On a Neumann boundary value problem for the Painlevé II equation in two-ion electro-diffusion

Abstract

A two-point Neumann boundary value problem for a two ion electro-diffusion model reducible to the Painlevé II equation is investigated. The problem is unconventional in that the model equation involves yet-to-be determined boundary values of the solution. In prior work by Thompson, the existence of a solution was established subject to an inequality on the physical parameters. Here, a two-dimensional shooting method is used to show that this restriction may be removed. A practical algorithm for the solution of the boundary value problem is presented in an appendix.

On Dirichlet two-point boundary value problems for the Ermakov–Painlevé IV equation

“…Despite these important commonalities investigations of Painlevé and Ermakov-Ray-Reid type systems have preceded independently until recently in [16] wherein hybrid Ermakov-Painlevé II symmetry reductions have been derived for N+1-dimensional resonant nonlinear Schrödinger systems. A range of boundary value problems for the Painlevé II equation has been investigated in [17][18][19][20][21], notably in the context of multi-ion electrodiffusion. In the present work, a class of boundary value problems for a hybrid Ermakov-Painlevé IV equation is investigated.…”

Bäcklund flux quantization in a model of electrodiffusion based on Painlevé II

“…The situation where j = 0 is of particular physical interest. For any value of j, Planck's solution (14) might be used as the zerothorder term in an asymptotic expansion, in powers of λ 2 , of a solution to the system (5) with λ = 0. This would require the methods of singular perturbation theory, as is clear from the way in which λ 2 appears multiplying the highest derivative in (13).…”

“…Flux quantization is a remarkable feature of the structure of solutions of (1) when grouped into sequences by Bäcklund transformations. To begin to explore the physical interpretation of this mathematical result, we reconsider Planck's electrically neutral solution (14) in the case E(x) ≡ 0 when it becomes an exact solution of (5), that is, when A + = A − = c 1 − c 0 = A. We take this as seed solution S in a doubly-infinite sequence Q S of exact solutions generated by Bäcklund transformations and their inverses.…”

“…Experiment with this technique using MATLAB shows that the number of excited states with positive concentrations obtained from (14) (in the case with E(x) ≡ 0), depends on the values of c 0 , c 1 and λ. Examples like this show that for any physically sensible seed solution S, we must expect that only a finite subsequence of Q S may correspond to physically meaningful situations; in other words, only a finite number of physically acceptable solutions will be generated from any acceptable seed solution by Bäcklund transformations and their inverses.…”

“…The MAPLE [15] routine dsolve solves (49) almost instantaneously when supplemented by BCs of the form (44) and (46), with the latter rewritten as Comparison with the behaviour of E(x) in cases with j 0 = 0, as in Fig. 3 below, suggests strongly that these known existence and monotonicity results [13,14] can be extended to such cases also.…”