Table 2 Classification of analytical solutions and their physical admissibility.

From: Optical soliton solutions of the stochastic generalized nonlinear Schrödinger equation with arbitrary refractive index in Itô sense

Solution family / Eq.

Parameter constraints (from text)

Physical status

Bright soliton (31)

\(a<0\), \((6m^{2}+5m+1)b>0\), \(\beta >0\), \(Q(\zeta )>0\)

Physically relevant

Dark soliton (39)

\(a>0\), \((6m^{2}+5m+1)b<0\), \(\beta >0\), \(Q(\zeta )>0\)

Physically relevant

Singular solitons (32), (40), (57)

Same sign constraints as parent families; profiles unbounded (csch/coth)

Nonphysical

Jacobi elliptic (\(0<M<1\))

Case-specific sign rules (see text), M real, \(Q(\zeta )>0\), no poles on real axis

Physically relevant (coherence degrades for large \(\delta\))

Trigonometric periodic (\(M=0\))

(33),(34): \(a>0\), \((6m^{2}+5m+1)b<0\); (45),(46): \(a<0\), \((6m^{2}+5m+1)b>0\); (52)–(54): \(a<0\), \((6m^{2}+5m+1)b<0\)

Physically relevant for low–moderate noise; destroyed for large \(\delta\)

Weierstrass elliptic (61)–(66)

Invariants \(r_2,r_3\) yield bounded \(\wp\); no poles; sign constraints in text

Physically relevant if bounded; coherence loss for large \(\delta\)

  1. “Physically relevant” means real, bounded, finite-energy envelopes without poles on the real axis. Large \(\delta\) impairs coherence even for admissible deterministic profiles.
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