We study local conservation laws for evolution equations in two independent variables. In particu... more We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives.
A new approach to the problem of group classification is applied to the class of first-order non-... more A new approach to the problem of group classification is applied to the class of first-order non-linear equations of the form $u_a u_a=F(t,u,u_t)$. It allowed complete solution of the group classification problem for a class of equations for functions depending on multiple independent variables, where highest derivatives enter nonlinearly. Equivalence groups of the class under consideration and algebraic properties of the symmetry algebra are studied. The class of equations considered presents generalisation of the eikonal and Hamilton-Jacobi equations. The paper contains the list of all non-equivalent equations from this class with symmetry extensions, and proofs of such non- equivalence. New first order non-linear equations possessing wide symmetry groups were constructed.
We perform the complete group classification in the class of cubic Schrödinger equations of the f... more We perform the complete group classification in the class of cubic Schrödinger equations of the form iψt + ψxx + ψ2ψ∗ + V (t, x)ψ = 0, where V is an arbitrary complex-valued potential depending on t and x. We construct all possible inequivalent potentials for which these equations ...
A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simp... more A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simple wave for a weakly dissipative fluid. This is a natural but nontrivial generalization of the so-called unidirectional quasi-simple wave. The method is more close to ...
We perform the complete group classification in the class of cubic Schr\"odinger equations o... more We perform the complete group classification in the class of cubic Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+\psi^2\psi^*+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x$. We construct all possible inequivalent potentials for which these equations have non-trivial Lie symmetries using algebraic and compatibility methods simultaneously. Our classification essentially amends earlier works on the subject.
A complete set of inequivalent realizations of three- and four-dimensional real unsolvable Lie al... more A complete set of inequivalent realizations of three- and four-dimensional real unsolvable Lie algebras in vector fields on a space of an arbitrary (finite) number of variables is obtained.
A complete set of inequivalent realizations of three- and four-dimensional real nonsolvable Lie a... more A complete set of inequivalent realizations of three- and four-dimensional real nonsolvable Lie algebras of vector fields on a space of an arbitrary (finite) number of variables is obtained.
We give a comprehensive analysis of interrelations between the basic concepts of the modern theor... more We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of
In the recent paper by Kudryashov [Commun. Nonlinear Sci. Numer. Simulat., 2009, V.14, 3507-3529]... more In the recent paper by Kudryashov [Commun. Nonlinear Sci. Numer. Simulat., 2009, V.14, 3507-3529] seven common errors in finding exact solutions of nonlinear differential equations were listed and discussed in detail. We indicate two more common errors concerning the similarity (equivalence with respect to point transformations) and linearizability of differential equations and then discuss the first of them. Classes of generalized KdV and mKdV equations with variable coefficients are used in order to clarify our conclusions. We investigate admissible point transformations in classes of generalized KdV equations, obtain the necessary and sufficient conditions of similarity of such equations to the standard KdV and mKdV equations and carried out the exhaustive group classification of a class of variable-coefficient KdV equations. Then a number of recent papers on such equations are commented using the above results. It is shown that exact solutions were constructed in these papers on...
Reduction operators (called also nonclassical or $Q$-conditional symmetries) of variable coeffici... more Reduction operators (called also nonclassical or $Q$-conditional symmetries) of variable coefficient semilinear reaction-diffusion equations with exponential source $f(x)u_t=(g(x)u_x)_x+h(x)e^{mu}$ are investigated using the algorithm involving a mapping between classes of differential equations, which is generated by a family of point transformations. A special attention is paid for checking whether reduction operators are inequivalent to Lie symmetry operators. The derived reduction operators
Reduction operators (called often nonclassical symmetries) of variable coefficient semilinear rea... more Reduction operators (called often nonclassical symmetries) of variable coefficient semilinear reaction-diffusion equations with power nonlinearity $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\neq0,1,2$) are investigated using the algorithm suggested in [O.O. Vaneeva, R.O. Popovych and C. Sophocleous, Acta Appl. Math., 2009, V.106, 1-46; arXiv:0708.3457].
Some aspects of $Q$-conditional symmetry and of its connections with reduction and compatibility ... more Some aspects of $Q$-conditional symmetry and of its connections with reduction and compatibility are discussed.
We study local conservation laws for evolution equations in two independent variables. In particu... more We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives.
A new approach to the problem of group classification is applied to the class of first-order non-... more A new approach to the problem of group classification is applied to the class of first-order non-linear equations of the form $u_a u_a=F(t,u,u_t)$. It allowed complete solution of the group classification problem for a class of equations for functions depending on multiple independent variables, where highest derivatives enter nonlinearly. Equivalence groups of the class under consideration and algebraic properties of the symmetry algebra are studied. The class of equations considered presents generalisation of the eikonal and Hamilton-Jacobi equations. The paper contains the list of all non-equivalent equations from this class with symmetry extensions, and proofs of such non- equivalence. New first order non-linear equations possessing wide symmetry groups were constructed.
We perform the complete group classification in the class of cubic Schrödinger equations of the f... more We perform the complete group classification in the class of cubic Schrödinger equations of the form iψt + ψxx + ψ2ψ∗ + V (t, x)ψ = 0, where V is an arbitrary complex-valued potential depending on t and x. We construct all possible inequivalent potentials for which these equations ...
A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simp... more A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simple wave for a weakly dissipative fluid. This is a natural but nontrivial generalization of the so-called unidirectional quasi-simple wave. The method is more close to ...
We perform the complete group classification in the class of cubic Schr\"odinger equations o... more We perform the complete group classification in the class of cubic Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+\psi^2\psi^*+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x$. We construct all possible inequivalent potentials for which these equations have non-trivial Lie symmetries using algebraic and compatibility methods simultaneously. Our classification essentially amends earlier works on the subject.
A complete set of inequivalent realizations of three- and four-dimensional real unsolvable Lie al... more A complete set of inequivalent realizations of three- and four-dimensional real unsolvable Lie algebras in vector fields on a space of an arbitrary (finite) number of variables is obtained.
A complete set of inequivalent realizations of three- and four-dimensional real nonsolvable Lie a... more A complete set of inequivalent realizations of three- and four-dimensional real nonsolvable Lie algebras of vector fields on a space of an arbitrary (finite) number of variables is obtained.
We give a comprehensive analysis of interrelations between the basic concepts of the modern theor... more We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of
In the recent paper by Kudryashov [Commun. Nonlinear Sci. Numer. Simulat., 2009, V.14, 3507-3529]... more In the recent paper by Kudryashov [Commun. Nonlinear Sci. Numer. Simulat., 2009, V.14, 3507-3529] seven common errors in finding exact solutions of nonlinear differential equations were listed and discussed in detail. We indicate two more common errors concerning the similarity (equivalence with respect to point transformations) and linearizability of differential equations and then discuss the first of them. Classes of generalized KdV and mKdV equations with variable coefficients are used in order to clarify our conclusions. We investigate admissible point transformations in classes of generalized KdV equations, obtain the necessary and sufficient conditions of similarity of such equations to the standard KdV and mKdV equations and carried out the exhaustive group classification of a class of variable-coefficient KdV equations. Then a number of recent papers on such equations are commented using the above results. It is shown that exact solutions were constructed in these papers on...
Reduction operators (called also nonclassical or $Q$-conditional symmetries) of variable coeffici... more Reduction operators (called also nonclassical or $Q$-conditional symmetries) of variable coefficient semilinear reaction-diffusion equations with exponential source $f(x)u_t=(g(x)u_x)_x+h(x)e^{mu}$ are investigated using the algorithm involving a mapping between classes of differential equations, which is generated by a family of point transformations. A special attention is paid for checking whether reduction operators are inequivalent to Lie symmetry operators. The derived reduction operators
Reduction operators (called often nonclassical symmetries) of variable coefficient semilinear rea... more Reduction operators (called often nonclassical symmetries) of variable coefficient semilinear reaction-diffusion equations with power nonlinearity $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\neq0,1,2$) are investigated using the algorithm suggested in [O.O. Vaneeva, R.O. Popovych and C. Sophocleous, Acta Appl. Math., 2009, V.106, 1-46; arXiv:0708.3457].
Some aspects of $Q$-conditional symmetry and of its connections with reduction and compatibility ... more Some aspects of $Q$-conditional symmetry and of its connections with reduction and compatibility are discussed.
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