Nonlocal PT-Symmetric Integrable Equations of Fourth-Order Associated with so(3, ℝ)

Integrable discrete PT symmetric model.

An exactly solvable discrete PT invariant nonlinear Schrödinger-like model is introduced. It is an integrable Hamiltonian system that exhibits a nontrivial nonlinear PT symmetry. A discrete one-soliton solution is constructed using a left-right Riemann-Hilbert formulation. It is shown that this pure soliton exhibits unique features such as power oscillations and singularity formation. The propo...

Symmetric Positive Solutions for Nonlinear Singular Fourth-Order Eigenvalue Problems with Nonlocal Boundary Condition

We investigate nonlinear singular fourth-order eigenvalue problems with nonlocal boundary condition u 4 t − λh t f t, u, u′′ 0, 0 < t < 1, u 0 u 1 ∫0a s u s ds, u′′ 0 u′′ 1 ∫1 0b s u ′′ s ds, where a, b ∈ L1 0, 1 , λ > 0, h may be singular at t 0 and/or 1. Moreover f t, x, y may also have singularity at x 0 and/or y 0. By using fixed point theory in cones, an explicit interval for λ is derived ...

Fourth Order Equations In

Collisional Dynamics of Solitons in the Coupled PT Symmetric Nonlocal Nonlinear Schrödinger Equations

We investigate the focussing coupled PT symmetric nonlocal nonlinear Schrödinger equation employing Darboux transformation approach. We find a family of exact solutions including pairs of Bright-Bright, Dark-Dark and Bright-Dark solitons in addition to solitary waves. We show that one can convert bright bound state onto a dark bound state in a two soliton solution by selectively finetuning the ...

Three symmetric positive solutions of fourth-order nonlocal boundary value problems

In this paper, we study the existence of three positive solutions for fourth-order singular nonlocal boundary value problems. We show that there exist triple symmetric positive solutions by using Leggett-Williams fixed-point theorem. The conclusions in this paper essentially extend and improve some known results. MSC: 34B16.