Analytical properties and numerical solutions of the derivative nonlinear Schrödinger equation

1988 ◽  
Vol 40 (3) ◽  
pp. 585-602 ◽  
Author(s):  
Silvina Ponce Dawson ◽  
Constantino Ferro Fontán
Keyword(s):  
Numerical Solutions ◽  
Temporal Evolution ◽  
Soliton Solutions ◽  
Nonlinear Schrödinger ◽  
Analytical Properties ◽  
Derivative Nonlinear Schrödinger Equation ◽  
The Stability ◽  
A Charge ◽  
The Relationship ◽  
Dnls Equation

From the analysis of the symmetries of the derivative nonlinear Schrödinger (DNLS) equation, we obtain a new constant of motion, which may be formally considered as a charge and which is related to the helicity of the physical System. From comparison of these symmetries and those of the soliton solutions, we draw conclusions about the number of constraints that must be imposed and the way a Liapunov functional must be constructed in order to study the solitons' stability. We also examine the relationship between the stability with respect to form and the symmetries that are broken by the soliton solutions. We complete the analysis with some numerical simulations: we solve the DNLS equation taking a slightly perturbed soliton as an initial condition and study its temporal evolution, finding that, as expected, they are stable with respect to form.

scholarly journals On the exact and numerical solutions to a nonlinear model arising in mathematical biology

2018 ◽  
Vol 22 ◽  
pp. 01061 ◽  
Author(s):  
Asif Yokus ◽  
Tukur Abdulkadir Sulaiman ◽  
Haci Mehmet Baskonus ◽  
Sibel Pasali Atmaca
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Hyperbolic Function ◽  
Von Neumann ◽  
Convection Diffusion Equation ◽  
Wolfram Mathematica ◽  
Forward Difference ◽  
Von Neumann Analysis ◽  
The Stability

This study acquires the exact and numerical approximations of a reaction-convection-diffusion equation arising in mathematical bi- ology namely; Murry equation through its analytical solutions obtained by using a mathematical approach; the modified exp(-Ψ(η))-expansion function method. We successfully obtained the kink-type and singular soliton solutions with the hyperbolic function structure to this equa- tion. We performed the numerical simulations (3D and 2D) of the obtained analytical solutions under suitable values of parameters. We obtained the approximate numerical and exact solutions to this equa- tion by utilizing the finite forward difference scheme by taking one of the obtained analytical solutions into consideration. We investigate the stability of the finite forward difference method with the equation through the Fourier-Von Neumann analysis. We present the L2 and L∞ error norms of the approximations. The numerical and exact approx- imations are compared and the comparison is supported by a graphic plot. All the computations and the graphics plots in this study are car- ried out with help of the Matlab and Wolfram Mathematica softwares. Finally, we submit a comprehensive conclusion to this study.

Oblique nonlinear Alfvén waves in strongly magnetized beam plasmas. Part 2. Soliton solutions and integrability

1993 ◽  
Vol 50 (3) ◽  
pp. 457-476 ◽  
Author(s):  
Bernard Deconinck ◽  
Peter Meuris ◽  
Frank Verheest
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Nonlinear Schrodinger Equation ◽  
Mhd Waves ◽  
Displacement Current ◽  
Nonlinear Schrödinger ◽  
Derivative Nonlinear Schrödinger Equation

Oblique propagation of MHD waves in warm multi-species plasmas with anisotropic pressures and different equilibrium drifts is described by a modified vector derivative nonlinear Schrödinger equation, if charge separation in Poisson's equation and the displacement current in Ampère's law are properly taken into account. This modified equation cannot be reduced to the standard derivative nonlinear Schrödinger equation, and hence requires a new approach to solitary-wave solutions, integrability and related problems. The new equation is shown to be integrable by the use of the prolongation method, and by finding a sufficient number of conservation laws, and possesses bright and dark soliton solutions, besides possible periodic solutions.

Dispersive shock waves propagating in the cubic-quintic derivative nonlinear Schrödinger equation

2010 ◽  
Vol 88 (1) ◽  
pp. 55-66 ◽  
Author(s):  
E. Kengne ◽  
A. Lakhssassi ◽  
T. Nguyen-Ba ◽  
R. Vaillancourt
Keyword(s):  
Shock Wave ◽  
Wave Propagation ◽  
Shock Waves ◽  
Input Signal ◽  
Electrical Transmission ◽  
Dissipative Term ◽  
Nonlinear Schrödinger ◽  
Electrical Transmission Line ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Dnls Equation

The propagation of a dispersive shock wave is studied in a quintic-derivative nonlinear Schrödinger (Q-DNLS) equation, which may describe, for example, the wave propagation on a discrete electrical transmission line. It is shown that a physical system described by a Q-DNLS equation without a dissipative term may support the propagation of shock waves. The influence of the derivative nonlinearity terms on the shock is analyzed. Using the found exact shock solutions of the Q-DNLS equation as the initial input signal, we investigate numerically the spatiotemporal stability of the shock signal in the network.

scholarly journals Constructions of the soliton solutions to the good Boussinesq equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mohammed Bakheet Almatrafi ◽  
Abdulghani Ragaa Alharbi ◽  
Cemil Tunç
Keyword(s):  
Traveling Wave ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Solution Stability ◽  
Exact Results ◽  
The Stability ◽  
Stability And Accuracy

Abstract The principal objective of the present paper is to manifest the exact traveling wave and numerical solutions of the good Boussinesq (GB) equation by employing He’s semiinverse process and moving mesh approaches. We present the achieved exact results in the form of hyperbolic trigonometric functions. We test the stability of the exact results. We discretize the GB equation using the finite-difference method. We also investigate the accuracy and stability of the used numerical scheme. We sketch some 2D and 3D surfaces for some recorded results. We theoretically and graphically report numerical comparisons with exact traveling wave solutions. We measure the $L_{2}$ L 2  error to show the accuracy of the used numerical technique. We can conclude that the novel techniques deliver improved solution stability and accuracy. They are reliable and effective in extracting some new soliton solutions for some nonlinear partial differential equations (NLPDEs).

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