A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations

2016 ◽  
Vol 71 (5) ◽  
pp. 475-480 ◽  
Author(s):  
Emrullah Yaşar ◽  
Sait San
Keyword(s):  
Conservation Laws ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Adjoint Equations ◽  
Nonlinear Wave Equations ◽  
Local Conservation ◽  
Physical Problems ◽  
Conservation Theorem

AbstractIn this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov’s new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer–Chree (PC) equation and the Kaup–Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.

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.

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.

scholarly journals Conservation Laws of Two(2+1)-Dimensional Nonlinear Evolution Equations with Higher-Order Mixed Derivatives

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Li-hua Zhang
Keyword(s):  
Conservation Laws ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Time Dependent ◽  
Mixed Derivatives ◽  
Paper Conservation ◽  
Conservation Theorem ◽  
Bbm Equation

In this paper, conservation laws for the(2+1)-dimensional ANNV equation and KP-BBM equation with higher-order mixed derivatives are studied. Due to the existence of higher-order mixed derivatives, Ibragimov’s “new conservation theorem” cannot be applied to the two equations directly. We propose two modification rules which ensure that the theorem can be applied to nonlinear evolution equations with any mixed derivatives. Formulas of conservation laws for the ANNV equation and KP-BBM equation are given. Using these formulas, many nontrivial and time-dependent conservation laws for these equations are derived.

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.

scholarly journals Conservation Laws of Nonlinear-Evolution Equations

1974 ◽  
Vol 52 (3) ◽  
pp. 886-889 ◽  
Author(s):  
K. Konno ◽  
H. Sanuki ◽  
Y. H. Ichikawa
Keyword(s):  
Conservation Laws ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations

An extension of the coupled derivative nonlinear Schrödinger hierarchy

2018 ◽  
Vol 32 (02) ◽  
pp. 1850016
Author(s):  
Siqi Xu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Compatibility Condition ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Matrix Spectral Problem

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

Sign in / Sign up

Export Citation Format

Share Document