Related papers: Nonautonomous symmetries of the KdV equation and s…
We solve the fifth-order Korteweg-de Vries (fKdV) equation which is a modified KdV equation perturbed by a fifth-order derivative term multiplied by a small parameter $\epsilon^2$, with $0< \epsilon \ll 1$. Unlike the KdV equation, the…
We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlev\'e equation with $E^{(1)}_6$ symmetry. We present a description of a set of symmetries of the reduced…
In connection with the problem of choosing the in- and out-states among the solutions of a wave equation with one-dimensional potential we study nonstationary and "stationary" families of complete sets. A nonstationary set consists of the…
In the recent paper [Stud. App. Math. 147 (2021) 752], squared eigenfunction symmetry constraint of the differential-difference modified Kadomtsev-Petviashvili (D$\Delta$mKP) hierarchy converts the D$\Delta$mKP system to the relativistic…
The construction of a solution of the perturbed KdV equation encounters obstacles to asymptotic integrability beyond the first order, when the zero-order approximation is a multiple-soliton wave. In the standard analysis, the obstacles lead…
We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko (GLM) equation. This is a linear integral equation in which…
We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a…
We develop a framework for the fifth-order Kadomtsev--Petviashvili equation on $\mathbb{T}_x \times \mathbb{R}_y$ within a mean-zero KP-adapted Sobolev scale. A localized high-order feedback acting on the periodic variable yields a…
The nature of transverse instabilities to dark solitons and dispersive shock waves for the (2+1)-dimensional defocusing nonlinear Schrodinger equation / Gross-Pitaevskii (NLS / GP) equation is considered. Special attention is given to the…
We classify the Lie point symmetries for the 2+1 nonlinear generalized Kadomtsev-Petviashvili equation by determine all the possible f(u) functional forms where the latter depends. For each case the one-dimensional optimal system is…
We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schr\"{o}dinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second order…
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…
The celebrated (1+1)-dimensional Korteweg de-Vries (KdV) equation and its (2+1)-dimensional extention, the Kadomtsev-Petviashvili (KP) equation, are two of the most important models in physical science. The KP hierarchy is explicitly…
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…
In this paper we study rotationally symmetric solutions of the Cahn-Hilliard equation in $\mathbb R^3$ constructed by the authors. These solutions form a one parameter family analog to the family of Delaunay surfaces and in fact the zero…
In this paper, we delve into the study of the generalized KP equation, which incorporates double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and…
In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of the initial-value problem with smooth initial conditions in an open sphere around the origin, where the internal and external damping…
The modified Kadomtsev-Petviashvili (mKP) equation is shown in this paper to be decomposable into the first two soliton equations of the 2N-coupled Chen-Lee-Liu and Kaup-Newell hierarchies by respectively nonlinearizing two sets of symmetry…
We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity,…
We revisit the symmetry structure of integrable PDEs, looking at the specific example of the KdV equation. We identify 4 nonlocal symmetries of KdV depending on a parameter, which we call generating symmetries. We explain that since these…