Related papers: The Painlev\'e Test of Higher Dimensional KdV Equa…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
A coupled KdV system with a free parameter proposed by Nutku and O\~{g}uz is considered. It is shown that the system passes the WTC's Painlev\'{e} test for arbitrary value of the parameter. A further analysis yields that the parameter can…
The discrete KdV (dKdV) equation, the pinnacle of discrete integrability, is often thought to possess the singularity confinement property because it confines on an elementary quadrilateral. Here we investigate the singularity structure of…
In this paper we discuss the Painlev\'e reductions of coupled KdV systems. We start by comparing the procedure with that of {\em stationary reductions}. Indeed, we see that exactly the same construction can be used at each step and parallel…
The coupled KdV-mKdV system arises as the classical part of one of superextensions of the KdV equation. For this system, we prove its complete integrability, i.e., existence of a recursion operator and of infinite series of symmetries.
A generalized Kadomtsev-Petviashvili equation (GKPE) $(u_t+u u_x + \beta(t)u +\gamma(t)u_{xxx})_x+\sigma(t)u_{yy}\ = \ 0$ is shown to admit an infinite-dimensional Lie group of symmetries when $\bt(t), \ga(t)$ and $\si(t)$ are arbitrary.…
A collection of miscellaneous continuous, semi-discrete, and discrete integrable systems can be associated with each integrable evolution equation of the KdV type. We give them for the Schwarz-KdV equation and generalize to the vector case.…
We analyze a variable coefficient coupled HI mKdV system that has shifted nonlocal reductions. The Weiss Tabor Carnevale test gives us coefficient restrictions to perform a time reparametrization to achieve an autonomous integrable model.…
The ``Painlev\'e analysis'' is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory…
An analysis of possible extension of the Painlev\'e test, to encompass the one-dimensional Vlasov equation, is performed. The extending requires a nontrivial generalization of the test. The proposed singularity analysis provides…
We investigate the Balitsky-Kovchegov (BK) equation for D=3 space-time dimensions, corresponding to one transverse coordinate, and we show that it can be solved analytically. The explicit solutions are found in the linear approximation and…
Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.
This paper concerns the discrete version of the Painlev\'e identification problem, i.e., how to recognize a certain recurrence relation as a discrete Painlev\'e equation. Often some clues can be seen from the setting of the problem, e.g.,…
The Korteweg-de Vries equation (KdV) and various generalized, most often semi- linear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general…
A nonlinear coupled system descriptive of multi-ion electrodiffusion is investigated and all parameters for which the system admits a single-valued general solution are isolated. This is achieved \textit{via} a method initiated by Painleve'…
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…
The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the…
We give a self-contained introduction to the relations between Integrable Systems and the Geometry of Riemann Surfaces. We start from a historical introduction to the topic of integrable systems. Afterwards, we study the polynomial…
We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg--de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations…
We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported…