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Modulation instability is one of the consequences of the water medium's inclination. It causes surface water waves to run into phenomena of splitting and merging in their propagation. An increase in wave amplitude follows this phenomenon, which can encourage the appearance of extreme waves. It is known that the Benjamin Bona Mahony (BBM) wave has modulation instability in its propagation, with the envelope evolving by the equation Nonlinear Schrodinger (NLS) equation dynamic. One of the NLS equation solution is known as Soliton on Finite Background (SFB). SFB is a continuation of the Benjamin-Feir nonlinear terms. Here, the probe of the BBM wave dynamics is conducted by transforming the complex amplitudes form of SFB variable into the polar form of displaced phase-amplitude. It was done to observe changes in the amplitude of the wave in a complex plane with phases that depend only on position. The description of the dynamics of the SFB can be illustrated through Argand diagrams. It was found that the modulation frequency affects the SFB phase: the smaller the modulation frequency, the higher the phase angle. Also, it is found that the phenomenon of SFB phase singularity occurs in extreme waves for certain frequency modulation intervals.

The velocity obstacle (VO) representation, a mathematical representation that supports separation assurance for autonomous vehicle navigation, can provide improved performance when the intended trajectory (intent) of the involved vehicles is shared among each other. This paper proposes a fully analytical closed mathematical form for VOs as an intuitive and flexible calculation method, which is able to incorporate time-varying concepts such as trajectory prediction errors. Several study cases for specific traffic situations are explored, and possibilities to extend the method to the third dimension are discussed. Analysis showed that the calculation of the VOs with intent information could show self-curve intersections when maneuvers at close distance are required. Further elimination of these intersections might be necessary for certain specific applications. [ABSTRACT FROM PUBLISHER]

Ultrashort optical pulses contain only a few optical cycles and exhibit broad spectra. Their carrier frequency is therefore not well defined and their description in terms of the standard slowly varying envelope approximation becomes questionable. Existing modeling approaches can be divided in two classes, namely generalized envelope equations, that stem from the nonlinear Schrödinger equation, and non-envelope equations which treat the field directly. Based on fundamental physical rules we will present an approach that effectively interpolates between these classes and provides a suitable setting for accurate and highly efficient numerical treatment of pulse propagation along nonlinear and dispersive optical media. [ABSTRACT FROM AUTHOR]

Abstract: Longitudinal and transverse beam dynamics during the bunch compression in a beam bunching system have been studied using disc model and envelope equation. The major contribution here is the inclusion of the effect of axial variation of the longitudinal electric field in the space charge term of beam envelope equation. Also included is a random Gaussian energy spread in the beam before the compression begins. In addition, the emittance growth resulting from the time dependant rf field in the transverse direction has been taken into account and the effective value of beam emittance has been used in the beam envelope equation. Finally, we have studied the beam bunching system of 100keV proton to be used in the injection line of the 10MeV high current compact cyclotron. Various numerical results of the study are presented for different geometrical and beam parameters. [Copyright &y& Elsevier]

We prove that the evolution of a slowly varying envelope of small amplitude of an underlying oscillating wave packet in the Fermi-Pasta-Ulam (FPU) system can be described approximately by the nonlinear Schrodinger equation. In contrast to other lattice equations for which this question has been addressed in the existing literature, the FPU system possesses a nontrivial quadratic resonance due to the curves of eigenvalues which vanish at the wave number k = 0. The proof of the error estimates is based on normal form transforms and a wave number-dependent scaling of the error function. [ABSTRACT FROM AUTHOR]

It holds in great generality that a plan is optimal for a dynamic programming problem, if and only if it is “thrifty” and “equalizing.” An alternative characterization of an optimal plan, that applies in many economic models, is that the plan must satisfy an appropriate Euler equation and a transversality condition. Here we explore the connections between these two characterizations. [ABSTRACT FROM AUTHOR]

Abstract: Padé approximation is superior to Taylor expansion when functions contain poles. This is especially important for response functions in complex frequency domain, where singularities are present and intimately related to resonances and absorption. Therefore, we introduce a rational Padé approximant for the complex medium refractive index . The approximant is calculated using only local information of medium dispersion properties close to a carrier frequency . In return it typically offers an accurate global representation of medium dispersion and absorption. Moreover, the fulfillment of the causality principle and the Kramers–Kronig relation can be established. In practice, our results are relevant if is known only for whereas optical field is spectrally broad such that (i) the resonance absorption becomes important and (ii) a traditional polynomial dispersion operator diverges and induces huge errors. As an exemplary application we use the approximant to derive a nonlocal envelope model for ultrashort pulses. The model provides a natural bridge between the commonly used local envelope equations and the most general non-envelope models operating directly with the electric field. [Copyright &y& Elsevier]