On transforms reducing one-dimensional systems of shallow-water to the wave equation with sound speed c 2 = x

AbstractIn this article, the generalized shallow water wave (GSWW) equation is studied from the perspective of one dimensional optimal systems and their conservation laws (Cls). Some reduction and a new exact solution are obtained from known solutions to one dimensional optimal systems. Some of the solutions obtained involve expressions with Bessel function and Airy function [1,2,3]. The GSWW is a nonlinear self-adjoint (NSA) with the suitable differential substitution, Cls are constructed using the new conservation theorem.

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Dynamical solitary interactions between lump waves and different forms of n-solitons (n→∞) for the (2+1)-dimensional shallow water wave equation

Partial Differential Equations in Applied Mathematics ◽

10.1016/j.padiff.2021.100026 ◽

2021 ◽

Vol 3 ◽

pp. 100026

Author(s):

Fahad Sameer Alshammari ◽

Md Fazlul Hoque ◽

Harun-Or-Roshid

Keyword(s):

Wave Equation ◽

Shallow Water ◽

Water Wave ◽

Shallow Water Wave

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Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

Journal of Applied Physics ◽

10.1063/5.0050895 ◽

2021 ◽

Vol 130 (2) ◽

pp. 025104

Author(s):

Misael Ruiz-Veloz ◽

Geminiano Martínez-Ponce ◽

Rafael I. Fernández-Ayala ◽

Rigoberto Castro-Beltrán ◽

Luis Polo-Parada ◽

...

Keyword(s):

Wave Equation ◽

One Dimensional ◽

The One ◽

Dimensional Wave

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Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽

10.3390/math9121439 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1439

Author(s):

Chaudry Masood Khalique ◽

Karabo Plaatjie

Keyword(s):

Differential Equation ◽

Wave Equation ◽

Shallow Water ◽

Water Wave ◽

Elliptic Integral ◽

Momentum Conservation ◽

Two Dimensional ◽

Order Ordinary Differential Equation ◽

Closed Form Solutions ◽

Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

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Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2019007 ◽

2020 ◽

Vol 26 ◽

pp. 7

Author(s):

Hui Wei ◽

Shuguan Ji

Keyword(s):

Wave Equation ◽

Periodic Solutions ◽

Spectral Properties ◽

Seismic Waves ◽

Forced Vibrations ◽

One Dimensional ◽

Semilinear Wave Equation ◽

Spatial Interval ◽

Rational Multiple ◽

Infinitely Many Periodic Solutions

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.

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