Related papers: Classification of Hamiltonian non-abelian Painlev\…
We find and study a six-parameter family of coupled Painlev\'e III systems in dimension six with affine Weyl group symmetry of type $D_6^{(1)}$. We also find and study its degenerate systems with affine Weyl group symmetry of types…
A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a…
Two approaches to the Painlev\'{e} I hierarchy are discussed: the isomonodromic construction based on meromorphic connections, and the minimal models construction based on a reduction of the KP hierarchy. An explicit correspondence between…
We prove that under a very general setting, a system of ODE passes the Painleve test if and only if there is a good change of variable, such that the pole singularity solutions are converted to regular power series, while the converted ODE…
We find a one-parameter family of polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry. We also show that this system can be obtained by the compatibility conditions for the linear differential equations in three…
In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type…
It is proved that the Painlev\'{e} VI equation $(PVI_{\al,\be,\ga,\de})$ for the special values of constants $(\al=\frac{\nu^2}{4},\be=-\frac{\nu^2}{4}, \ga=\frac{\nu^2}{4},\de=\f1{2}-\frac{\nu^2}{4})$ is a reduced hamiltonian system. Its…
Kaup-Newell soliton hierarchy is derived from a kind of Lax pairs different from the original ones. Binary nonlinearization procedure corresponding to the Bargmann symmetry constraint is carried out for those Lax pairs. The proposed Lax…
The H\'enon--Heiles system in the general form is studied. In a nonintegrable case new solutions have been found as formal Laurent series, depending on three parameters. One of parameters determines a location of the singularity point,…
Time independent Hamiltonians of the physical type H = (P_1^2+P_2^2)/2+V(Q_1,Q_2) pass the Painleve' test for only seven potentials $V$, known as the He'non-Heiles Hamiltonians, each depending on a finite number of free constants. Proving…
A Lax pair for the additive difference Painlev\'e equation of type $E_7^{(1)}$ is explicitly obtained as certain linear difference equations of scalar form. The compatibility of the Lax pair is proved by using certain characterization of…
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 apply Painlev\'e test to the most general variable coefficient nonlinear Schrodinger (VCNLS) equations as an attempt to identify integrable classes and compare our results versus those obtained by the use of other tools like…
A kind of Bargmann symmetry constraints involving Lax pairs and adjoint Lax pairs is proposed for soliton hierarchy. The Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative finite dimensional integrable…
In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlev\'e equations by using a reduced hypercube structure. In particular, we consider the $A_5^{(1)}$-surface $q$-Painlev\'e system which has the…
We study the system of first order PDEs for pseudo-Riemannian metrics governing the Hamiltonian formalism for systems of hydrodynamic type. In the diagonal setting the integrability conditions ensure the compatibility of this system and,…
There is a lack of knowledge about the topological invariants of non-linear $d$-dimensional systems with a periodic potential. We study these systems through a classification of the linearized NLS/GP equation around their soliton solutions.…
It is shown that a generalization of the Painlev\'e-II equation (P-II) to a system of coupled equations with symmetry breaking terms is integrable. A Lax pair for this system is used to relate the asymptotic behavior of the solutions at…
Stability criteria are given for linear periodic Hamiltonian systems with impulse effect. A Lyapunov type inequality and a disconjugacy criterion are also established. The results improve the ones in the literature for such systems.
A new system of coupled higher-order nonlinear Schroedinger equations is proposed which passes the Painleve test for integrability well. A Lax pair and a multi-field generalization are obtained for the new system.