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.

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.

SYMMETRY AND CONSERVATION LAW CLASSIFICATION AND EXACT SOLUTIONS TO THE GENERALIZED KdV TYPES OF EQUATIONS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250188 ◽  
Author(s):  
HANZE LIU ◽  
JIBIN LI ◽  
LEI LIU
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Conservation Law ◽  
Vector Fields ◽  
Nonlinear Partial Differential Equations ◽  
Second Order ◽  
Analytic Method ◽  
Composite Function ◽  
Symmetry And Conservation Law

In this paper, complete geometric symmetry and conservation law classification of the generalized KdV types of equations are investigated. All of the geometric vector fields and second-order multipliers for the equations are obtained, and the corresponding conservation laws of the equations are presented explicitly. These comprise all of the second-order conservation laws for the equations. Furthermore, an analytic method is developed for dealing with the exact solutions to the generalized nonlinear partial differential equations with composite function terms.

scholarly journals Derivation of Conservation Laws for the Magma Equation Using the Multiplier Method: Power Law and Exponential Law for Permeability and Viscosity

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
N. Mindu ◽  
D. P. Mason
Keyword(s):  
Conservation Laws ◽  
Power Law ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Multiplier Method ◽  
Reduction Theorem ◽  
Double Reduction ◽  
Exponential Law ◽  
Spatial Derivatives ◽  
Derivatives Of

The derivation of conservation laws for the magma equation using the multiplier method for both the power law and exponential law relating the permeability and matrix viscosity to the voidage is considered. It is found that all known conserved vectors for the magma equation and the new conserved vectors for the exponential laws can be derived using multipliers which depend on the voidage and spatial derivatives of the voidage. It is also found that the conserved vectors are associated with the Lie point symmetry of the magma equation which generates travelling wave solutions which may explain by the double reduction theorem for associated Lie point symmetries why many of the known analytical solutions are travelling waves.

scholarly journals Conservation laws for even order systems of polyharmonic map type

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Frédéric Louis de Longueville ◽  
Andreas Gastel
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Weak Solutions ◽  
Conservation Law ◽  
Second Order ◽  
Critical Nonlinearity ◽  
Natural Conditions ◽  
Quasilinear Systems ◽  
Even Order

AbstractFollowing Rivière’s study of conservation laws for second order quasilinear systems with critical nonlinearity and Lamm/Rivière’s generalization to fourth order, we consider similar systems of order 2m. Typical examples are m-polyharmonic maps. Under natural conditions, we find a conservation law for weak solutions on 2m-dimensional domains. This implies continuity of weak solutions.

Conservation Laws in Systems With Variable Mass

1993 ◽  
Vol 60 (4) ◽  
pp. 954-958 ◽  
Author(s):  
L. Cveticanin
Keyword(s):  
Variational Principle ◽  
Dynamic System ◽  
Conservation Laws ◽  
Dynamic Systems ◽  
Conservation Law ◽  
Variable Mass ◽  
Mass Variation ◽  
Noether’S Theorem ◽  
Noether's Theorem

In this paper, a method for obtaining conservation laws of dynamic systems with variable mass is developed. It is based on Noether’s theorem to the existence of conservation laws and D’Alembert’s variational principle. In the general case, a dynamic system with variable mass is purely nonconservative. Noether’s identity for such a case is expanded by the terms that describe the mass variation. If Noether’s identity if satisfied, a conservation law exists. Two groups of systems with variable mass are considered: a nonlinear vibrating machine and a rotor with variable mass. For these systems, conservation laws are obtained using the procedure developed in this paper.

scholarly journals SYMMETRIES, NEWTONOID VECTOR FIELDS AND CONSERVATION LAWS IN THE LAGRANGIAN k-SYMPLECTIC FORMALISM

2012 ◽  
Vol 24 (10) ◽  
pp. 1250030 ◽  
Author(s):  
LUCÍA BUA ◽  
IOAN BUCATARU ◽  
MODESTO SALGADO
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
Vector Fields ◽  
Partial Differential ◽  
Noether’S Theorem ◽  
Dynamical Symmetries ◽  
Noether's Theorem ◽  
Configuration Manifold

In this paper, we study symmetries, Newtonoid vector fields, conservation laws, Noether's theorem and its converse, in the framework of the k-symplectic formalism, using the Frölicher–Nijenhuis formalism on the space of k1-velocities of the configuration manifold.For the case k = 1, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's theorem and its converse. For the case k > 1, we provide a new proof for Noether's theorem, which shows that, in the k-symplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.

scholarly journals The dispersionless Veselov–Novikov equation: symmetries, exact solutions, and conservation laws

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Oleg I. Morozov ◽  
Jen-Hsu Chang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Second Order ◽  
Invariant Solutions ◽  
Arbitrary Degree ◽  
Spatial Variables ◽  
Particular Solutions ◽  
Novikov Equation

AbstractWe study symmetries, invariant solutions, and conservation laws for the dispersionless Veselov–Novikov equation. The emphasis is placed on cases when the odes involved in description of the invariant solutions are integrable by quadratures. Then we find some non-invariant solutions, in particular, solutions that are polynomials of an arbitrary degree $$N \ge 3$$ N ≥ 3 with respect to the spatial variables. Finally we compute all conservation laws that are associated to cosymmetries of second order.

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