Journal of Nonlinear Mathematical Physics

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained is revealed. Various contexts are considered: algebraic and Diophantine equations, nonlinear Sturm–Liouville problems, dynamical systems (with continuous and with discrete time), nonlinear partial differential equations, analytical geometry, functional equations. While this transformation, in one or another context, is certainly known to many, it does not seem to be as universally known as it deserves to be, for instance it is not routinely taught in basic University courses (to the best of our knowledge). The main purpose of this paper is to bring about a change in this respect; but we also hope that some of the findings reported herein — and the multitude of analogous findings easily obtainable via this technique — will be considered remarkable by the relevant experts, in spite of their elementary origin.

In this paper we make a full analysis of the symmetry reductions of a beam equation by using the classical Lie method of infinitesimals and the nonclassical method. We consider travelling wave reductions depending on the form of an arbitrary function. We have found several new classes of solutions that have not been considered before: solutions expressed in terms of Jacobi elliptic functions, Wadati solitons and compactons. Several classes of coherent structures are displayed by some of the solutions: kinks, solitons, two humps compactons.

The classical Lie method is applied to a nonisospectral problem associated with a system of partial differential equations in 2 + 1 dimensions (Maccari A, J. Math. Phys.12 (1998) 6547–6551.). Identification of the classical Lie symmetries provides a set of reductions that give rise to different nontrivial spectral problems in 1 + 1 dimensions. The form in which the spectral parameter of the 1 + 1 Lax pair is introduced is carefully described.

We apply a list of criteria which leads to a class of fifth-order symmetry-integrable evolution equations. The recursion operators for this class are given explicitly. Multipotentialisations are then applied to the equations in this class in order to extend this class of integrable equations.

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-Bäcklund transformations for the equations/systems. In this respect we define the triangular-auto-Bäcklund transformation and derive its connections to the converse problem. Several explicit examples are given. In particular, we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.

Nonlinear n-dimensional second-order diffusion equations admitting maximal Lie algebras of point symmetries are considered. Examples of invariant solutions, as well as of solutions on invariant subspaces for some nonlinear operators, are constructed for arbitrary n. A complete description of all possible types of invariant solutions is given in the case n = 2 for the equation possessing an infinitely dimensional symmetry algebra. The results obtained are generalized for the hyperbolic and other fourth-order parabolic equations of thin film and nonlinear dispersion type.

In [19] we derive nonlocal symmetries for ordinary differential equations by embedding the given equation in an auxiliary system. Since the nonlocal symmetries of the ODE's are local symmetries of the associated auxiliary system this result provides an algorithmic method to derive this kind of nonlocal symmetries. In this work we show some classes of ordinary differential equations which do not admit any Lie symmetry unless some conditions are satisfied but for which we have derived nonlocal symmetries. These nonlocal symmetries allow us to reduce the order for these equations even if these equations do not admit point symmetries.

The inheritance of symmetries of partial differential equations occurs in a different manner from that of ordinary differential equations. In particular, the Lie algebra of the symmetries of a partial differential equation is not sufficient to predict the symmetries that will be inherited by a resulting reduced partial (or ordinary) differential equation. We show how this suggests a possible source of Type I hidden symmetries of partial differential equations as well as provide interesting consequences for solutions of partial differential equations.

The paper is devoted to the Lie group analysis of a nonlinear equation arising in metallurgical applications of Magnetohydrodynamics. Self-adjointness of the basic equations is investigated. The analysis reveals two exceptional values of the exponent playing a significant role in the model.

An application of approximate transformation groups to study dynamics of a system with distinct time scales is discussed. The utilization of the Krylov–Bogoliubov–Mitropolsky method of averaging to find solutions of the Lie equations is considered. Physical illustrations from the plasma kinetic theory demonstrate the potentialities of the suggested approach. Several examples of invariant solutions for the system of the Vlasov-Maxwell equations for the two-component (electron-ion) plasma are presented.

Lie point symmetry group classification of a scalar stochastic differential equation (SDE) with one-dimensional Brownian motion is presented. First we prove that the admitted symmetry group is at most three-dimensional. Then the classification is carried out with the help of Lie algebra realizations by vector fields.

We study conditions for a sequence of Appell polynomials to be a basis on a sequence space.

The manuscript is devoted to nonisentropic solutions of simple wave type of the gas dynamics equations. For an isentropic flow these equations (in one-dimensional and steady two-dimensional cases) are reduced to the equations written in the Riemann invariants. The system written in the Riemann invariants is hyperbolic and homogeneous. It allows obtaining simple waves, which are also called Riemann waves. For nonisentropic flows there are no Riemann invariants. The question is: what solutions could substitute the Riemann waves? By the method of differential constraints such types of solutions are found here. For these classes of solutions one can integrate the gas dynamics equations: finite formulas with one parameter are obtained. These solutions have some properties similar to simple Riemann waves. For example, they describe a nonisentropic rarefaction wave. The rarefaction waves play the main role in many applications such as the problem of pulling a piston, decay of arbitrary discontinuity and others.

In the literature, the generalized Sundman transformation has been used for obtaining necessary and sufficient conditions for a single second- and third-order ordinary differential equation to be equivalent to a linear equation in the Laguerre form. As far as we are aware, the generalized Sundman transformation has not been applied to a system of equations. The motivation of this work is then to expand the application of the generalized Sundman transformation to a system of ordinary differential equations, in particular, to a system of two second-order ordinary differential equations.

We study the class of the ordinary differential equations of the form ẍ + a₂(t, x)ẋ² + a₁(t, x)ẋ + a₀(t, x) = 0, that admit v = ∂_x as λ-symmetry for some λ = α(t, x)ẋ + β(t, x). This class coincides with the class of the second-order equations that have first integrals of the form C(t) + 1/(A(t, x)ẋ + B(t, x)). We provide a method to calculate the functions A, B and C that define the first integral. Some relationships with the class of equations linearizable by local and a specific type of nonlocal transformations are also presented.

We briefly derive the infinite set of multi-point correlation equations based on the Navier–Stokes equations for an incompressible fluid. From this we reconsider the previously derived set of Lie symmetries, i.e. those directly induced by the ones from classical mechanics and also new symmetries. The latter are denoted statistical symmetries and have no direct counterpart in classical mechanics. Finally, we considerably extend the set of symmetries by Lie algebra methods and give the corresponding commutator tables. Due to the infinite dimensionality of the multi-point correlation equations completeness of its symmetries is not proven yet and is still an open question.