Abstract
The multitemporal nonlinear Schrödinger PDE (with oblique derivative) was stated for the first time in our research group as a universal amplitude equation which can be derived via a multiple scaling analysis in order to describe slow modulations of the envelope of a spatially and temporarily oscillating wave packet in space and multitime (an equation which governs the dynamics of solitons through meta-materials). Now we exploit some hypotheses in order to find important explicit families of exact solutions in all dimensions for the multitime nonlinear Schrödinger PDE with a multitemporal directional derivative term. Using quite effective methods, we discovered families of ODEs and PDEs whose solutions generate solutions of multitime nonlinear Schrödinger PDE. Each new construction involves a relatively small amount of intermediate calculations.
Highlights
In order to define the multitime nonlinear Schrödinger PDE, we need: (i) Two intervals, I1 ⊆ R, I2 ⊆ [0, ∞), (ii) an open subset D ⊆ Rm, (iii) the function f : I1 × I2 → R, and (iv) the vector field h = : I1 × D → Rm
Remark 1. (a) If u is a solution of multitime NLSE (1), −u is a solution of the multitime
(b) If u is a solution of the multitime NLSE (1), for any constant k ∈ R, the function φ := eiku is a solution of the multitime NLSE (1)
Summary
In order to define the multitime nonlinear Schrödinger PDE (multitime NLSE), we need: (i) Two intervals, I1 ⊆ R, I2 ⊆ [0, ∞), (ii) an open subset D ⊆ Rm, (iii) the function f : I1 × I2 → R, and (iv) the vector field h = (hα) : I1 × D → Rm For possible numerical solutions we can use the techniques from our papers [16,17,18,19], using either discretization with respect to space variable x, or in relation to multitime variable t, or both
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