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.

Nonlinear travelling internal waves with piecewise-linear shear profiles

2018 ◽  
Vol 856 ◽  
pp. 984-1013 ◽  
Author(s):  
K. L. Oliveras ◽  
C. W. Curtis
Keyword(s):  
Piecewise Linear ◽  
Travelling Waves ◽  
Tangential Velocity ◽  
Travelling Wave ◽  
Numerical Continuation ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Two Fluids ◽  
Wave Solutions ◽  
Linear Shear

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

scholarly journals Analytic Travelling Wave Solutions and Numerical Analysis of Fisher’s Equation via Explicit-Implicit FDM

2019 ◽  
pp. 1-13
Author(s):  
Asif Ahmed ◽  
Md. Kamrujjaman
Keyword(s):  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Second Phase ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Nonlinear Parabolic ◽  
Dimensional Equation ◽  
Two Phases

We study the nonlinear parabolic Fisher’s equations for travelling wave solutions. The analyses focus on to describe the analytic solution in the spatial pattern of travelling wave solutions; especially the solutions are characterized in invariant with respect to translation in space. There are two phases in the work: in the first stage, we analyze dimensional reaction-diffusion equation with logistic type growth while in the second phase the non-dimensional equation known as Fishers’ equation is studied numerically. To investigate the results numerically, we select the explicit-implicit finite difference method (FDM) and the approximate solutions are compared with the exact solution in different time steps.

scholarly journals A Newton-Type Approach to Approximate Travelling Wave Solutions of a Schrödinger-Benjamin-Ono System

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Juan Carlos Muñoz Grajales
Keyword(s):  
Iterative Method ◽  
Solitary Waves ◽  
Numerical Scheme ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Newton’S Iterative Method ◽  
Wave Solutions

We introduce a Newton’s iterative method to approximate periodic and nonperiodic travelling wave solutions of the Schrödinger-Benjamin-Ono system derived by M. Funakoshi and M. Oikawa. We analyze numerically the influence of the model’s parameters on these solutions and illustrate the collision of two unequal-amplitude solitary waves propagating with different speeds computed by using the proposed numerical scheme.

New Exact Travelling Wave Solutions of Nonlinear Coagulation Problem with Mass Loss

2010 ◽  
Vol 65 (3) ◽  
pp. 209-214
Author(s):  
El-Said A. El-Wakil ◽  
Essam M. Abulwafa ◽  
Mohammed A. Abdou
Keyword(s):  
Differential Equation ◽  
Mass Loss ◽  
Evolution Equations ◽  
Expansion Method ◽  
Approximate Solutions ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Travelling Wave Solutions ◽  
Rational Solutions ◽  
Wave Solutions

This paper suggests a generalized F-expansion method for constructing new exact travelling wave solutions of a nonlinear coagulation problem with mass loss. This method can be used as an alternative to obtain analytical and approximate solutions of different types of kernel which are applied in physics. The nonlinear kinetic equation, which is an integro differential equation, is transformed into a differential equation using Laplace’s transformation. The inverse Laplace transformation of the solution gives the size distribution function of the system. As a result, many exact travelling wave solutions are obtained which include new periodic wave solutions, trigonometric function solutions, and rational solutions. The method is straightforward and concise,and it can also be applied to other nonlinear evolution equations arising in mathematical physics.

scholarly journals Analysis of a Fourier-Galerkin numerical scheme for a 1D Benney-Luke-Paumond equation

2016 ◽  
Vol 49 (2) ◽  
pp. 213 ◽  
Author(s):  
Juan Carlos Muñoz Grajales
Keyword(s):  
Water Waves ◽  
Numerical Scheme ◽  
Approximate Solutions ◽  
Spatial Period ◽  
Initial Value ◽  
Fully Discrete ◽  
Numerical Solver ◽  
Periodic Cauchy Problem ◽  
Periodic Setting ◽  
Shallow Channel

We study convergence of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a nonlinear Benney-Luke-Paumond equation that models long water waves with small amplitude propagating over a shallow channel with at bottom. The accuracy of the numerical solver is checked using some exact solitary wave solutions. In order to apply the Fourier-spectral scheme in a non periodic setting, we approximate the initial value problem with x ∈ R by the corresponding periodic Cauchy problem for x ∈ [0, L], with a large spatial period L.

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.

Viscoplastic fluid displacements in horizontal narrow eccentric annuli: stratification and travelling wave solutions

2008 ◽  
Vol 605 ◽  
pp. 293-327 ◽  
Author(s):  
M. CARRASCO-TEJA ◽  
I. A. FRIGAARD ◽  
B. R. SEYMOUR ◽  
S. STOREY
Keyword(s):  
Steady State ◽  
Sufficient Conditions ◽  
Travelling Wave ◽  
Viscoplastic Fluid ◽  
Critical Conditions ◽  
Travelling Wave Solutions ◽  
Viscous Forces ◽  
Two Fluids ◽  
Wave Solutions ◽  
Eccentric Annuli

We consider laminar displacement flows in narrow eccentric annuli, oriented horizontally, between two fluids of Herschel–Bulkley type, (i.e. including Newtonian, power-law and Bingham models). This situation is modelled via a Hele-Shaw approach. Whereas slumping and stratification would be expected in the absence of any imposed flow rate, for a displacement flow we show that there are often steady-state travelling wave solutions in this displacement. These may exist even at large eccentricities and for large density differences between the fluids. When heavy fluids displace light fluids, annular eccentricity opposes buoyancy and steady states are more prevalent than when light fluids displace heavy fluids. For large ratios of buoyancy forces to viscous forces we derive a lubrication-style displacement model. This simplification allows us to find necessary and sufficient conditions under which a displacement can be steady, which can be expressed conveniently in terms of a consistency ratio. It is interesting that buoyancy does not appear in the critical conditions for a horizontal well. Instead a competition between fluid rheologies and eccentricity is the determining factor. Buoyancy acts only to determine the axial length of the steady-state profile.

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.

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