A full analysis of all regimes for optical dispersive shock wave (DSW) propagation in nematic liquid crystals is undertaken. These dispersive shock waves are generated from step initial conditions for the optical field and are resonant in that linear diffractive waves are in resonance with the DSW, resulting in a resonant linear wavetrain propagating ahead of it. It is found that there are six regimes, which are distinct and require different solution methods. In previous studies, the same solution method was used for all regimes, which does not yield solutions in full agreement with numerical solutions. Indeed, the standard DSW structure disappears for sufficiently large initial jumps. Asymptotic theory, approximate methods or Whitham modulation theory are used to find solutions for these resonant dispersive shock waves in a given regime. The solutions are found to be in excellent agreement with numerical solutions of the nematic equations in all regimes. It is found that for small initial jumps, the resonant wavetrain is unstable, but that it stabilises above a critical jump height. It is additionally found that the DSW is unstable, except for small jump heights for which there is no resonance and large jump heights for which there is no standard DSW structure.

The structure of optical dispersive shock waves in nematic liquid crystals is investigated as the power of the optical beam is varied, with six regimes identified, which complements previous work pertinent to low power beams only. It is found that the dispersive shock wave structure depends critically on the input beam power. In addition, it is known that nematic dispersive shock waves are resonant and the structure of this resonance is also critically dependent on the beam power. Whitham modulation theory is used to find solutions for the six regimes with the existence intervals for each identified. These dispersive shock wave solutions are compared with full numerical solutions of the nematic equations, and excellent agreement is found.

This paper is about the fractional Schrödinger equation expressed in terms of the Caputo time-fractional and quantum Riesz-Feller space fractional derivatives for particle moving in a potential field. The cases of free particle (zero potential) and a linear potential are considered. For free particle, the solution is obtained in terms of the Fox H-function. For the case of a linear potential, the separation of variables method allows the fractional Schrödinger equation to be split into space fractional and time fractional ones. By using the Fourier and Mellin transforms for the space equation and the contour integrals technique for the time equation, the solutions are obtained also in terms of the Fox H-function. Moreover, some recent results related to the time fractional equation have been revised and reconsidered. The results obtained in this paper contain as particular cases already known results for the fractional Schrödinger equation in terms of the quantum Riesz space-fractional derivative and standard Laplace operator.

The addition of higher order asymptotic corrections to the Korteweg–de Vries equation results in the extended Korteweg–de Vries (eKdV) equation. These higher order terms destabilize the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and result in the emission of resonant radiation. In broad terms, there are three possible dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dispersive shock wave (CDSW) and travelling dispersive shock wave (TDSW). While there are existing solutions for the RDSW and TDSW regimes obtained using modulation theory, there is no existing solution for the CDSW regime. Modulation theory and the associated concept of a Whitham shock are used to obtain this CDSW solution. In addition, it is found that the resonant wavetrain emitted by the eKdV equation with water wave coefficients has a minimal amplitude. This minimal amplitude is explained based on the developed Whitham modulation theory.

This paper is about the fractional Schrödinger equation (FSE) expressed in terms of the quantum Riesz-Feller space fractional and the Caputo time fractional derivatives. The main focus is on the case of time independent potential fields as a Dirac-delta potential and a linear potential. For such type of potential fields the separation of variables method allows to split the FSE into space fractional equation and time fractional one. The results obtained in this paper contain as particular cases already known results for FSE in terms of the quantum Riesz space fractional derivative and standard Laplace operator. ∞ −∞ e ipx/ℏψ (p, t) dp Key words and phrases. Fractional calculus, fractional Schrödinger equation, quantum Riesz-Feller space fractional derivative, Caputo time fractional derivative, Fourier transform in momentum representation.