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Nail H. Ibragimov

Publications and source records attributed to Nail H. Ibragimov.

14 recordsLinked to original sources
arXiv2015

Symmetries and conservation laws of the generalized Krichever-Novikov equation

A computational classification of contact symmetries and higher-order local symmetries that do not commute with $t,x$, as well as local conserved densities that are not invariant under $t,x$ is carried out for a generalized version of the Krichever-Novikov equation. Several new results are obtained. First, the Krichever-Novikov equation is explicitly shown to have a local conserved density that contains $t,x$. Second, apart from the dilational point symmetries known for special cases of the Krichever-Novikov equation and its generalized version, no other local symmetries with low differential order are found to contain $t,x$. Third, the basic Hamiltonian structure of the Krichever-Novikov equation is used to map the local conserved density containing $t,x$ into a nonlocal symmetry that contains $t,x$. Fourth, a recursion operator is applied to this nonlocal symmetry to produce a hierarchy of nonlocal symmetries that have explicit dependence on $t,x$. When the inverse of the Hamiltonian map is applied to this hierarchy, only trivial conserved densities are obtained.

nlin.SI↗
arXiv2012

Group Analysis of the Novikov Equation

We find the Lie point symmetries of the Novikov equation and demonstrate that it is strictly self-adjoint. Using the self-adjointness and the recent technique for constructing conserved vectors associated with symmetries of differential equations, we find the conservation law corresponding to the dilations symmetry and show that other symmetries do not provide nontrivial conservation laws. Then we investigat the invariant solutions.

math-ph↗
arXiv2011

Evolution equations of the second and third order with Lie-Bäcklund symmetries

Chapter 4 of my book [1] is dedicated to Lie-Bäcklund group analysis of various types of differential equations. The Russian edition of this book had an Addendum (pages 262-266) containing a summary of new results (obtained by the end of 1982) on classification of evolution equations of the second and third order possessing Lie-Bäcklund symmetries. The \textit{Addendum} was not included in the English translation of the book by technical reasons. I present here the missing translation. In order to make the text self-contained, I have added in the translation the equations from the main body of the book used in the Addendum.

math-ph↗
arXiv2011

Nonlinear self-adjointness in constructing conservation laws

The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint nonlinear equations. The class of nonlinearly self-adjoint equations includes, in particular, all linear equations. Conservation laws associated with symmetries can be constructed for all nonlinearly self-adjoint differential equations and systems. The number of equations in systems can be different from the number of dependent variables.

math-ph↗
arXiv2011

Group Analysis of Nonlinear Internal Waves in Oceans. I: Self-adjointness, conservation laws, invariant solutions

The paper is devoted to the group analysis of equations of motion of two-dimensional uniformly stratified rotating fluids used as a basic model in geophysical fluid dynamics. It is shown that the nonlinear equations in question have a remarkable property to be self-adjoint. This property is crucial for constructing conservation laws provided in the present paper. Invariant solutions are constructed using certain symmetries. The invariant solutions are used for defining internal wave beams.

math-ph↗
arXiv2011

Group Analysis of Nonlinear Internal Waves in Oceans. II: The symmetries and rotationally invariant solution

74J30The maximal group of Lie point symmetries of a system of nonlinear equations used in geophysical fluid dynamics is presented. The Lie algebra of this group is infinite-dimensional and involves three arbitrary functions of time. The invariant solution under the rotation and dilation is constructed. Qualitative analysis of the invariant solution is provided and the energy of this solution is presented.

math-ph↗
arXiv2011

Memoir on Integration of Ordinary Differential [1.2ex] Equations by Quadrature

The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations reducible to algebraic equations is found. It depends on two arbitrary functions. The method for solving all these equations is given. The new class contains the constant coefficient equations and Euler's equations as particular cases.

math.CA↗
arXiv2011

Nonlinear self-adjointness and conservation laws

The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the strict self-adjointness and quasi self-adjointness introduced earlier by the author. It is shown that the equations possessing the nonlinear self-adjointness can be written equivalently in a strictly self-adjoint form by using appropriate multipliers. All linear equations possess the property of nonlinear self-adjointness, and hence can be rewritten in a nonlinear strictly self-adjoint. For example, the heat equation $u_t - Δu = 0$ becomes strictly self-adjoint after multiplying by $u^{-1}.$ Conservation laws associated with symmetries can be constructed for all differential equations and systems having the property of nonlinear self-adjointness.

math-ph↗
arXiv2007

Linearization of fourth-order ordinary differential equations by point transformations

We present here the solution of the problem on linearization of fourth-order equations by means of point transformations. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation.

math.CA↗
arXiv2007

Conservation laws for the Maxwell-Dirac equations with a dual Ohm's law

Using a general theorem on conservation laws for arbitrary differential equations proved by Ibragimov, we have derived conservation laws for Dirac's symmetrized Maxwell-Lorentz equations under the assumption that both the electric and magnetic charges obey linear conductivity laws (dual Ohm's law). We find that this linear system allows for conservation laws which are non-local in time.

math-ph↗
arXiv2006

Group classification of the Sachs equations for a radiating axisymmetric, non-rotating, vacuum space-time

We carry out a Lie group analysis of the Sachs equations for a time-dependent axisymmetric non-rotating space-time in which the Ricci tensor vanishes. These equations, which are the first two members of the set of Newman-Penrose equations, define the characteristic initial-value problem for the space-time. We find a particular form for the initial data such that these equations admit a Lie symmetry, and so defines a geometrically special class of such spacetimes. These should additionally be of particular physical interest because of this special geometric feature.

gr-qc↗