Related papers: Nonlinear self-adjointness in constructing conserv…
Symmetry invariant local interaction of a many body system leads to global constraints. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry.
Evidently, the linear superposition principle can not be exactly established as a general principle in the presence of nonlinearity, and, at the first glance, there is no expectation for it to hold even approximately. In this letter, it is…
A theorem due to Nail H. Ibragimov (2007) provides a connection between symmetries and conservation laws for arbitrary differential equations. The theorem is valid for any system of differential equations provided that the number of…
We present some results in the analysis of non-compact differential equations on unbounded domains.
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
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…
The models of the non-linear optics in which solitons were appeared are considered. These models are of paramount importance in studies of non-linear wave phenomena. The classical examples of phenomena of this kind are the self-focusing,…
We use the Lagrange-Noether methods to derive the conservation laws for models in which matter interacts nonminimally with the gravitational field. The nonminimal coupling function can depend arbitrarily on the gravitational field strength.…
This paper deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. When the poles form a symmetric structure, it is natural we wonder…
A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical…
We construct an infinite hierarchy of nonlocal conservation laws for the $ABC$ equation $A u_t\,u_{xy}+B u_x\,u_{ty}+C u_y\,u_{tx} = 0$, where $A,B,C$ are constants and $A+B+C\neq 0$, using a nonisospectral Lax pair. As a byproduct, we…
A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider these equations as a data-structure, from which mathematical properties can be…
A new method for the Lie group classification of differential equations is proposed. It is based of the determination of all possible cases of linear dependence of certain indeterminate appearing in the determining equations of symmetries…
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd…
We demonstrate a kind of linear superposition for a large number of nonlinear equations, both continuum and discrete. In particular, we show that whenever a nonlinear equation admits solutions in terms of Jacobi elliptic functions…
We present an infinite series of autonomous discrete equations on the square lattice possessing hierarchies of autonomous generalized symmetries and conservation laws in both directions. Their orders in both directions are equal to $\kappa…
A non-linear differential equation arising from a stochastic process known as branching Brownian motion is considered. We find an explicit solution and show the uniqueness of the solution under some boundedness conditions using…
We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same…