The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.
SUMMARYA Fourier-Galerkin spectral technique for solving coupled higher-order initial-boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so-called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived.The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth-order algebraic for both linear and nonlinear problems. Though algebraic, the fifth-order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems.An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement.The new technique is applied to the G-jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters.
In this Letter, an investigation is performed on the utilization of nematic liquid crystal (NLC) cells in the design of leaky-wave antennas (LWAs) for millimeter-wave (mm-wave) radiation in order to dynamically control its beam scanning capability at a single frequency. A NLC compound is sandwiched between two single-sided copper-plated substrates allowing a traveling wave to be guided through a substrate-integrated waveguide. The tuning capabilities of the structure, based on the use of K15 or GT7-29001 as the middle layer, were evaluated for different biasing conditions demonstrating the associated dynamic scanning of the main beam. A quasi-periodic LWA was designed to operate in the 5G mm-wave band, thus supporting a fast-wave propagation with tunable phase constant and dynamic beam steering at a single frequency. The simulated results clearly illustrate a dynamic beam scanning range of 45° through the use of an external bias voltage ranging between 0 and 40 V. These results are quite promising creating a fertile ground for the utilization of NLCs in the design and fabrication of LWAs for 5G wireless communication networks.
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