Related papers: On Integrable Ermakov-Painlev\'{e} IV Systems
In this letter, the integrability aspects of a generalized Fisher type equation with modified diffusion in (1+1) and (2+1) dimensions are studied by carrying out a singularity structure and symmetry analysis. It is shown that the Painlev\'e…
This paper investigates quasi-homogeneous integrable systems by analyzing their Laurent series solutions near movable singularities, motivated by patterns observed in Kovalevskaya exponents of four-dimensional Painlev\'e-type equations. We…
We show that the one dimensional unitary matrix model with potential of the form $a U + b U^2 + h.c.$ is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space…
We discuss global tensor invariants of a rigid body motion in the cases of Euler, Lagrange and Kovalevskaya. These invariants are obtained by substituting tensor fields with cubic on variable components into the invariance equation and…
In this paper we present a formally fourth-order accurate hybrid-variable method for the Euler equations in the context of method of lines. The hybrid-variable (HV) method seeks numerical approximations to both cell-averages and nodal…
The Li\'enard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Li\'enard equation and some equations from the…
In this paper we establish B\"{a}cklund transformations between solutions of several cases of classical isotropic MHD and plasma equilibria and corresponding anisotropic equilibria. The transformations appear to be infinite-dimensional and…
The dynamics of nonlinear reaction-diffusion systems is dominated by the onset of patterns and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized…
The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…
We investigate the existence of Liouville integrable cosmological models in hybrid metric-Palatini theory. Specifically we use the symmetry conditions for the existence of quadratic in the momentum conservation laws for the field equations…
It is proved that for a given truncated Painlev\'e expansion of an arbitrary nonlinear Painlev\'e integrable system, the residue with respect to the singularity manifold is a nonlocal symmetry. The residual symmetries can be localized to…
We propose an extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. This extension allows us to fill in the gap in cluster construction of the $q$-difference Painlev\'e equations, showing…
The problem of construction of the boundary conditions for the Toda lattice compatible with its higher symmetries is considered. It is demonstrated that this problem is reduced to finding of the differential constraints consistent with the…
We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper…
In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order…
We construct a new series of multicomponent integrable PDE systems that contain as particular example (with appropriately chosen parameters) many famous integrable systems including KdV, coupled KdV, Harry Dym, coupled Harry Dym,…
We present certain general structures related to the solutions of Painlev\'e equation II and to the solutions of the differential equation satisfied by the corresponding Hamiltonian equations, together with the tau functions. By taking…
We establish a matrix generalization of the ultradiscrete fourth Painlev\'e equation (ud-PIV). Well-defined multicomponent systems that permit ultradiscretization are obtained using an approach that relies on a group defined by constraints…
The article is devoted to the existence of solutions of a certain system of quadratic integral equations in H^1(R, R^N). We show the existence of a perturbed solution by using a fixed point technique in the Sobolev space on the real line.
In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…