Travelling wave solutions for inhomogeneous acoustic gravity waves

1979 ◽  
Vol 34 (3-4) ◽  
pp. 279-283 ◽  
Author(s):  
G. C. Das
Keyword(s):  
Gravity Waves ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Acoustic Gravity Waves ◽  
Wave Solutions

scholarly journals Error Analysis of an Implicit Spectral Scheme Applied to the Schrödinger-Benjamin-Ono System

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Juan Carlos Muñoz Grajales
Keyword(s):  
Gravity Waves ◽  
Numerical Scheme ◽  
Water Regime ◽  
Approximate Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Fully Discrete ◽  
Two Fluids ◽  
Wave Solutions ◽  
Numerical Solver

We develop error estimates of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a coupled nonlinear Schrödinger-Benjamin-Ono system that describes the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. The accuracy of the numerical solver is checked using some exact travelling wave solutions of the system.

scholarly journals The instability of periodic surface gravity waves

2011 ◽  
Vol 675 ◽  
pp. 141-167 ◽  
Author(s):  
BERNARD DECONINCK ◽  
KATIE OLIVERAS
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Travelling Wave ◽  
Coordinate Frame ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Full Spectrum ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Non Local

Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this paper, we discuss the stability of periodic travelling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem, modified from that of Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, p. 313), restricted to a one-dimensional surface. Transforming the non-local formulation to a travelling coordinate frame, we obtain a new formulation for the stationary solutions in the travelling reference frame as a single equation for the surface in physical coordinates. We demonstrate that this equation can be used to numerically determine non-trivial travelling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically examine the spectral stability of the periodic travelling wave solutions by extending Fourier–Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin–Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wavelength of the perturbation and are difficult to find numerically. To address this problem, we propose a strategy to estimate a priori the location in the complex plane of the eigenvalues associated with the instability.

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

scholarly journals Singularity analysis and analytic solutions for the Benney–Gjevik equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andronikos Paliathanasis ◽  
Genly Leon ◽  
P. G. L. Leach
Keyword(s):  
Evolution Equations ◽  
Singularity Analysis ◽  
Analytic Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Painlevé Test ◽  
Wave Solutions ◽  
Algebraic Solutions

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

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