Orbital Stability or Instability of Solitary Waves to Generalized Boussinesq Equation with Quadratic-cubic Nonlinearity

2018 ◽  
Author(s):  
Milena Dimova ◽  
Natalia Kolkovska ◽  
Nikolay Kutev
Keyword(s):  
Solitary Waves ◽  
Boussinesq Equation ◽  
Orbital Stability ◽  
Cubic Nonlinearity ◽  
Generalized Boussinesq Equation

scholarly journals Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1398
Author(s):  
Natalia Kolkovska ◽  
Milena Dimova ◽  
Nikolai Kutev
Keyword(s):  
Solitary Waves ◽  
Orbital Stability ◽  
Second Derivative ◽  
Cubic Nonlinearity ◽  
Abstract Theory ◽  
Power Type ◽  
Analytical Formulas ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Double Dispersion

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

scholarly journals Analytical and Numerical Results on Global Dynamics of the Generalized Boussinesq Equation with Cubic Nonlinearity and External Excitation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mingyuan Li ◽  
Wei Zhang ◽  
Qiliang Wu
Keyword(s):  
Theoretical Prediction ◽  
Boussinesq Equation ◽  
Harmonic Excitation ◽  
Reductive Perturbation Method ◽  
Global Dynamics ◽  
Cubic Nonlinearity ◽  
Smale Horseshoe ◽  
Equivalent Wave ◽  
Generalized Boussinesq Equation ◽  
Perturbed Equation

This paper analytically and numerically presents global dynamics of the generalized Boussinesq equation (GBE) with cubic nonlinearity and harmonic excitation. The effect of the damping coefficient on the dynamical responses of the generalized Boussinesq equation is clearly revealed. Using the reductive perturbation method, an equivalent wave equation is then derived from the complex nonlinear equation of the GBE. The persistent homoclinic orbit for the perturbed equation is located through the first and second measurements, and the breaking of the homoclinic structure will generate chaos in a Smale horseshoe sense for the GBE. Numerical examples are used to test the validity of the theoretical prediction. Both theoretical prediction and numerical simulations demonstrate the homoclinic chaos for the GBE.

scholarly journals Solitary-wave propagation and interactions for a sixth-order generalized Boussinesq equation

2005 ◽  
Vol 2005 (9) ◽  
pp. 1435-1448 ◽  
Author(s):  
Bao-Feng Feng ◽  
Takuji Kawahara ◽  
Taketomo Mitsui ◽  
Youn-Sha Chan
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Shooting Method ◽  
Boussinesq Equation ◽  
Initial Conditions ◽  
Phase Plane Analysis ◽  
Plane Analysis ◽  
Implicit Finite Difference Scheme ◽  
Implicit Finite Difference ◽  
Generalized Boussinesq Equation

We study the solitary waves and their interaction for a six-order generalized Boussinesq equation (SGBE) both numerically and analytically. A shooting method with appropriate initial conditions, based on the phase plane analysis around the equilibrium point, is used to construct the solitary-wave solutions for this nonintegrable equation. A symmetric three-level implicit finite difference scheme with a free parameterθis proposed to study the propagation and interactions of solitary waves. Numerical simulations show the propagation of a single solitary wave of SGBE, and two solitary waves pass by each other without changing their shapes in the head-on collisions.

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