Quantum mechanics

Gaussian

Quintic function

Mathematical physics

Constant coefficients

Stability (learning theory)

Soliton

Computer science

Physics

Nonlinear system

Mathematics

Mathematical analysis

Combinatorics

Programming language

Machine learning

Constant (computer programming)

Nonlinear Dynamics

Gaussian spatial soliton solutions of both the constant-coefficient and variable-coefficient (2 + 1)-dimensional nonlinear Schrödinger equations in quintic–septimal nonlinear materials with different diffractions are presented under two kinds of $${\mathcal {P}}{\mathcal {T}}$$ P T -symmetric potentials. The linear stability analysis and direct numerical simulation are jointly utilized to investigate the stability for analytical solutions of the constant-coefficient equation. Results from the linear stability analysis and the direct numerical simulation possess a high degree of consistency, that is, the stable case for Gaussian spatial solitons of the constant-coefficient equation appears only in the defocusing quintic and focusing septimal nonlinear material. Moreover, reconstruction of stable Gaussian spatial solitons of the variable-coefficient equation is studied based on the expression of the effective propagation distance Z(z) by choosing an appropriate form of diffraction $$\beta _1(z)$$ β 1 ( z ) .

Reconstruction of stability for Gaussian spatial solitons in quintic–septimal nonlinear materials under $${{\varvec{\mathcal {P}}}}{\varvec{\mathcal {T}}}$$ P T -symmetric potentials