Orbital Stability and Instability of Solitary Waves for a Class of Dispersive Symmetric Regularized Long-Wave Equation

2019 ◽  
Vol 16 (4) ◽  
Author(s):  
Sevdzhan Hakkaev
Keyword(s):  
Wave Equation ◽  
Solitary Waves ◽  
Orbital Stability ◽  
Long Wave ◽  
Stability And Instability ◽  
Regularized Long Wave Equation

scholarly journals Numerical investigation of nonlinear generalized regularized long wave equation via delta-shaped basis functions

2020 ◽  
Vol 10 (2) ◽  
pp. 244-258 ◽  
Author(s):  
Ömer Oruç
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Finite Difference ◽  
Solitary Waves ◽  
Basis Functions ◽  
Long Wave ◽  
Highly Nonlinear ◽  
Regularized Long Wave Equation ◽  
Waves Interaction ◽  
Grlw Equation

In this study we will investigate generalized regularized long wave (GRLW)equation numerically. The GRLW equation is a highly nonlinear partialdifferential equation. We use finite difference approach for timederivatives and linearize the nonlinear equation. Then for space discretizationwe use delta-shaped basis functions which are relatively few studiedbasis functions. By doing so we obtain a linear system of equationswhose solution is used for constructing numerical solution of theGRLW equation. To see efficiency of the proposed method four classictest problems namely the motion of a single solitary wave, interactionof two solitary waves, interaction of three solitary waves and Maxwellianinitial condition are solved. Further, invariants are calculated.The results of numerical simulations are compared with exact solutionsif available and with finite difference, finite element and some collocationmethods. The comparison indicates that the proposed method is favorableand gives accurate results.

scholarly journals Orbital Stability of Solitary Waves for Generalized Symmetric Regularized-Long-Wave Equations with Two Nonlinear Terms

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Weiguo Zhang ◽  
Xu Chen ◽  
Zhengming Li ◽  
Haiyan Zhang
Keyword(s):  
Explicit Expression ◽  
Solitary Waves ◽  
Wave Speed ◽  
Wave Equations ◽  
Sufficient Conditions ◽  
Orbital Stability ◽  
Long Wave ◽  
Stability And Instability ◽  
Stability Of Solitary Waves

This paper investigates the orbital stability of solitary waves for the generalized symmetric regularized-long-wave equations with two nonlinear terms and analyzes the influence of the interaction between two nonlinear terms on the orbital stability. SinceJis not onto, Grillakis-Shatah-Strauss theory cannot be applied on the system directly. We overcome this difficulty and obtain the general conclusion on orbital stability of solitary waves in this paper. Then, according to two exact solitary waves of the equations, we deduce the explicit expression of discriminationd′′(c)and give several sufficient conditions which can be used to judge the orbital stability and instability for the two solitary waves. Furthermore, we analyze the influence of the interaction between two nonlinear terms of the equations on the wave speed interval which makes the solitary waves stable.

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