Abstract
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.
Highlights
Scaling symmetries have wide applications in science and in engineering and are far from being a special case in physics - they can be found, for instance, in quantum physics, fluid mechanics, turbulence, elasticity, heat diffusion, convection, filtration, gas dynamics, and in the theory of detonation and combustion
In this paper we seek for conservation laws associated with scaling symmetries for scalinginvariant subclasses of a generalized vcK(m, n)
We highlight that conservation laws associated with scaling symmetries can be obtained via multipliers method by computing fluxes with no integration involved [3, 8, 14]
Summary
Scaling symmetries have wide applications in science and in engineering and are far from being a special case in physics - they can be found, for instance, in quantum physics, fluid mechanics, turbulence, elasticity, heat diffusion, convection, filtration, gas dynamics, and in the theory of detonation and combustion (see [12] and references therein). The relations between Ibragimov’s approach and the direct method are well known [4,46,47], and the latter is largely employed to build up conservation laws for nonlinearly self-adjoint equations [7, 13, 15, 22, 30, 44]. In this paper we seek for conservation laws associated with scaling symmetries for scalinginvariant subclasses of a generalized vcK(m, n). To this end, we shall consider the expanded form of vcK(m, n) (1.1), ut + α0(t, x, u) + α1(t, x, u)ux + α2(t, x, u)u2x + α3(t, x, u)u3x +α4(t, x, u)uxuxx + α5(t, x, u)uxx + α6(t, x, u)uxxx = 0 ,. We highlight that conservation laws associated with scaling symmetries can be obtained via multipliers method by computing fluxes with no integration involved [3, 8, 14]
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