Scaling Symmetries and Conservation Laws for Variable-coefficients Nonlinear Dispersive Equations

Scaling symmetries arise in different branches of physics, and symmetry-based approaches are powerful tools for studying scaling-invariant models since they can provide conservation laws that are not obvious by inspection. In this framework, the class of variable-coefficients nonlinear dispersive equations vc$K(m,n)$, which contains several important evolution equations modeling nonlinear phenomena, is considered. For some of its scaling-invariant subclasses, we study its nonlinear self-adjointness and construct eight new local conservation laws associated with scaling symmetries by using a general theorem on conservation laws and the multipliers method. The property of scale invariance of those equations led to five conservation laws with a direct physical interpretation: energy, center of mass, and mass are the conserved quantities obtained in some cases.

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Local conservation laws of second-order evolution equations

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/41/36/362002 ◽

2008 ◽

Vol 41 (36) ◽

pp. 362002 ◽

Cited By ~ 16

Author(s):

Roman O Popovych ◽

Anatoly M Samoilenko

Keyword(s):

Conservation Laws ◽

Evolution Equations ◽

Second Order ◽

Local Conservation

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A review on some results on local conservation laws for certain evolution equations

TEMA (São Carlos) ◽

10.5540/tema.2013.014.01.0109 ◽

2013 ◽

Vol 14 (1) ◽

pp. 109-118 ◽

Cited By ~ 2

Author(s):

I. L. Freire ◽

J. C. S. Sampaio

Keyword(s):

Conservation Laws ◽

Evolution Equations ◽

Local Conservation ◽

Fifth Order

In this work we revisit some recent results on conservation laws for a class of fifth-order evolution equations up to fifth-order.

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Corrigendum to “On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models” [Communications in Nonlinear Science and Numerical Simulation 19(2) (2014) 350--360]

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2015.06.024 ◽

2016 ◽

Vol 30 (1-3) ◽

pp. 401-402

Author(s):

Igor Leite Freire ◽

Júlio Cesar Santos Sampaio

Keyword(s):

Numerical Simulation ◽

Conservation Laws ◽

Evolution Equations ◽

Local Conservation ◽

Nonlinear Science

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On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2013.06.010 ◽

2014 ◽

Vol 19 (2) ◽

pp. 350-360 ◽

Cited By ~ 16

Author(s):

Igor Leite Freire ◽

Júlio Cesar Santos Sampaio

Keyword(s):

Conservation Laws ◽

Evolution Equations ◽

Local Conservation

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Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.611 ◽

2014 ◽

Vol 760 ◽

pp. 368-386 ◽

Cited By ~ 20

Author(s):

Alexei F. Cheviakov ◽

Martin Oberlack

Keyword(s):

Conservation Laws ◽

Conservation Law ◽

Evolution Equations ◽

Stokes Equations ◽

Three Dimensional ◽

Infinite Family ◽

Time Dependent ◽

Local Conservation ◽

Direct Construction ◽

Primitive Variables

AbstractLocal conservation laws are systematically constructed for three-dimensional time-dependent viscous and inviscid incompressible fluid flows, in primitive variables and vorticity formulation, using the direct construction method. Complete sets of local conservation laws in primitive variables are derived for the case of conservation law multipliers depending on derivatives up to the second order. In the vorticity formulation, there exists an infinite family of vorticity-dependent conservation laws involving an arbitrary differentiable function of space and time, holding for both viscous and inviscid cases. The infinite conservation law family is used to generate further independent hierarchies of conservation laws that essentially involve vorticity and arbitrary flow parameters, which are determined by known evolution equations such as those for momentum, energy or helicity, though not necessarily in the form of a conservation law. The new conservation laws are not restricted to any reduced flow geometry such as planar or axisymmetric limits. Examples are considered.

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Explicit solutions of two nonlinear dispersive equations with variable coefficients

Physics Letters A ◽

10.1016/j.physleta.2008.10.031 ◽

2008 ◽

Vol 372 (47) ◽

pp. 7001-7006 ◽

Cited By ~ 1

Author(s):

Shaoyong Lai ◽

Xiumei Lv ◽

Yonghong Wu

Keyword(s):

Variable Coefficients ◽

Explicit Solutions ◽

Dispersive Equations ◽

Nonlinear Dispersive Equations

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A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations

Zeitschrift für Naturforschung A ◽

10.1515/zna-2016-0057 ◽

2016 ◽

Vol 71 (5) ◽

pp. 475-480 ◽

Cited By ~ 4

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.

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