Conservation laws for a coupled variable-coefficient modified Korteweg–de Vries system in a two-layer fluid model

Bozhkov, Yuri; Dimas, S.; Ibragimov, Nail H.

doi:10.1016/j.cnsns.2012.09.015

Cited by 24 publications

(12 citation statements)

References 7 publications

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“…Using the Noether operators (14) and (15), the symmetries (23)-(25), (27), and the formal Lagrangian (5) with the function v(t, x) given by (30), four new conservation laws have been found for the subdiffusion equation (1) with the Caputo fractional derivative. The corresponding conserved vectors are presented in Table 3, where function Φ(t) is defined as…”

Section: Conservation Laws For Nonlinear Tfde With the Caputo Fractiomentioning

confidence: 99%

Conservation laws for time-fractional subdiffusion and diffusion-wave equations

2015

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Laboratory "Group analysis of mathematical models in natural and engineering sciences", AbstractThe concept of nonlinear self-adjointness is employed to construct the conservation laws for fractional evolution equations using its Lie point symmetries. The approach is demonstrated on subdiffusion and diffusionwave equations with the Riemann-Liouville and Caputo time-fractional derivatives. It is shown that these equations are nonlinearly self-adjoint and therefore desired conservation laws can be obtained using appropriate formal Lagrangians. Fractional generalizations of the Noether operators are also proposed for the equations with the Riemann-Liouville and Caputo timefractional derivatives of order α ∈ (0, 2). Using these operators and formal Lagrangians, new conserved vectors have been constructed for the linear and nonlinear fractional subdiffusion and diffusion-wave equations corresponding to its Lie point symmetries.

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Section: Conservation Laws For Nonlinear Tfde With the Caputo Fractiomentioning

confidence: 99%

Conservation laws for time-fractional subdiffusion and diffusion-wave equations

2015

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“…Some years ago, Ibragimov presented a general theorem for constructing conservation laws based on the self-adjointness concept [23,24]; later on, it was generalized to nonlinear selfadjointness [25,26]. Concerning classes of third-order nonlinear evolution equations, there are several works devoted to classify them as nonlinearly self-adjoint and to construct conservation laws via Ibragimov's theorem [10,11,18,19,20,21,41,42]. The relations between Ibragimov's approach and the direct method are well known [4,46,47], and the latter is also largely employed to build up conservation laws for nonlinearly self-adjoint equations [7,13,15,22,30,44].…”

Section: Introductionmentioning

confidence: 99%

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

Silva

Souza

2019

Tend. Mat. Apl. Comput.

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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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“…To cite some of them; for instance, up to our knowledge, the first paper dealing with some self-adjointness and systems of PDEs was [22]. Nonlinear self-adjointness of a system of coupled modified KdV equations was studied in [23]. Further examples can be found in [4].…”

Section: Introductionmentioning

confidence: 99%