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.

Existence of Multiple Solutions for Second-Order Problem with Stieltjes Integral Boundary Condition

“…for any ω ∈ W . Notice that the critical points of Γ are the generalized solutions of problem (1). Therefore, we just validate that Y and Φ accord with the conditions of Theorem 6.…”

“…In the past few decades, the boundary value problems have appealed to many scholars in the mathematical field. Generally speaking, the boundary value problems mostly involve in two-point [1][2][3][4][5], three-point [6][7][8], and multipoint [9][10][11]. Many physical phenomena were formulated as nonlocal mathematical models with integral boundary conditions [12][13][14][15][16][17][18][19][20][21], such as fractional differential equation [22][23][24][25][26][27][28][29][30], nonlinear singular parabolic equation [31], and general second-order equation [19,[32][33][34][35][36][37][38][39][40].…”

“…To the best of my knowledge, no one use the theory of critical point and variational method to deal with the existence of solution of problem (1).…”

On integrable Ermakov–Painlevé IV systems

“…Dirichlet two-point boundary value problems for both the Ermakov-Painlevé II and Ermakov-Painlevé IV equations have been recently investigated in [1,2]. In [1], the hybrid Ermakov-Painlevé II equation was linked to its integrable Painlevé XXXIV avatar in the context of a three-ion reduction of the classical Nernst-Planck system as derived in [25].…”

On Hybrid Ermakov-Painlevé Systems. Integrable Reduction

“…The work of [2,[36][37][38] on Ermakov-Painlevé II systems has recently been augmented by the introduction in [47] of prototype Ermakov-Painlevé IV systems via a symmetry reduction of a coupled derivative resonant NLS triad. Dirichlet type two-point boundary value problems for a single hybrid Ermakov-Painlevé IV equation have been investigated with regard to existence and uniqueness properties in [3].…”