Generalised conservation laws, reductions and exact solutions of the $$K(m,n)$$ equations via double reduction theory

Pramana ◽  
2021 ◽  
Vol 95 (1) ◽  
Author(s):  
A Iqbal ◽  
I Naeem
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Reduction Theory ◽  
Double Reduction

scholarly journals Conservation Laws and Exact Solutions of Generalized Nonlinear System and Nizhink-Novikov-Veselov Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
A. A. Zaidi ◽  
M. D. Khan ◽  
I. Naeem
Keyword(s):  
Nonlinear System ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Symmetries ◽  
Reduction Theory ◽  
Lie Point Symmetries ◽  
Double Reduction

The Lie symmetries, conservation laws, and exact solutions of a generalized nonlinear system and a (2+1)-dimensional generalized Nizhink-Novikov-Veselov (NNV) equation, arising in the study of hydrodynamics, are investigated. The multiplier approach is employed to compute the conservation laws for systems under consideration. The Lie point symmetries are derived and the association between symmetries and conserved vectors are established using symmetries conservation laws relationship. The double reduction theory is utilized which results in the reduction and exact solutions of models under investigation. All cases are discussed in detail and new solutions are determined.

scholarly journals Exact Solutions of Generalized Modified Boussinesq, Kuramoto-Sivashinsky, and Camassa-Holm Equations via Double Reduction Theory

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Zulfiqar Ali ◽  
Syed Husnine ◽  
Imran Naeem
Keyword(s):  
Exact Solutions ◽  
Fourth Order ◽  
Arbitrary Function ◽  
Mixed Derivative ◽  
Reduction Theory ◽  
Double Reduction ◽  
Derivative Term ◽  
Divergence Property

We find exact solutions of the Generalized Modified Boussinesq (GMB) equation, the Kuromoto-Sivashinsky (KS) equation the and, Camassa-Holm (CH) equation by utilizing the double reduction theory related to conserved vectors. The fourth order GMB equation involves the arbitrary function and mixed derivative terms in highest derivative. The partial Noether’s approach yields seven conserved vectors for GMB equation and one conserved for vector KS equation. Due to presence of mixed derivative term the conserved vectors for GMB equation derived by the Noether like theorem do not satisfy the divergence relationship. The extra terms that constitute the trivial part of conserved vectors are adjusted and the resulting conserved vectors satisfy the divergence property. The double reduction theory yields two independent solutions and one reduction for GMB equation and one solution for KS equation. For CH equation two independent solutions are obtained elsewhere by double reduction theory with the help of conserved Vectors.

scholarly journals Double Reduction Analysis of Benjamin, DGH and Generalized DGH Equations

2020 ◽  
Vol Volume 14 ◽  
Keyword(s):  
Evolution Equation ◽  
Exact Solutions ◽  
Higher Order ◽  
Reduction Theory ◽  
Double Reduction ◽  
Linear Evolution ◽  
Holm Equation ◽  
Linear Evolution Equation ◽  
Non Linear ◽  
Benjamin Equation

The exact solutions of non-linear evolution equation, Benjamin equation, Dullin-Gottwald-Holm (DGH) equation and generalized Dullin-Gottwald-Holm equation are established using the conserved vectors. The multiplier approach is applied to construct the conserved vectors for equations under consideration. For non-linear evolution equation three conserved vectors and for Benjamin equation four conserved vectors are obtained. The conserved vectors for DGH and generalized DGH equations were reported in [1]. The higher order multiplier is considered for DGH equation and a new conserved vector is found. The double reduction theory is utilized to obtain various exact solutions for Benjamin equation, DGH equation and generalized DGH equation.

Solutions for ultra-broad beam propagation in a planar waveguide with Kerr-like nonlinearity

2018 ◽  
Vol 27 (03) ◽  
pp. 1850032 ◽  
Author(s):  
Sameerah Jamal
Keyword(s):  
Helmholtz Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Conservation Law ◽  
Planar Waveguide ◽  
Nonlinear Optical ◽  
Optical Systems ◽  
Double Reduction ◽  
Nonlinear Media ◽  
Broad Beam

We consider nonlinear optical systems describing the propagation of beams in Kerr-type nonlinear media, experiencing diffraction in transverse and longitudinal directions. The first model investigated, under nonparaxial approximation, is the complex nonlinear Helmholtz equation, recast into a coupled, real system of partial differential equations. We construct and apply its conserved vectors to determine exact solutions. This approach of a double reduction combines a point symmetry with a particular conservation law to enact a reduction and derive solutions. A second model is also studied, that is related to the Maxwell’s equations under paraxial approximation — a version of the nonlinear Schrödinger equation. We show that when the paraxial effect vanishes, a number of additional exact solutions and conservation laws are admitted.

scholarly journals Some Reduction and Exact Solutions of a Higher-Dimensional Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Guangming Wang ◽  
Zhong Han
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Conservation Law ◽  
Second Order ◽  
Reduction Theorem ◽  
Double Reduction ◽  
Noether’S Theorem ◽  
Higher Dimensional ◽  
Dimensional Equation ◽  
Nonlinear Ode

The conservation laws of the(3+1)-dimensional Zakharov-Kuznetsov equation were obtained using Noether’s theorem after an interesting substitutionu=vxto the equation. Then, with the aid of an obtained conservation law, the generalized double reduction theorem was applied to this equation. It can be verified that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions of the Zakharov-Kuznetsov equation were constructed after solving the reduced equation.

scholarly journals Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
R. Naz ◽  
Z. Ali ◽  
I. Naeem
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
First Integral ◽  
Second Order ◽  
Lie Symmetries ◽  
Reduction Theorem ◽  
Explicit Solutions ◽  
Double Reduction ◽  
Zk Equation ◽  
New Exact Solutions

We study here the Lie symmetries, conservation laws, reductions, and new exact solutions of (2+1) dimensional Zakharov-Kuznetsov (ZK), Gardner Kadomtsev-Petviashvili (GKP), and Modified Kadomtsev-Petviashvili (MKP) equations. The multiplier approach yields three conservation laws for ZK equation. We find the Lie symmetries associated with the conserved vectors, and three different cases arise. The generalized double reduction theorem is then applied to reduce the third-order ZK equation to a second-order ordinary differential equation (ODE) and implicit solutions are established. We use the Sine-Cosine method for the reduced second-order ODE to obtain new explicit solutions of ZK equation. The Lie symmetries, conservation laws, reductions, and exact solutions via generalized double reduction theorem are computed for the GKP and MKP equations. Moreover, for the GKP equation, some new explicit solutions are constructed by applying the first integral method to the reduced equations.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

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