Abstract. In this article, we first employ the concentration compactness techniques to prove existence and stability results of standing waves for nonlinear fractional Schrödinger-Choquard equation, and the constants a, λ are nonnegative satisfying a + λ > 0. We then extend the arguments to establish similar results for coupled standing waves of nonlinear fractional Schrödinger systems of Choquard type. The same argument works for equations with an arbitrary number of combined nonlinearities and when |x| β−N is replaced by a more general convolution potential K : R N → [0, ∞) under certain assumptions. The arguments can be applied and the results are identical for the case α = 1 as well.

In this paper we establish existence and stability results concerning fully nontrivial solitarywave solutions to 3-coupled nonlinear Schrödinger systemwhere uj are complex-valued functions of (x, t) ∈ R 2 and a kj are positive constants satisfying a kj = a jk (symmetric attractive case). Our approach improves many of the previous known results. In all methods used previously to study solitary waves, which we are aware of, the variational problem has consisted of finding the extremum of an energy functional subject to the constraints that were not independently chosen. Here we study a problem of minimizing the energy functional subject to three independent L 2 mass constraints and establish existence and stability results for a true three-parameter family of solitary waves.

Abstract. This paper proves existence and stability results of solitary-wave solutions of a system of 2-coupled nonlinear Schrödinger equations with power-type nonlinearities arising in several models of modern physics. The existence of vector solitary-wave solutions (i.e, both components are nonzero) is established via variational methods. The set of minimizers is shown to be stable and further information about the structures of this set are given. The results extend stability results previously obtained by Cipolatti and Zumpichiatti [14], Nguyen and Wang [31,32], and Ohta [33].

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Physics based SRP (Solar Radiation Pressure) models using ray tracing methods are powerful tools when modelling the forces on complex real world space vehicles. Currently high resolution (1 mm) ray tracing with secondary intersections is done on high performance computers at UCL (University College London). This study introduces the BVH (Bounding Volume Hierarchy) into the ray tracing approach for physics based SRP modelling and makes it possible to run high resolution analysis on personal computers. The ray tracer is both general and efficient enough to cope with the complex shape of satellites and multiple reflections (three or more, with no upper limit). In this study, the traditional ray tracing technique is introduced in the first place and then the BVH is integrated into the ray tracing. Four aspects of the ray tracer were tested for investigating the performance including runtime, accuracy, the effects of multiple reflections and the effects of pixel array resolution.Test results in runtime on GPS IIR and Galileo IOV (In Orbit Validation) satellites show that the BVH can make the force model computation 30-50 times faster. The ray tracer has an absolute accuracy of several nanonewtons by comparing the test results for spheres and planes with the analytical computations. The multiple reflection effects are investigated both in the intersection number and acceleration on GPS IIR, Galileo IOV and Sentinel-1 spacecraft. Considering the number of intersections, the 3rd reflection can capture 99.12%, 99.14%, and 91.34% of the total reflections for GPS IIR, Galileo IOV satellite bus and the Sentinel-1 spacecraft respectively. In terms of the multiple reflection effects on the acceleration, the secondary reflection effect for Galileo IOV satellite and Sentinel-1 can reach 0.2 nm/s 2 and 0.4 nm/s 2 respectively. The error percentage in the accelerations magnitude results show that the 3rd reflection should be considered in order to make it less than 0.035%. The pixel array resolution tests show that the dimensions of the components have to be considered when choosing the spacing of the pixel in order not to miss some components of the satellite in ray tracing. This paper presents the first systematic and quantitative study of the secondary and higher order intersection effects. It shows conclusively the effect is non-negligible for certain classes of misson.

We introduce a simple single-band receiver clock jump and cycle slip (CJCS) detection and correction algorithm suitable for a standalone single-frequency Global Navigation Satellite System (GNSS) receiver. The real-time algorithm involves using an adaptive time differencing technique for the generation of adaptive differenced sequences of single-frequency code and phase observations. The sequences are used for determining thresholds and for the detection and determination of a receiver clock jump and cycle slips. The cycle slip values are fixed by rounding-up float values obtained via weighted least squares adjustment, following the elimination of the receiver's high-order clock drift at every epoch. The performance of this new technique was investigated with simulated cycle slip values and with different types of receiver clock jumps at millisecond and microsecond levels. It achieved 100% detection and correction of all types of receiver clock jumps; between 97 to 100% cycle slip detection; and between 96.9 to 100% cycle slip correction including cycle slips of ±1 cycle, for different rates of observations acquired by different fixed and mobile GNSS receivers. The algorithm thus facilitates precise timing and positioning on standalone low-cost single-frequency GNSS devices. unwanted frequent re-initializations, i.e., an overall reduced level of performance. Here, the authors present a new algorithm for dealing with this CJCS problem on a single-frequency GNSS receiver. Several single-frequency cycle slip correction methods, which are either code-phase based, phase-only based or doppler-inclusive based, exist. A code-phase method is presented in Fath-Allah (2010). This method is limited by the usual code-level errors that often prevent the detection of small cycle slip values. Phase-only methods found in Jia and Wu (2001) and Cosser et al. (2004) are based on 3rd-order differencing or fitting, which have no means of detecting and eliminating receiver clock jumps or achieving reliable detection of cycle slips in data sets with low observation rates. The Doppler-inclusive methods, such as found in Ren et al. (2011), combine Doppler measurements with phase and/or code observations. One drawback with the Doppler-inclusive technique is the inability to detect receiver clock jumps, plus the fact that not all single-frequency GNSS chip sets provide Doppler measurements. Known single-frequency techniques for addressing clock jumps are found in Momoh and Ziebart (2012), which involves phase differencing and code-based thresholding; and in Deo and El-Mowafy (2015), which is based on extrapolation and spline fitting of combined code and phase observations. These two methods are suitable for homogeneous clock jumps v J is at least half the number of satellites used in the clock jump detection, a clock jump is confirmed and its value, * J is computed as ** () v J mode J  (12) where mode is the arithmetic mode operator. Outliers, if any, are excluded from the values in * v J before the mode value is computed. There is no cloc...

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