Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part II: Comparisons of local error estimation and step-selection strategies for nonlinear Schrödinger and wave equations

2019 ◽  
Vol 234 ◽  
pp. 55-71 ◽  
Author(s):  
Winfried Auzinger ◽  
Iva Březinová ◽  
Harald Hofstätter ◽  
Othmar Koch ◽  
Michael Quell
Keyword(s):  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Splitting Methods ◽  
Nonlinear Evolution Equations ◽  
Local Error ◽  
Local Error Estimation ◽  
Adaptive Integration ◽  
Selection Strategies ◽  
Step Selection

scholarly journals A note on improved F -expansion method combined with Riccati equation applied to nonlinear evolution equations

2014 ◽  
Vol 1 (2) ◽  
pp. 140038 ◽  
Author(s):  
Md. Shafiqul Islam ◽  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
Antonio Mastroberardino
Keyword(s):  
Riccati Equation ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Wave Equations ◽  
Travelling Wave Solutions

The purpose of this article is to present an analytical method, namely the improved F -expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.

scholarly journals Quasilinear wave equations and related nonlinear evolution equations

1981 ◽  
Vol 84 ◽  
pp. 31-83 ◽  
Author(s):  
Yoshio Yamada
Keyword(s):  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Quasilinear Wave Equations

In this paper we consider the relations between quasilinear wave equations

Functional variable method to study nonlinear evolution equations

2013 ◽  
Vol 3 (3) ◽  
Author(s):  
Mostafa Eslami ◽  
Mohammad Mirzazadeh
Keyword(s):  
Exact Solutions ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Variable Method ◽  
Nonlinear Wave Equations ◽  
Wave Solutions ◽  
Functional Variable ◽  
Functional Variable Method

AbstractThe functional variable method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations. This method presents a wider applicability for handling nonlinear wave equations. In this paper, the functional variable method is used to construct exact solutions of Davey-Stewartson equation, generalized Zakharov equation, K(m, n) equation with generalized evolution term, (2 + 1)-dimensional long-wave-short-wave resonance interaction equation and nonlinear Schrödinger equation with power law nonlinearity. The obtained solutions include solitary wave solutions, periodic wave solutions.

The Exp-function Method for the Riccati Equation and Exact Solutions of Dispersive Long Wave Equations

2008 ◽  
Vol 63 (10-11) ◽  
pp. 663-670 ◽  
Author(s):  
Sheng Zhang ◽  
Wei Wang ◽  
Jing-Lin Tong
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Function Method ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
New Exact Solutions ◽  
Long Wave

In this paper, the Exp-function method is used to seek new generalized solitonary solutions of the Riccati equation. Based on the Riccati equation and one of its generalized solitonary solutions, new exact solutions with three arbitrary functions of the (2+1)-dimensional dispersive long wave equations are obtained. Compared with the tanh-function method and its extensions, the proposed method is more powerful. It is shown that the Exp-function method provides a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics.

Notiz: A Symbolic Computation-Based Method and Two Nonlinear Evolution Equations for Water Waves

1997 ◽  
Vol 52 (3) ◽  
pp. 295-296
Author(s):  
Yi-Tian Gao ◽  
Bo Tian
Keyword(s):  
Symbolic Computation ◽  
Solitary Waves ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Long Wave ◽  
Shallow Water Wave Equations

Abstract A symbolic-computation-based method, which has been newly proposed, is considered for a (2+1)-dimensional generalization of shallow water wave equations and a coupled set of the (2 +1)-dimensional integrable dispersive long wave equations. New sets of soliton-like solutions are constructed, along with solitary waves.

Lump solutions and interaction phenomena of the (3 + 1)-dimensional nonlinear evolution equations

2019 ◽  
Vol 29 (9) ◽  
pp. 3417-3436 ◽  
Author(s):  
Jin-Jin Mao ◽  
Shou-Fu Tian ◽  
Xing-Jie Yan ◽  
Tian-Tian Zhang
Keyword(s):  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Direct Method ◽  
Elastic Interaction ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Wave Equations ◽  
Lump Solutions ◽  
Content Type ◽  
Dimensional Variable

Purpose The purpose of this study is to examine the lump solutions of the (3 + 1)-dimensional nonlinear evolution equations by considering a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and a (3 + 1)-dimensional variable-coefficient generalized B-type Kadomtsev–Petviashvili (vcgBKP) equation as examples. Design/methodology/approach Based on Hirota’s bilinear theory, a direct method is used to examine the lump solutions of these two equations. Findings The complete non-elastic interaction solutions between a lump and a stripe are also discussed for the equations, which show that the lump solitons are swallowed by the stripe solitons. Originality/value The dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional KP-type nonlinear wave equations.

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