*two independent small parameters*, which represent the ratio of two characteristic time-scales and the spatial amplitude of oscillations; the related scaling of the variables and parameters uses the Strouhal number;

A variable-coefficient Korteweg–de Vries equation is used to model the deformation of nonlinear periodic and solitary water waves propagating on a unidirectional background current, which is either flowing in the same direction as the waves, or is opposing them. As well as the usual form of the Korteweg–de Vries equation, an additional term is needed when the background current has vertical shear. This term, which has hitherto been often neglected in the literature, is linear in the wave amplitude and represents possible nonconservation of wave action. An additional feature is that horizontal shear in the background current is inevitably accompanied by a change in total fluid depth, to conserve mass, and this change in depth is a major factor in the deformation of the waves. Using a combination of asymptotic analyses and numerical simulations, it is found that waves grow on both advancing and opposing currents, but the growth is greater when the current is opposing.

]]>According to Dyson's threefold way, from the viewpoint of global time reversal symmetry, there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary, and symplectic ensembles, denoted COE, CUE, and CSE, respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability , we take up the problem of calculating the first two terms in the scaled large *N* expansion of the spacing distributions. It is well known that the leading term admits a characterization in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterizations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE, there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case, some further statistics are similarly analyzed.

We study planar nematic equilibria on a two-dimensional annulus with strong and weak tangent anchoring, in the Oseen–Frank theoretical framework. We analyze a radially invariant defect-free state and compute analytic stability criteria for this state in terms of the elastic anisotropy, annular aspect ratio, and anchoring strength. In the strong anchoring case, we define and characterize a new spiral-like equilibrium which emerges as the defect-free state loses stability. In the weak anchoring case, we compute stability diagrams that quantify the response of the defect-free state to radial and azimuthal perturbations. We study sector equilibria on sectors of an annulus, including the effects of weak anchoring and elastic anisotropy, giving novel insights into the correlation between preferred numbers of boundary defects and the geometry. We numerically demonstrate that these sector configurations can approximate experimentally observed equilibria with boundary defects.

]]>In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space , then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the defocusing CSP equation from the single-component extended Kadomtsev-Petviashvili (KP) hierarchy by the reduction method. As a by-product, the *N*-dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.

In a series of recent works by Demirkaya et al., stability analysis for the static kink solutions to the one-dimensional continuous and discrete Klein–Gordon equations with a -symmetric perturbation has been performed. In the present paper, we study two-dimensional (2D) quadratic operator pencil with a small localized perturbation. Such an operator pencil is motivated by the stability problem for the static kink in 2D Klein–Gordon field taking into account spatially localized -symmetric perturbation, which is in the form of viscous friction. Viscous regions with positive and negative viscosity coefficient are balanced. For the considered operator pencil, we show that its essential spectrum has certain critical points generating eigenvalues under the perturbation. Our main results are sufficient conditions ensuring the existence or absence of such eigenvalues as well as the asymptotic expansions for these eigenvalues if they exist.

]]>The aim of this paper is: using the two-timing method to study and classify the multiplicity of distinguished limits and asymptotic solutions for the advection equation with a general oscillating velocity field. Our results are:

- (i)the dimensionless advection equation that contains
*two independent small parameters*, which represent the ratio of two characteristic time-scales and the spatial amplitude of oscillations; the related scaling of the variables and parameters uses the Strouhal number; - (ii)an infinite sequence of
*distinguished limits*has been identified; this sequence corresponds to the successive degenerations of a*drift velocity*; - (iii)we have derived the averaged equations and the oscillatory equations for the first
*four distinguished limits*; derivations are performed up to the fourth orders in small parameters; - (v)we have shown, that
*each distinguished limit*generates an infinite number of*parametric solutions*; these solutions differ from each other by the slow time-scale and the amplitude of the prescribed velocity; - (vi)we have discovered the inevitable presence of
*pseudo-diffusion*terms in the averaged equations, pseudo-diffusion appears as a Lie derivative of the averaged tensor of quadratic displacements; we have analyzed the matrix of pseudo-diffusion coefficients and have established its degenerated form and hyperbolic character; however, for one-dimensional cases, the pseudo-diffusion can appear as ordinary diffusion; - (vii)the averaged equations for four different types of oscillating velocity fields have been considered as the examples of different drifts and pseudo-diffusion;
- (viii)our main methodological result is the introduction of a logical order into the area and classification of an infinite number of asymptotic solutions; we hope that it can help in the study of the similar problems for more complex systems;
- (ix)our study can be used as a test for the validity of the two-timing hypothesis, because in our calculations we do not employ any additional assumptions.

A nonlocal nonlinear Schrödinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced “potential” is symmetric thus the nonlocal NLS equation is also symmetric. In this paper, new *reverse space-time* and *reverse time* nonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space-time, and in some cases reverse time, nonlocal NLS, modified Korteweg-deVries (mKdV), sine-Gordon, (1 + 1) and (2 + 1) dimensional three-wave interaction, derivative NLS, “loop soliton,” Davey–Stewartson (DS), partially symmetric DS and partially reverse space-time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space-time and reverse time nonlocal discrete nonlinear Schrödinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painlevé type equations are derived from the reverse space-time and reverse time nonlocal NLS equations.

Employing matrix formulation and decomposition technique, we theoretically provide essential necessary and sufficient conditions for the existence of general analytical solutions for *N*-dimensional damped compressible Euler equations arising in fluid mechanics. We also investigate the effect of damping on the solutions, in terms of density and pressure. There are two merits of this approach: First, this kind of solutions can be expressed by an explicit formula and no additional constraint on the dimension of the damped compressible Euler equations is needed. Second, we transform analytically the process of solving the Euler equations into algebraic construction of an appropriate matrix . Once the required matrix is chosen, the solution is obtained directly. Here, we overcome the difficulty of solving matrix differential equations by utilizing decomposition and reduction techniques. In particular, we find two important solvable relations between the dimension of the Euler equations and the pressure parameter: in the damped case and for no damping. These two cases constitute a full range of solvable parameter . Special cases of our results also include several interesting conclusions: (1) If the velocity field is a linear transformation on the Euclidean spatial vector , then the pressure *p* is a quadratic form of . (2) The damped compressible Euler equations admit the Cartesian solutions if is an antisymmetric matrix. (3) The pressure *p* possesses radially symmetric forms if is an antisymmetrical orthogonal matrix.

We study the propagation of weakly nonlinear waves in nonideal fluids, which exhibit mixed nonlinearity. A method of multiple scales is used to obtain a transport equation from the Navier–Stokes equations, supplemented by the equation of state for a van der Waals fluid. Effects of van der Waals parameters on the wave evolution, governed by the transport equation, are investigated.

]]>The squared singular values of the product of *M* complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G-functions, or equivalently hypergeometric functions , also referred to as hyper-Bessel functions. In the case it is well known that the corresponding gap probability for no squared singular values in (0, *s*) can be evaluated in terms of a solution of a particular sigma form of the Painlevé III' system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalized this formalism to general , but has not exhibited its reduction. After detailing the necessary working in the case , we consider the problem of reducing the 12 coupled differential equations in the case to a single differential equation for the resolvent. An explicit fourth-order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third-order nonlinear equation. The small and large *s* asymptotics of the fourth-order equation are discussed, as is a possible relationship of the systems to so-called four-dimensional Painlevé-type equations.

This article deals with a forced Burgers equation (FBE) subject to the initial function, which is continuous and summable on . Large time asymptotic behavior of solutions to the FBE is determined with precise error estimates. To achieve this, we construct solutions for the FBE with a different initial class of functions using the method of separation of variables and Cole–Hopf like transformation. These solutions are constructed in terms of Hermite polynomials with the help of similarity variables. The constructed solutions would help us to pick up an asymptotic approximation and to show that the magnitude of the difference function of the true and approximate solutions decays algebraically to 0 for large time.

]]>The soliton solutions of the Degasperis–Procesi equations are constructed by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by Hirota's method.

]]>We mainly study a system of two coupled nonlinear Schrödinger equations where one equation includes gain and the other one includes losses. This model constitutes a generalization of the model of pulse propagation in birefringent optical fibers. We aim in this study at partially answering a question of some authors in [1]: “Is the *H*^{1}-norm of the solution globally bounded in the Manakov case, when ?” We found that in the Manakov case, and when , the solution stays in , and also that the *H*^{1}-norm of the solution cannot blow up in finite time. In the Manakov case, an estimate of the total energy is provided, which is different from that has been given in [1]. These results are corroborated by numerical results that have been obtained with a finite element solver well adapted for that purpose.