A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wilton's ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extreme standing waves. In shallow water, we find that standing waves take the form of counterpropagating solitary waves that repeatedly collide quasi-elastically. In deep water with surface tension, we find that standing waves resemble counter-propagating depression waves. We also discuss existence and non-uniqueness of solutions, and smooth versus erratic dependence of Fourier modes on wave amplitude and fluid depth.In the numerical method, robustness is achieved by posing the problem as an overdetermined nonlinear system and using either adjoint-based minimization techniques or a quadratically convergent trust-region method to minimize the objective function. Efficiency is achieved in the trust-region approach by parallelizing the Jacobian computation so the setup cost of computing the Dirichlet-to-Neumann operator in the variational equation is not repeated for each column. Updates of the Jacobian are also delayed until the previous Jacobian ceases to be useful. Accuracy is maintained using spectral collocation with optional mesh refinement in space, a high order Runge-Kutta or spectral deferred correction method in time, and quadruple-precision for improved navigation of delicate regions of parameter space as well as validation of double-precision results. Implementation issues for GPU acceleration are briefly discussed, and the performance of the algorithm is tested for a number of hardware configurations.
The spin splitting in GaN-based heterostructures has been investigated by means of circular photogalvanic effect experiments under uniaxial strain. The ratios of Rashba and Dresselhaus spin-orbit coupling coefficients (R/D ratios) have been measured in AlxGa1−xN/GaN heterostructures with various Al compositions. It is found that the R/D ratio increases from 4.1 to 19.8 with the Al composition of the AlxGa1−xN barrier varied from 15% to 36%. The Dresselhaus coefficient of bulk GaN is experimentally obtained to be 0.4 eV Å3. The results indicate that the spin splitting in GaN-based heterostructures can be modulated effectively by the polarization-induced electric fields.
Spectra of the interband spin photocurrent due to Rashba and Dresselhaus spin splittings have been experimentally investigated in InGaAs/AlGaAs quantum wells at room temperature. The Rashba- and Dresselhaus-induced circular photogalvanic effect (CPGE) spectra are found to be quite similar in the spectral regions corresponding to the transitions 1e1hh (the first conduction to the first valence sub-band of heavy hole) and 1e2hh. The ratio of Rashba- and Dresselhaus-induced CPGE currents for the transition 1e1hh is estimated to be 4.95. The magnitude of the Rashba-induced CPGE current is up to several tens of nA/W for the transition 1e1hh, which is 1 order of magnitude larger than that obtained in GaN/AlGaN superlattices. Comparing the CPGE spectrum with reflectance-difference and photoreflectance spectra, we find that the large Rashba spin splitting is mainly induced by a large indium atom segregation effect and by the internal field in the quantum wells.
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