Solutions of nonlinear Schrödinger equation for interfacial waves propagating between two ideal fluids

2009 ◽  
Vol 87 (6) ◽  
pp. 675-684 ◽  
Author(s):  
A. M. Abourabia ◽  
M. A. Mahmoud ◽  
G. M. Khedr
Keyword(s):  
Dispersion Relation ◽  
Multiple Scale ◽  
Transformation Method ◽  
Wave Number ◽  
Sine Gordon Equation ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Inviscid Fluids ◽  
Ideal Fluids ◽  
Infinite Extent

We present solutions to the problem of waves propagating at an interface between two inviscid fluids of infinite extent and differing densities. The method of multiple scale is employed to obtain a dispersion relation and nonlinear Schrödinger (NLS) equation, which describes the behavior of the system for the fluid interface. The dispersion relation of the model NLS equation is studied. The solutions of the NLS equation are derived analytically by using the complex tanh-function method and the function transformation method into a sine-Gordon equation. Also, diagrams are drawn to illustrate the elevation of the interface, the slip velocity, and the conservation of power. We observe that the elevation of the interface is in the form of traveling quasi-solitary waves that decrease as the wave number increases. We see that the slip velocities also bring a nonlinear and periodic characters. Finally, we observe that the conservation of power is in the form of traveling waves. Also, as the wave number increases, the conservation of power is more accurate in fluctuating around zero.

Solutions of solitary-wave for the variable-coefficient nonlinear Schrödinger equation with two power-law nonlinear terms

2018 ◽  
Vol 32 (28) ◽  
pp. 1850310 ◽  
Author(s):  
Le Xin ◽  
Ying Kong ◽  
Lijia Han
Keyword(s):  
Solitary Wave ◽  
Power Law ◽  
Optical Lattice ◽  
Variable Coefficient ◽  
Transformation Method ◽  
Potential Space ◽  
Nls Equation ◽  
Quadratic Potential ◽  
Nonlinear Schrödinger ◽  
Constant Coefficients

In this paper, we consider the variable-coefficient power-law nonlinear Schrödinger equations (NLSEs) with external potential as well as the gain or loss function. First, we generalize the similarity transformation method, which converts the variable coefficient NLSE with two power-law nonlinear terms to the autonomous dual-power NLS equation with constant coefficients. Then, we obtain the exact solutions of the variable-coefficient NLSE. Moreover, we discuss the solitary-wave solutions for equations with vanishing potential, space-quadratic potential and optical lattice potential, which are applied to many branches of physics.

A Direct Transformation Method and its Application to Variable Coefficient Nonlinear Equations of Schrödinger Type

2009 ◽  
Vol 64 (11) ◽  
pp. 697-708 ◽  
Author(s):  
Li-Hua Zhang ◽  
Xi-Qiang Liu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Variable Coefficient ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Transformation Method ◽  
Nonlinear Evolution Equations ◽  
Nls Equation ◽  
Direct Transformation ◽  
Nonlinear Schrödinger

In this paper, the generalized variable coefficient nonlinear Schrödinger (NLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients are directly reduced to simple and solvable ordinary differential equations by means of a direct transformation method. Taking advantage of the known solutions of the obtained ordinary differential equations, families of exact nontravelling wave solutions for the two equations have been constructed. The characteristic feature of the direct transformation method is, that without much extra effort, we circumvent the integration by directly reducing the variable coefficient nonlinear evolution equations to the known ordinary differential equations. Another advantage of the method is that it is independent of the integrability of the given nonlinear equation. The method used here can be applied to reduce other variable coefficient nonlinear evolution equations to ordinary differential equations.

scholarly journals Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water

2017 ◽  
Vol 24 (2) ◽  
pp. 255-264 ◽  
Author(s):  
Anatoly Abrashkin ◽  
Efim Pelinovsky
Keyword(s):  
Wave Number ◽  
Lagrangian Coordinates ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Wave Packet Dynamics ◽  
Lagrangian Solution ◽  
Rotational Wave ◽  
Special Cases ◽  
Lagrangian Coordinate ◽  
Rotational Waves

Abstract. The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift of the wave number in the carrier wave and in variation in the coefficient multiplying the nonlinear term. In the case of vorticity dependence on the vertical Lagrangian coordinate only (Gouyon waves), the shift of the wave number and the respective coefficient are constant. When the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally inhomogeneous. There are special cases (e.g., Gerstner waves) in which the vorticity is proportional to the squared wave amplitude and nonlinearity disappears, thus making the equations for wave packet dynamics linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables may be obtained from the Lagrangian solution by simply changing the horizontal coordinates.

scholarly journals The Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water: rogue wave aspect

2016 ◽  
Author(s):  
Anatoly Abrashkin ◽  
Efim Pelinovsky
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Wave Number ◽  
Lagrangian Coordinates ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Lagrangian Solution

Abstract. The nonlinear Schrödinger equation (NLS equation) describing weakly rotational wave packets in an infinity-depth fluid in the Lagrangian coordinates is derived. The vorticity is assumed to be an arbitrary function of the Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is proved that the modulation instability criteria of the low-vorticity waves and deep water potential waves coincide. All the known solutions of the NLS equation for rogue waves are applicable to the low-vorticity waves. The effect of vorticity is manifested in a shift of the wave number in the carrier wave. In case of vorticity dependence on the vertical Lagrangian coordinate only (the Gouyon waves) this shift is constant. In a more general case, where the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally heterogeneous. There is a special case with the Gerstner waves where the vorticity is proportional to the square of the wave amplitude, and the resulting non-linearity disappears, thus making the equations of the dynamics of the Gerstner wave packet linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables can be obtained from the Lagrangian solution by the ordinary change of the horizontal coordinates.

The effect of an axial electric field on the nonlinear stability between two uniform stream flows of finitely conducting cylinders

2003 ◽  
Vol 81 (6) ◽  
pp. 805-821 ◽  
Author(s):  
Abdel Raouf F Elhefnawy ◽  
Galal M Moatimid ◽  
Abd Elmonem Khalil Elcoot
Keyword(s):  
Dispersion Relation ◽  
Electric Field ◽  
Circular Cross Section ◽  
Multiple Scale ◽  
Physical Parameters ◽  
Linear Dispersion ◽  
Order Problem ◽  
Nonlinear Schrödinger ◽  
Axial Electric Field ◽  
The Stability

Weakly nonlinear streaming instability of two conducting fluids with an interface is presented for cylinders of circular cross section. The two fluids are subjected to a uniform axial electric field. Gravitational effects are neglected. The method of multiple scale perturbation is used to obtain a dispersion relation for the first-order problem and two nonlinear Schrödinger equations for the higher orders. The nonlinear Schrödinger equation, generally, describes the competition between nonlinearity and a linear dispersion relation. One of these equations is used to determine the nonlinear cutoff electric field separating stable and unstable disturbances, while the other is used to analyze the stability of the system. The stability criterion is expressed theoretically in terms of various parameters of the problem. Stability diagrams are obtained for different sets of physical parameters. New instability regions in the parameter space, which appear due to nonlinear effects, are indicated. PACS Nos.: 47.20, 47.55.C, 47.65

New Rational Homoclinic Solution and Rogue Wave Solution for the Coupled Nonlinear Schrödinger Equation

2014 ◽  
Vol 69 (8-9) ◽  
pp. 441-445 ◽  
Author(s):  
Long-Xing Li ◽  
Jun Liu ◽  
Zheng-De Dai ◽  
Ren-Lang Liu
Keyword(s):  
Schrödinger Equation ◽  
Wave Solution ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Homoclinic Solution ◽  
Nonlinear Schrodinger Equation ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Rogue Wave Solution

In this work, the rational homoclinic solution (rogue wave solution) can be obtained via the classical homoclinic solution for the nonlinear Schrödinger (NLS) equation and the coupled nonlinear Schrödinger (CNLS) equation, respectively. This is a new way for generating rogue wave comparing with direct constructing method and Darboux dressing technique

Analytical simulations of the Fokas system; extension (2 + 1)-dimensional nonlinear Schrödinger equation

2021 ◽  
Author(s):  
Mostafa M. A. Khater
Keyword(s):  
Optical Fibers ◽  
Pulse Propagation ◽  
Dynamical Behavior ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Contour Plots ◽  
System Extension ◽  
Inverse Spectral Method ◽  
Nonlinear Pulse Propagation ◽  
The Stability

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

Note on magneto-hydrodynamic gravitational stability of a double fluids interface

2008 ◽  
Vol 86 (3) ◽  
pp. 477-485
Author(s):  
Ahmed E Radwan ◽  
Mourad F Dimian
Keyword(s):  
Dispersion Relation ◽  
Present Model ◽  
The Self ◽  
Wave Number ◽  
Symmetric Mode ◽  
Fluid Interface ◽  
Unperturbed State ◽  
Gravitational Stability ◽  
The Stability ◽  
Range Force

The magneto–gravitational stability of double-fluid interface is discussed. The pressure in the unperturbed state is not constant because the self-gravitating force is a long-range force. The dispersion relation is derived and discussed. The self-gravitating model is unstable in the symmetric mode m = 0 (m is the transverse wave number), while it is stable in all other states. The effects of the densities, the liquid-fluid radii ratios, and the electromagnetic force on the stability of the present model are identified for all wavelengths.PACS Nos.: 47.35.Tv, 47.65.–d, 04.40.–b

Mixed interaction solutions for the coupled nonlinear Schrödinger equations

2021 ◽  
pp. 2150004
Author(s):  
Yaning Tang ◽  
Jiale Zhou
Keyword(s):  
Darboux Transformation ◽  
Rogue Waves ◽  
Transformation Method ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Interaction Solutions ◽  
Nonlinear Schrodinger Equations

We investigate the mixed interaction solutions of the coupled nonlinear Schrödinger equations (CNLSE) through the Darboux transformation method. First of all, we derive the nonsingular localized wave solutions for two cases of CNLSE by the Darboux transformation method and matrix analysis method. Furthermore, we take a limit technique to deduce rogue waves and divide the rogue waves into four categories through analyzing their dynamic behaviors. Based on the obtained theorems, the Darboux transformations are presented to solve interaction solutions between distinct nonlinear waves. In this paper, we mainly study four types. Finally, the dynamic characteristics of the constructed these solutions are analyzed by sequences of interesting figures plotted with the help of Maple.

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