We first derive Alber’s equation for the Wigner distribution function using the fourth-order nonlinear Schrödinger equation, and on the basis of this equation we next analyse the stability of the narrowband approximation of the Joint North Sea Wave Project spectrum. Therefore, one interesting result of this study concerns the effect of modulational instability obtained from the fourth-order nonlinear Schrödinger equation. The analysis is restricted to one horizontal direction, parallel to the direction of wave motion, to take advantage of potential flow theory. We find that shear currents considerably modify the instability behaviours of weakly nonlinear waves. The key point of this study is that the present fourth-order analysis shows considerable deviations in the modulational instability properties from the third-order analysis and reduces the growth rate of instability. Moreover, we present here a connection between the random and deterministic properties of a random wavetrain for vanishing spectrum bandwidth.

Complicated option pricing models attract much attention in financial industries, as they produce relatively better accurate values by taking into account more realistic assumptions such as market liquidity, uncertain volatility and so forth. We propose a new hybrid method to accurately explore the behaviour of the nonlinear pricing model in illiquid markets, which is important in financial risk management. Our method is based on the Newton iteration technique and the Fréchet derivative to linearize the model. The linearized equation is then discretized by a differential quadrature method in space and a quadratic trapezoid rule in time. It is observed through computations that the accurate solutions for the model emerge using very few grid points and time elements, compared with the finite difference method in the literature. Furthermore, this method also helps to avoid consideration of the convergence issues of the Newton approach applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. It is important to note that the Fréchet derivative supports to enhance the convergence order of the proposed iterative scheme.

The hydroelastic interaction between water waves and multiple submerged porous elastic plates of arbitrary lengths in deep water is examined using the Galerkin approximation technique. We observe the influence of flexible porous plates of arbitrary lengths by analysing the reflection coefficient, dissipated energy and wave forces acting on the plates. Results are presented for various values of angle of incidence, separation lengths of plates, porosity levels, submergence depth and flexural rigidity. The convergence and accuracy of the method are verified by comparing the results with existing literature. The significant impact of flexural rigidity in the presence of porosity on wave reflection, dissipated energy and wave forces is demonstrated. Moreover, a notable reduction in wave load is observed with an increase in the number of plates.