This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order $k$ variational equations, for arbitrary values of $k$, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear springpendulum problem for the values of the parameter that can not be decided using first order variational equations.

In this paper the problem of classification of integrable natural Hamiltonian systems with $n$ degrees of freedom given by a Hamilton function, which is the sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k > 2$, is investigated. It is assumed that the potential is not generic. Except for some particular cases a potential $V$ is not generic if it admits a nonzero solution of equation $V'(\bf d)=0$. The existence of such a solution gives very strong integrability obstructions obtained in the frame of the Morales–Ramis theory. This theory also gives additional integrability obstructions which have the form of restrictions imposed on the eigenvalues $(\lambda_1, \ldots, \lambda_n)$ of the Hessian matrix $V''(\bf d)$ calculated at a nonzero $\bf d \in \mathbb{C}^n$ satisfying $V'(\bf d)=\bf d$. In our previous work we showed that for generic potentials some universal relations between $(\lambda_1, \ldots, \lambda_n)$ calculated at various solutions of $V'(\bf d)={\bf d}$ exist. These relations allow one to prove that the number of potentials satisfying the necessary conditions for the integrability is finite. The main aim of this paper was to show that relations of such forms also exist for nongeneric potentials. We show their existence and derive them for the case $n = k = 3$ applying the multivariable residue calculus. We demonstrate the strength of the results analyzing in details the nongeneric cases for $n = k = 3$. Our analysis covers all the possibilities and we distinguish those cases where known methods are too weak to decide if the potential is integrable or not. Moreover, for $n = k = 3$, thanks to this analysis, a three-parameter family of potentials integrable or superintegrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.

We consider the Euler approach to constructing to investigating of the superintegrable systems related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stäckel systems.

Some methods to look for exact solutions of nonlinear differential equations are discussed. It is shown that many popular methods are equivalent to each other. Several recent publications with "new" solitary wave solutions for the Kuramoto–Sivashinsky equation are analyzed. We demonstrate that all these solutions coincide with the known ones.

An ensemble of convective thermals is considered in the surface layer of penetrative turbulent convection over a homogeneous heated horizontal surface. An integral model of an unsteady spontaneous jet having an exact self-similar solution is proposed to describe the dynamics of an isolated convective element. A statistical model for an ensemble of convective elements using a hydrodynamic analogy of the isolated spontaneous jet equations is suggested. It is supposed that motion of the elements of an ensemble corresponds to a statistic invariant that combines the squared velocity and the diameter of the jet. Using the combination of the statistic invariant of an ensemble and the Boltzmann distributions on squares of velocities, the size distribution of spontaneous jets in a convective surface layer of the atmosphere is constructed, which agrees with available experimental data.