scholarly journals Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg–de Vries equations

2007 ◽  
Vol 235 (1) ◽  
pp. 1-30 ◽  
Author(s):  
Jaime Angulo Pava
Keyword(s):  
Nonlinear Stability ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Periodic Traveling Wave

scholarly journals Existence of periodic traveling wave solution to the forced generalized nearly concentric Korteweg-de Vries equation

2000 ◽  
Vol 24 (6) ◽  
pp. 371-377 ◽  
Author(s):  
Kenneth L. Jones ◽  
Xiaogui He ◽  
Yunkai Chen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Integral Equations ◽  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
De Vries ◽  
Periodic Traveling Wave

This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg-de Vries equation in the form of(uη+u/(2η)+[f(u)]ξ+uξξξ)ξ+uθθ/η2=h0. The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili equation,(ut+[f(u)]x+uxxx)x+uyy=h0, and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous functions. The Schauder's fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.

ON THE INSTABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS—REVISITED

2021 ◽  
pp. 1-23
Author(s):  
FÁBIO NATALI ◽  
SABRINA AMARAL
Keyword(s):  
Traveling Wave ◽  
Orbital Stability ◽  
Traveling Wave Solutions ◽  
Spectral Stability ◽  
Dispersive Equations ◽  
Periodic Waves ◽  
General Proof ◽  
Wave Solutions ◽  
Periodic Traveling Wave

Abstract The purpose of this paper is to present an extension of the results in [8]. We establish a more general proof for the moving kernel formula to prove the spectral stability of periodic traveling wave solutions for the regularized Benjamin–Bona–Mahony type equations. As applications of our analysis, we show the spectral instability for the quintic Benjamin–Bona–Mahony equation and the spectral (orbital) stability for the regularized Benjamin–Ono equation.

scholarly journals Anti-periodic traveling wave solutions to a class of higher-order Kadomtsev-Petviashvili-Burgers equations

1994 ◽  
Vol 7 (1) ◽  
pp. 1-12
Author(s):  
Sergiu Aizicovici ◽  
Yun Gao ◽  
Shih-Liang Wen
Keyword(s):  
Traveling Wave ◽  
Continuous Dependence ◽  
Traveling Wave Solutions ◽  
Higher Order ◽  
Two Dimensional ◽  
Burgers Equations ◽  
Wave Solutions ◽  
Periodic Traveling Wave ◽  
Continuous Dependence On Data

We discuss the existence, uniqueness, and continuous dependence on data, of anti-periodic traveling wave solutions to higher order two-dimensional equations of Korteweg-deVries type.

Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics

2020 ◽  
Vol 34 (29) ◽  
pp. 2050282
Author(s):  
Asıf Yokuş ◽  
Doğan Kaya
Keyword(s):  
Traveling Wave ◽  
Numerical Solutions ◽  
Traveling Wave Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Traveling Wave Solution ◽  
Von Neumann ◽  
Wave Solutions ◽  
De Vries ◽  
The Stability

The traveling wave solutions of the combined Korteweg de Vries-modified Korteweg de Vries (cKdV-mKdV) equation and a complexly coupled KdV (CcKdV) equation are obtained by using the auto-Bäcklund Transformation Method (aBTM). To numerically approximate the exact solutions, the Finite Difference Method (FDM) is used. In addition, these exact traveling wave solutions and numerical solutions are compared by illustrating the tables and figures. Via the Fourier–von Neumann stability analysis, the stability of the FDM with the cKdV–mKdV equation is analyzed. The [Formula: see text] and [Formula: see text] norm errors are given for the numerical solutions. The 2D and 3D figures of the obtained solutions to these equations are plotted.

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