In this paper, we consider a supersymmetric AKNS spectral problem. Two elementary and a binary Darboux transformations are constructed. By means of reductions, Darboux and Bäcklund transformations are given for the supersymmetric modified Korteweg-de Vries, sinh-Gordon, and nonlinear Schrödinger equations. These Darboux and Bäcklund transformations are adopted for the constructions of integrable discrete super systems, and both semidiscrete and fully discrete systems are presented. Also, the continuum limits of the relevant discrete systems are worked out.

]]>The Whitham modulation theory for periodic traveling waves of PDEs generated by a Lagrangian produces first-order dispersionless PDEs that are, generically, either hyperbolic or elliptic. In this paper, degeneracy of the Whitham equations is considered where one of the characteristic speeds is zero. In this case, the Whitham equations are no longer valid. Reformulation and rescaling show that conservation of wave action morphs into the Korteweg–de Vries (KdV) equation on a longer time scale thereby generating dispersion in the Whitham modulation equations even for finite amplitude waves.

]]>The problem of a weak shock, reflected and diffracted by a wedge, is studied for the two-dimensional compressible Euler system. Some recent developments are overviewed and a perspective is presented within the context of a real gas, modeled by the van der Waals equation of state. The regular reflection configuration and the detachment criterion are studied in the light of real gas effects. Some basic features of the phenomenon and the nature of the self-similar flow pattern are explored using asymptotic expansions. The analysis presented here predicts several inviscid flow properties of the real gases undergoing shock reflection–diffraction phenomenon.

]]>We examine the impact of small parity-time () symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing perturbations offer potentially novel means for the control of light in honeycomb lattices.

]]>The paper investigates the integrable discretization of a negative order AKNS equation. Two semidiscrete and one fully discrete versions of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented. Dynamics of one-soliton and two-soliton solutions of spacial-discrete system are characterized.

]]>In this paper, we derive a new representation for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by Howls 1992. Using this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the incomplete gamma function with large *a* and fixed positive λ, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingle's formal result regarding the exponentially improved version of the asymptotic series of .

We consider systems of differential equations with quadratic nonlinearities having applications for biochemistry and population dynamics, which may have a large dimension *n*. Due to the complexity of these systems, reduction algorithms play a crucial role in study of their large time behavior. Our approach aims to reduce a large system to a smaller one consisting of *m* differential equations, where . Under some restrictions (that allow us to separate slow and fast variables in the system) we obtain a new system of differential equations, involving slow variables only. This reduction is feasible from a computational point of view for large *n* that allows us to investigate sensitivity of dynamics with respect to random variations of parameters. We show that the quadratic systems are capable to generate all kinds of structurally stable dynamics including chaos.

On reexamining the hydrodynamic instability, Yu [1] showed that when the fully dynamical interactions are duly accounted for, and proper mathematical analysis is carried out, the positive feedback between the wave and evolving current can initiate and sustain rip current circulations with scales comparable to field observations on alongshore uniform beaches. In this study, we extend that analysis to consider nonplanar beaches, and to include a new branch of unstable modes that correspond to alongshore propagating horizontal circulations with the magnitudes of the flow growing in time. The latter has not previously been studied. These propagating unstable modes have typical time periods of tens of minutes and alongshore propagation speeds of a few centimeters per second. The physical implications of their spatial and slow time oscillations are discussed, as of relevance to occurrence and recurrence of transient rips, alongshore migration of rip currents and very low frequency pulsations in surf zone eddy circulations.

]]>In the present study, oblique surface wave scattering by a submerged vertical flexible porous plate is investigated in both the cases of water of finite and infinite depths. Using Green's function technique, the boundary value problem is converted into a system of three Fredholm type integral equations. Various integrals associated with the integral equations are evaluated using appropriate Gauss quadrature formulae and the system of integral equations are converted into a system of algebraic equations. Further, using Green's second identity, expressions for the reflection and transmission coefficients are obtained in terms of the velocity potential and its normal derivative. Energy balance relations for wave scattering by flexible porous plates and permeable membrane barriers are derived using Green's identity and used to check the correctness of the computational results. From the general formulation of the submerged plate, wave scattering by partial plates such as (i) surface-piercing and (ii) bottom-standing plates are studied as special cases. Further, oblique wave scattering by bottom-standing and surface-piercing porous membrane barriers are studied in finite water depth as particular cases of the flexible plate problem. Various numerical results are presented to study the effect of structural rigidity, angle of incidence, membrane tension, structural length, porosity and water depth on wave scattering. It is found that wave reflection is more for a surface-piercing flexible porous plate in infinite water depth compared to finite water depth and opposite trend is observed for a submerged flexible porous plate. For a surface-piercing nonpermeable membrane, zeros in transmission coefficient are observed for waves of intermediate water depth which disappear with the inclusion of porosity. The study reveals that porosity has small influence on the wave-induced excitation of the structure with higher flexibility but it tends to reduce the deflection of a stiffer structure. In case of partial flexible plates and membrane barriers, irrespective of the gap length, full transmission occurs due to wave diffraction through the gap in the very long wave regime while, full reflection occurs by complete flexible impermeable barriers for similar wave condition.

]]>The Kidder problem is with and where . This looks challenging because of the square root singularity. We prove, however, that for all . Other very simple but very accurate curve fits and bounds are given in the text; . Maple code for a rational Chebyshev pseudospectral method is given as a table. Convergence is geometric until the coefficients are when the coefficients . An initial-value problem is obtained if is known; the slope Chebyshev series has only a fourth-order rate of convergence until a simple change-of-coordinate restores a geometric rate of convergence, empirically proportional to . Kidder's perturbation theory (in powers of α) is much inferior to a delta-expansion given here for the first time. A quadratic-over-quadratic Padé approximant in the exponentially mapped coordinate predicts the slope at the origin very accurately up to about . Finally, it is shown that the singular case can be expressed in terms of the solution to the Blasius equation.

]]>This paper concerns the separable solutions of a generalized Burgers equation. Existence of separable solutions to the generalized Burgers equation is proved under certain conditions. A careful numerical study shows that these separable solutions of the generalized Burgers equation describe the large time asymptotic behavior of solutions of initial boundary value problems.

]]>A numerical method is proposed for computing time-periodic and relative time-periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameters in terms of the solution through quasi-Rayleigh quotients, so that the resulting integrodifferential equation is for the time-periodic solution only. Then this equation is computed in the combined spatiotemporal domain as a boundary value problem by Newton-conjugate-gradient iterations. The proposed method applies to both stable and unstable time-periodic solutions; its numerical accuracy is spectral; it is fast-converging; its memory use is minimal; and its coding is short and simple. As numerical examples, this method is applied to the Kuramoto–Sivashinsky equation and the cubic-quintic Ginzburg–Landau equation, whose time-periodic or relative time-periodic solutions with spatially periodic or spatially localized profiles are computed. This method also applies to systems of ordinary differential equations, as is illustrated by its simple computation of periodic orbits in the Lorenz equations. MATLAB codes for all numerical examples are provided in the Appendices to illustrate the simple implementation of the proposed method.

]]>A new method for analyzing linear elliptic partial differential equations in the interior of a convex polygon was developed in the late 1990s. This method does not rely on the classical approach of separation of variables and on the use of classical integral transforms and therefore is well suited for the investigation of the biharmonic equation. Here, we present a novel integral representation of the solution of the biharmonic equation in the interior of a convex polygon. This representation contains certain free parameters and therefore is more general than the one presented in [1]. For a given boundary value problem, by choosing these free parameters appropriately, one can obtain the simplest possible representation for the solution. This representation still involves certain unknown boundary values, thus for this formula to become effective it is necessary to characterize the unknown boundary values in terms of the given boundary conditions. This requires the investigation of certain relations refereed to as the *global relations*. A general approach for analyzing these relations is illustrated by solving several problems formulated in the interior of a semistrip. In addition, for completeness, similar results are presented for the Poisson equation by employing an integral representation for the Laplace equation which is more general than the one derived in the late 1990s.