.jde.2019.10.042
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.