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

Abstract

Two-point boundary value problems of Dirichlet-type are investigated for a hybrid Ermakov–Painlevé IV equation. Existence and uniqueness results are established in terms of the Painlevé parameters. In addition, it is shown how Ermakov invariants may be used to systematically obtain solutions of a coupled Ermakov–Painlevé IV system in terms of seed solutions of the canonical integrable Painlevé IV equation.

Appendix: Coupled Ermakov–Painlevé IV system

The importance of the Ermakov–Painlevé equation (1.1) as a canonical form is emphasised here by establishing its role in the construction of solution sets of an overarching coupled Ermakov–Painlevé IV system. It is remarked that Ermakov–Painlevé IV type systems have recently derived in [22] as symmetry reductions of coupled derivative nonlinear Schrödinger systems such as arise in plasma physics.

Here, we consider the hybrid two-component Ermakov–Painlevé IV system

$$\begin{aligned}&\displaystyle u_{xx} - \left[ \dfrac{3}{4} (u^2 + v^2 )^2 + 2x ( u^2 + v^2 ) + x^2 - \alpha \right] u = \dfrac{\beta }{2u^3}, \nonumber \\&\displaystyle v_{xx} - \left[ \dfrac{3}{4} (u^2 + v^2 )^2 + 2x ( u^2 + v^2 ) + x^2 - \alpha \right] v = \dfrac{\beta }{2v^3}. \end{aligned}$$

(5.1)

This is seen to admit a characteristic Ermakov invariant

$$\begin{aligned} ( u_x v - v_x u )^2 + \frac{\beta }{2}\ \left[ \left( \frac{u}{v} \right) ^2 + \left( \frac{v}{u} \right) ^2 \right] = \mathcal {E} \end{aligned}$$

(5.2)

together with the (non-local) Hamiltonian

$$\begin{aligned} u^2_x + v^2_x - \frac{1}{4} \Sigma ^3 - 2 \int x \Sigma d \Sigma - \int (x^2 - \alpha ) d \Sigma + \frac{\beta }{2}\ \left[ \frac{1}{u^2} + \frac{1}{v^2} \right] = 2 \mathcal {H}. \end{aligned}$$

(5.3)

where $\Sigma :=u^2+v^2$.

The identity

$$\begin{aligned} \left( u^2 + v^2\right) \left( u^2_x + v^2_x\right) - \left( u_x v - v_x u\right) ^2 \equiv \left( uu_x + vv_x\right) ^2 \end{aligned}$$

(5.4)

is now used together with the Ermakov and Hamiltonian invariant relations (5.2), (5.3) to show that

$$\begin{aligned}&\Sigma \ \left[ \ 2 \mathcal {H} + \dfrac{1}{4} \Sigma ^3 + 2 \displaystyle \int x \Sigma d \Sigma + \displaystyle \int (x^2 - \alpha ) d \Sigma - \dfrac{\beta }{2}\ \left( \dfrac{1}{u^2} + \dfrac{1}{v^2} \right) \ \right] \nonumber \\&\quad \quad \quad - \left[ \mathcal {E} - \dfrac{\beta }{2} \left( \left( \dfrac{u}{v} \right) ^2 + \left( \dfrac{v}{u} \right) ^2 \right) \right] = \dfrac{1}{4} \left( \dfrac{d \Sigma }{dx} \right) ^2, \end{aligned}$$

(5.5)

whence,

$$\begin{aligned} \Sigma \ \left[ \ 2 \mathcal {H} + \dfrac{1}{4} \Sigma ^3 + 2 \int x \Sigma d \Sigma + \int (x^2 - \alpha ) d \Sigma \ \right] - \mathcal {E} - \beta = \dfrac{1}{4} \left( \frac{d \Sigma }{dx} \right) ^2. \end{aligned}$$

(5.6)

The latter, in turn, leads to a Painlevé IV equation in $\Sigma$, namely

$$\begin{aligned} \Sigma _{xx} = \frac{1}{2} \frac{\Sigma ^2_x}{\Sigma } + \frac{3}{2} \Sigma ^3 + 4 x \Sigma ^2 + 2 ( x^2 - \alpha ) \Sigma + \frac{2}{\Sigma } ( \mathcal {E} + \beta ) \end{aligned}$$

(5.7)

which, in terms of $\Lambda = \Sigma ^{1/2}$ produces the canonical Ermakov–Painlevé IV equation

$$\begin{aligned} \Lambda _{xx} - \left[ \ \frac{3}{4} \Lambda ^4 + 2 x \Lambda ^2 + x^2 - \alpha \ \right] \Lambda = \frac{\mathcal {E} + \beta }{\Lambda ^3}. \end{aligned}$$

(5.8)

Importantly, the latter together with each in turn of the constituent nonlinear equations in the original Ermakov–Painlevé IV system (5.1) constitute Ermakov–Ray-Reid systems. This result in an additional pair of Ermakov–type invariant relations, viz

$$\begin{aligned} \left( u_x \Lambda - \Lambda _x u\right) ^2 + \left( \mathcal {E} + \beta \right) \left( \frac{u}{\Lambda } \right) ^2 + \frac{\beta }{2} \left( \frac{\Lambda }{u} \right) ^2 = \mathbb {R}_\mathrm{I}, \end{aligned}$$

(5.9)

and

$$\begin{aligned} ( v_x \Lambda - \Lambda _x v )^2 + ( \mathcal {E} + \beta ) \left( \frac{v}{\Lambda } \right) ^2 + \frac{\beta }{2} \left( \frac{\Lambda }{v} \right) ^2 = \mathbb {R}_\mathrm{II}. \end{aligned}$$

(5.10)

Thus, on introduction of the new independent variable $\bar{x}$ according to

$$\begin{aligned} d \bar{x} = \Lambda ^{-2} dx, \end{aligned}$$

(5.11)

and new dependent variables $U,\ V$ via

$$\begin{aligned} U = \left( \frac{u}{\Lambda } \right) ^2, \quad V = \left( \frac{v}{\Lambda } \right) ^2 \end{aligned}$$

(5.12)

it is seen that (5.9) and (5.10), in turn, yield

$$\begin{aligned} d U^{1/2}/ d \bar{x} = \pm \sqrt{ \mathbb {R}_\mathrm{I} - ( \mathcal {E} + \beta ) U - \frac{\beta }{2} U^{-1}}, \end{aligned}$$

(5.13)

and

$$\begin{aligned} d V^{1/2}/ d \bar{x} = \pm \sqrt{ \mathbb {R}_\mathrm{II} - ( \mathcal {E} + \beta ) V - \frac{\beta }{2} V^{-1}}. \end{aligned}$$

(5.14)

The latter pair show that, if $\mathcal {E} + \beta > 0$ then

$$\begin{aligned} U = \frac{1}{2( \mathcal {E} + \beta )}\ \left[ \mathbb {R}_\mathrm{I} + \sqrt{ \mathbb {R}^2_\mathrm{I} - 2 \beta ( \mathcal {E} + \beta )} \sin \left( \pm 2 \sqrt{ \mathcal {E} + \beta }\ \bar{x} + \mathbb {K}_\mathrm{I} \right) \right] \end{aligned}$$

(5.15)

and

$$\begin{aligned} V = \frac{1}{2( \mathcal {E} + \beta )}\ \left[ \mathbb {R}_\mathrm{II} + \sqrt{ \mathbb {R}^2_\mathrm{II} - 2 \beta ( \mathcal {E} + \beta )} \sin \left( \pm 2 \sqrt{ \mathcal {E} + \beta }\ \bar{x} + \mathbb {K}_\mathrm{II} \right) \right] \end{aligned}$$

(5.16)

where $\mathbb {K}_\mathrm{I}$ and $\mathbb {K}_\mathrm{II}$ are constants of integration and it is required that $\mathbb {R}^2_\mathrm{I} > 2 \beta (\mathcal {E} + \beta ),\,\mathbb {R}^2_\mathrm{II} > 2 \beta (\mathcal {E} + \beta )$. The relations (5.15), (5.16) are subject to the constraint $U + V = 1$ and the latter relation may be used to calculate $V$ in terms of $U$ as given by (5.15) without recourse to (5.14). Moreover, addition of the integrals of motion (5.9), (5.10) together with use of the identity (5.4) and of the Ermakov invariant relation (5.2) is seen to impose the constraint

$$\begin{aligned} \mathbb {R}_\mathrm{I} + \mathbb {R}_\mathrm{II} = 2 \left( \mathcal {E} + \beta \right) . \end{aligned}$$

(5.17)

In summary, solution pairs $\{u, v\}$ of the Ermakov–Painlevé IV system (5.1) may be constructed in terms of a seed solution $\Sigma$ of the canonical Painlevé IV equation (5.7) via relation

$$\begin{aligned} u = \pm \sqrt{ \frac{1}{2 (\mathcal {E} + \beta )} \left( \mathbb {R}_\mathrm{I} + \sqrt{ \mathbb {R}^2_\mathrm{I} - 2\beta (\mathcal {E} + \beta )} \sin \left( \pm 2 \sqrt{\mathcal {E} + \beta } \bar{x} + \mathbb {K}_\mathrm{I} \right) \right) } \Sigma \end{aligned}$$

(5.18)

together with

$$\begin{aligned} v = \pm \sqrt{ \Sigma - u^2} \end{aligned}$$

(5.19)

where $\bar{x}$ is obtained by integration of the relation (5.11), namely

$$\begin{aligned} d \bar{x} = \Sigma ^{-1} dx \end{aligned}$$

(5.20)

It is remarked that such seed solutions $\Sigma$ of Painlevé IV may be generated, in particular, via the application of established Bäcklund transformations (Bassom et al. [24], Gromak [26]).