scholarly journals Control and Stabilization of High-Order KdV Equation Posed on the Periodic Domain

2018 ◽  
Vol 31 (1) ◽  
pp. 29-46
Author(s):  
Zhao Xiangqing and Bai Meng
Keyword(s):  
Kdv Equation ◽  
High Order ◽  
Periodic Domain

A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg–de Vries (KdV) equation

2021 ◽  
Author(s):  
Murat Polat ◽  
Ömer Oruç
Keyword(s):  
Conservation Laws ◽  
Lie Group ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
High Order ◽  
Basis Functions ◽  
Numerical Integrator ◽  
The Kdv Equation ◽  
De Vries ◽  
Novel Method

In this work, we develop a novel method to obtain numerical solution of well-known Korteweg–de Vries (KdV) equation. In the novel method, we generate differentiation matrices for spatial derivatives of the KdV equation by using delta-shaped basis functions (DBFs). For temporal integration we use a high order geometric numerical integrator based on Lie group methods. This paper is a first attempt to combine DBFs and high order geometric numerical integrator for solving such a nonlinear partial differential equation (PDE) which preserves conservation laws. To demonstrate the performance of the proposed method we consider five test problems. We reckon [Formula: see text], [Formula: see text] and root mean square (RMS) errors and compare them with other results available in the literature. Besides the errors, we also monitor conservation laws of the KDV equation and we show that the method in this paper produces accurate results and preserves the conservation laws quite good. Numerical outcomes show that the present novel method is efficient and reliable for PDEs.

High-order breathers, lumps and hybrid solutions to the (2+1)-dimensional fifth-order KdV equation

2019 ◽  
Vol 33 (22) ◽  
pp. 1950255 ◽  
Author(s):  
Wen-Tao Li ◽  
Zhao Zhang ◽  
Xiang-Yu Yang ◽  
Biao Li
Keyword(s):  
Kdv Equation ◽  
Perturbation Expansion ◽  
Soliton Solutions ◽  
High Order ◽  
Bilinear Method ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Fifth Order ◽  
Parameter Constraints

In this paper, the (2+1)-dimensional fifth-order KdV equation is analytically investigated. By using Hirota’s bilinear method combined with perturbation expansion, the high-order breather solutions of the fifth-order KdV equation are generated. Then, the high-order lump solutions are also derived from the soliton solutions by a long-wave limit method and some suitable parameter constraints. Furthermore, we extend this method to obtain hybrid solutions by taking long-wave limit for partial soliton solutions. Finally, the dynamic behavior of these solutions is presented in the figures.

scholarly journals Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function

2021 ◽  
Vol 24 (5) ◽  
pp. 1507-1534
Author(s):  
Uyen Le ◽  
Dmitry E. Pelinovsky
Keyword(s):  
Green Function ◽  
Linear Operator ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Maximum Point ◽  
Periodic Domain ◽  
Caputo Fractional Derivatives ◽  
The Green Function

Abstract The linear operator c + (−Δ) α/2, where c > 0 and (−Δ) α/2 is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg–de Vries equation. We establish a relation of the Green function of this linear operator with the Mittag–Leffler function, which was previously used in the context of the Riemann–Liouville and Caputo fractional derivatives. By using this relation, we prove that the Green function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every c > 0 and every α ∈ (0, 2]. On the other hand, we argue from numerical approximations that in the case of α ∈ (2, 4], the Green function is positive and single-lobe for small c and non-positive and non-single lobe for large c.

scholarly journals Kato smoothing properties  of a class of nonlinear dispersive wave equations on a periodic domain

2021 ◽  
Author(s):  
Bingyu Zhang ◽  
Shu-Ming Sun ◽  
Xin Yang ◽  
Ning Zhong
Keyword(s):  
Cauchy Problem ◽  
Wave Equations ◽  
Kdv Equation ◽  
Dispersive Equations ◽  
Periodic Domain ◽  
Smoothing Property ◽  
The Kdv Equation ◽  
The Cauchy Problem ◽  
Real Value ◽  
Nonlinear Dispersive Wave

The solutions of the Cauchy problem of the KdV equation on a periodic domain $\T$,  \[ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in \T, \ t\in \R,\]  possess neither  the sharp Kato smoothing property,  \[ \phi \in H^s (\T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (\T, L^2 (0,T)),\]  nor the Kato smoothing property,  \[ \phi \in H^s (\T) \implies u\in L^2 (0,T; H^{s+1} (\T)).\]  Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain $\T$,  \[ u_t +uu_x +u_{xxx} - g(x) (g(x) u)_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in \T, \  t>0 \, ,\ \qquad (1) \]  where $g\in C^{\infty} (\T)$ is  a  real value function with  the support  \[ \mbox{$\omega = \{ x\in \T, \  g(x) \ne 0\}$.}\]  It is shown  that    \begin{itemize}  \item[(1)]  if $\omega\ne \emptyset$,   then the solutions of  the Cauchy problem (1) possess the Kato smoothing property;   \item[(2)] if     $g$ is a nonzero constant function,  then the solutions of  the Cauchy problem (1) possess the  sharp Kato smoothing property.   \end{itemize}

scholarly journals Symplectic multiquadric quasi-interpolation approximations of KdV equation

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5161-5171
Author(s):  
Shengliang Zhang ◽  
Liping Zhang
Keyword(s):  
Kdv Equation ◽  
Order Approximation ◽  
High Order ◽  
Basis Functions ◽  
High Order Accuracy ◽  
Approximation Properties ◽  
High Order Approximation ◽  
Shape Preserving ◽  
Order Accuracy ◽  
Long Time

Radial basis functions quasi-interpolation is very useful tool for the numerical solution of differential equations, since it possesses shape-preserving and high-order approximation properties. Based on multiquadric quasi-interpolations, this study suggests a meshless symplectic procedure for KdV equation. The method has a number of advantages over existing approaches including no need to solve a resultant full matrix, accuracy and ease of implementation. We also present a theoretical framework to show the conservativeness and convergence of the proposed method. As the numerical experiments show, it not only offers a high order accuracy but also has a good property of long-time tracking capability.

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