Related papers: The connecting solution of the Painlev\'e phase tr…
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
We consider the problem of the determination of the isotropy classes of the orbit spaces of all the real linear groups, with three independent basic invariants satisfying only one independent relation. The results are obtained in the $\hat…
We explain the relation between the mixed mKdV/sinh-Gordon model and the Kudryashov's equation. Then, we use the mixed AKNS/Lund-Regge model to find a system of ODEs which is candidate to define a new transcendental function. We also…
In this paper the entanglement and quantum phase transition of the anisotropic s=1/2 XY model are studied by using the quantum renormalization group method. By solving the renormalization equations, we get the trivial fixed point and the…
We study the global flow of the anisotropic Manev problem, which describes the planar motion of two bodies under the influence of an anisotropic Newtonian potential with a relativistic correction term. We first find all the heteroclinic…
A well-known conjecture of De Giorgi -- motivated by analogy with the Bernstein problem for minimal surfaces -- asserts the rigidity of monotone solutions to the Allen--Cahn equation in $\mathbb{R}^{d+1}$, with $d\leq 7$. We establish close…
Compositions of rational transformations of independent variables of linear matrix ordinary differential equations (ODEs) with the Schlesinger transformations ($RS$-transformations) are used to construct algebraic solutions of the sixth…
We construct a solution of an analog of the Schr\"{o}dinger equation for the Hamiltonian $ H_I (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ in the Painleve I hierarchy. This solution is produced by an explicit…
Some new Hamiltonian systems of quasi-Painlev\'e type are presented and the analogue of Okamoto's space of initial conditions computed. Using the geometric approach that was introduced originally for the identification problem of Painlev\'e…
Consider the linear stochastic differential equation (SDE) on $\mathbb{R}^n$: \[\mathrm {d}{X}_t=AX_t\,\mathrm{d}t+B\,\mathrm{d}L_t,\] where $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ real matrix and $L_t$ is a L\'{e}vy…
We derive determinant representations and nonlinear differential equations for the scaled 2-point functions of the 2D Ising model on the cylinder. These equations generalize well-known results for the infinite lattice (Painlev\'e III…
The frustrated magnet $\alpha$-RuCl3 is one of the prime candidates for realizing a Kitaev quantum spin liquid (QSL). However, the existence of a field-induced intermediate QSL phase in this material remains under debate. Here, we employ…
In this paper, we study special solutions of five autonomous integrable partial difference equations (P$\Delta$Es). More precisely, we show that these P$\Delta$Es admit special solutions that are described by non-autonomous ordinary…
We examine quantum extensions of the continuous Painlev\'e equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlev\'e equations II, IV and V. From their auto-B\"acklund…
The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite 2D channel.…
A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found…
The isotropic XY model $(s=1/2)$ in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of $N$ segments with $n$…
The Yang-Baxter Equation (YBE) plays a crucial role for studying integrable many-body quantum systems. Many known YBE solutions provide various examples ranging from quantum spin chains to superconducting systems. Models of solvable…
We investigate entanglement and quantum phase transition (QPT) in a two-dimensional Heisenberg anisotropic spin-1/2 XY model, using quantum renormalization group method (QRG) on a square lattice of $N\times N$ sites. The entanglement…
We study a family of unbounded solutions to the Korteweg-de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second…