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Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with \(\mathcal{PT}\)-symmetric harmonic potential via deep learning. (English) Zbl 1472.81089

Summary: In this paper, we investigate the logarithmic nonlinear Schrödinger (LNLS) equation with the parity-time \((\mathcal{PT})\)-symmetric harmonic potential, which is an important physical model in many fields such as nuclear physics, quantum optics, magma transport phenomena, and effective quantum gravity. Three types of initial value conditions and periodic boundary conditions are chosen to solve the LNLS equation with \(\mathcal{PT}\)-symmetric harmonic potential via the physics-informed neural networks (PINNs) deep learning method, and these obtained results are compared with ones deduced from the Fourier spectral method. Moreover, we also investigate the effectiveness of the PINNs deep learning for the LNLS equation with \(\mathcal{PT}\) symmetric potential by choosing the distinct space widths or distinct optimized steps. Finally, we use the PINNs deep learning method to effectively tackle the data-driven discovery of the LNLS equation with \(\mathcal{PT} \)-symmetric harmonic potential such that the coefficients of dispersion and nonlinear terms or the amplitudes of \(\mathcal{PT}\)-symmetric harmonic potential can be approximately found.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
35Q41 Time-dependent Schrödinger equations and Dirac equations
81P68 Quantum computation
68T05 Learning and adaptive systems in artificial intelligence
35C06 Self-similar solutions to PDEs
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