arXiv subjects

Andrew Pickering

Publications and source records attributed to Andrew Pickering.

5 recordsLinked to original sources

arXiv2026

Asymptotic Analysis of discrete nonlinear localised modes in a Kagome lattice

We describe a nonlinear kagome lattice with nonlinear dynamics described by Klein-Gordon interactions with a scalar unknown at each node, such as might occur in a nonlinear electrical lattice. We show that the dispersion relation has three bands - a flat band and two other surfaces which may meet in Dirac points or be separated by a gap. By using multiple scales asymptotic methods, we find a variety of reductions to nonlinear Schrodinger (NLS) systems, some of which are similar to those obtained previously, and have the Townes soliton as a solution. We find a novel system of coupled NLS equations, by forming an asymptotic expansion for small amplitude weakly nonlinear waves around the point where the flat band meets the upper surface of the dispersion relation. We analyse this 2+1 dimensional system using Lie symmetries, and find further reductions to more complicated solitary wave solutions. Numerical simulations of the wave are also presented.

nlin.PS↗

arXiv2019

Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations

In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation $$ \partial_t u=(u^m)_{xx}+a(x)(u^m)_x+b(x)u^m, $$ posed for $x\in\real$, $t\geq0$ and $m>1$, where $a$, $b$ are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium type $$ f(y)\partial_{\tau}\theta=(\theta^m)_{yy}, $$ posed in the half-line $y\in[0,\infty)$ with $\tau\geq0$, $m>1$ and suitable density functions $f(y)$. We apply this correspondence to the case of constant coefficients $a(x)=1$ and $b(x)=K>0$. For this case, we prove that compactly supported solutions to the first equation blow up in finite time, together with their interfaces, as $x\to-\infty$. We then establish the large time behavior of solutions to a homogeneous Dirichlet problem associated to the first equation on a bounded interval. We also prove a finite time blow-up of the interfaces for compactly supported solutions to the second equation when $f(y)=y^{-\gamma}$ with $\gamma>2$.

math.AP↗

arXiv2005

Bäcklund transformations for fourth Painlevé hierarchies

Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé ($P_{IV}$) hierarchies recently found by the same authors [{\em Publ. Res. Inst. Math. Sci. (Kyoto)} {\bf 37} 327--347 (2001)]. We show how the known pattern of BTs for $P_{IV}$ can be extended to our $P_{IV}$ hierarchies. Remarkably, the BTs required to do this are precisely the Miura maps of the dispersive water wave hierarchy. We also obtain the important result that the fourth Painlevé equation has only one nontrivial fundamental BT, and not two such as is frequently stated.

nlin.SI↗

arXiv1999

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleve classification.

solv-int↗

arXiv1998

Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, P2. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive auto-Bäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of P34 equations, and a little known Bäcklund transformation for P2 itself. We then use our results on Bäcklund transformations to obtain, for each member of the P2 hierarchy, a sequence of special integrals.

solv-int↗