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We provide the differential equations that generalize the Newtonian N-body problem of celestial mechanics to spaces of constant Gaussian curvature, k, for all k real. In previous studies, the equations of motion made sense only for k…
In the 2-dimensional curved 3-body problem, we prove the existence of Lagrangian and Eulerian homographic orbits, and provide their complete classification in the case of equal masses. We also show that the only non-homothetic hyperbolic…
In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential…
The connection between differential geometry of curves and the (2+1)-dimensional integrable spin system - the M-III equation is established. Using the presented geometrical formalism the L-equivalent counterpart of the M-III equation is…
We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves…
We consider the problem of integrability of the Poisson equations describing spatial motion of a rigid body in the classical nonholonomic Suslov problem. We obtain necessary conditions for their solutions to be meromorphic and show that…
This work deals with traveling waves in the two-dimensional Galileon theory. We use the Hirota procedure to calculate one-Galileon, two-Galileon, three-Galileon and breather-like Galileon solutions in the theory under consideration. The…
A possible model for quantum kinematics of a test particle in a curved space-time is proposed. Every reasonable neighbourhood V_e of a curved space-time can be equipped with a nonassociative binary operation called the geodesic…
Given a stochastic differential equation (SDE) in $\mathbb{R}^n$ whose solution is constrained to lie in some manifold $M \subset \mathbb{R}^n$, we propose a class of numerical schemes for the SDE whose iterates remain close to $M$ to high…
We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along null geodesics. For the Cauchy problem, we give a new…
We study the local and shifted nonlocal reductions of the integrable coupled Kundu type system. We then consider particular cases of this system; namely Chen-Lee-Liu, Gerdjikov-Ivanov, and Kaup-Newell systems. We obtain one- and two-soliton…
We propose a generalization of two classes of Lie-Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie-Hamilton systems on Riemannian spaces (flat $2$-torus, product of hyperbolic lines, sphere and…
We construct Gaussian invariant measures for the two-dimensional Euler equation on the plane. We show the existence of solution with initial conditions in the support of the measures, namely $H^\beta_{loc}(\R^2)$ with $\beta<-1$. Uniqueness…
Most evolution equations %or wave equations are partially integrable and, in order to explicitly integrate all possible cases, there exist several methods of complex analysis, but none is optimal. The theory of Nevanlinna and Wiman-Valiron…
The validity of Kepler Laws for the {\it spherical Kepler problem} -- namely, the problem of the motion of a particle on the unit sphere {in $\mathbb R^3$} undergoing an attraction by another particle in the sphere, tangent to the geodesic…
We establish sharp lower and upper bounds for the number of integral points near dilations of a space curve with nowhere vanishing torsion.
Geometric representations of solutions provides intuitive physical insights. To which end studying dynamics of Quantum systems via $su (n)$ Lie algebra proves to be convenient way of obtaining geometric solution. In this paper link is…
We extend to any simply connected K\"ahler manifold with non-positive sectional curvature some conditions for interpolation in $\mathbb{C}$ and in the unit disk given by Berndtsson, Ortega-Cerd\`a and Seip. The main tool is a comparison…
Kinematic algebras can be realised on geometric spaces and constrain the physical models that can live on these spaces. Different types of kinematic algebras exist and we consider the interplay of these algebras for non-relativistic limits…
Liquid-vapor flows with phase transitions have a wide range of applications. Isothermal two-phase flows described by a single set of isothermal Euler equations, where the mass transfer is modeled by a kinetic relation, have been…