Three-dimensional spatiotemporal vector solitary waves in coupled nonlinear Schrödinger equations with variable coefficients

Spatiotemporal soliton structures in (3 $$+$$ + 1)-dimensional $$\mathcal {PT}$$ PT -symmetric nonlinear couplers with gain and loss

“…with the modulation depth of the pulse q ∈ [0, 1], the topological charge m and P m l (cos θ) is the associated Legendre polynomials with the degree l and order m satisfying l ≥ m ≥ 0 [32].…”

“…Based on the Hirota bilinear method [32], substituting U = r l g(r,z) f (r,z) with the complex and real functions g(r, z) and f (r, z) in Eq. (6), we have…”

“…In the following, we expand functions g(r, z) and f (r, z) as power series of a small expansion parameter ε. In order to get the first-order solution, substituting function g(r, z) truncated as g(r, z) = εg 1 (r, z) and f (r, z) as f (r, z) = 1 + ε 2 f 2 (r, z) [32] in Eqs. (7) and (8) yields…”

“…We discuss the dynamical evolutions of spatiotemporal solitons in a periodic modulation system [32,33] β(z) = β 0 cos(σ z), χ (z) = χ 0 cos(δz),…”

“…Many researchers have pointed that the introduction of sign-changing coefficients improves the stability of solitons [32,33]. We fix constants as g 10 = 1, g 20 = 3, c = 1.5, q = 0.95.…”