We present an approach for analyzing initial-boundary value problems which are formulated on the finite interval (, where *L* is a positive constant) for integrable equation whose Lax pairs involve 3 × 3 matrices. Boundary value problems for integrable nonlinear evolution partial differential equations (PDEs) can be analyzed by the unified method introduced by Fokas and developed by him and his collaborators. In this paper, we show that the solution can be expressed in terms of the solution of a 3 × 3 Riemann–Hilbert problem (RHP). The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions , and , which in turn are defined in terms of the initial values, boundary values at , and boundary values at , respectively. However, these spectral functions are not independent; they satisfy a global relation. Here, we show that the characterization of the unknown boundary values in terms of the given initial and boundary data is explicitly described for a nonlinear evolution PDE defined on the interval. Also, we show that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained for the case of boundary value problems formulated on the half-line.

We study the asymptotic behavior of sharp front solutions arising from the nonlinear diffusion equation , where the diffusivity is an exponential function . This problem arises, for example, in the study of unsaturated flow in porous media where θ represents the liquid saturation. For physical parameters corresponding to actual porous media, the diffusivity at the residual saturation is so that the diffusion problem is nearly degenerate. Such problems are characterized by wetting fronts that sharply delineate regions of saturated and unsaturated flow, and that propagate with a well-defined speed. Using matched asymptotic expansions in the limit of large β, we derive an analytical description of the solution that is uniformly valid throughout the wetting front. This is in contrast with most other related analyses that instead truncate the solution at some specific wetting front location, which is then calculated as part of the solution, and beyond that location, the solution is undefined. Our asymptotic analysis demonstrates that the solution has a four-layer structure, and by matching through the adjacent layers, we obtain an estimate of the wetting front location in terms of the material parameters describing the porous medium. Using numerical simulations of the original nonlinear diffusion equation, we demonstrate that the first few terms in our series solution provide approximations of physical quantities such as wetting front location and speed of propagation that are more accurate (over a wide range of admissible β values) than other asymptotic approximations reported in the literature.

]]>Regular roguing is an effective method to control plant virus diseases. In this paper, a compartmental mathematical model is established to represent the dynamics of plant disease in a periodic environment, including impulsive roguing control strategy. The basic reproductive number and its relation to the persistence of the disease is discussed via using next infection operation. Numerical simulations are performed to demonstrate the theoretical findings, and to illustrate the effect of control measures. Our results show that, (i) when the infection rate is high, it may be impossible to eradicate the disease by simply roguing the infectious plant, so how to effectively identify the latent plant is a key issue in disease control, (ii) increasing the replanting rate is bad for disease control, (iii) the published autonomous research model with continuous roguing may overestimate the infectious risk inherent to impulsive control.

]]>Large-scale simulations in spherical geometries require associated quadrature formulas. Classical approaches based on tabulated weights are limited to specific quasi-uniform distributions of relatively low numbers of nodes. By using a radial basis function-generated finite differences (RBF-FD)-based approach, the proposed algorithm creates quadrature weights for *N* arbitrarily scattered nodes in only operations.

We explore the possibility of multi-site breather states in a nonlinear Klein–Gordon lattice to become nonlinearly unstable, *even if* they are found to be spectrally stable. The mechanism for this nonlinear instability is through the resonance with the wave continuum of a multiple of an internal mode eigenfrequency in the linearization of excited breather states. For the nonlinear instability, the internal mode must have its Krein signature opposite to that of the wave continuum. This mechanism is not only theoretically proposed, but also numerically corroborated through two concrete examples of the Klein–Gordon lattice with a soft (Morse) and a hard (ϕ^{4}) potential. Compared to the case of the nonlinear Schrödinger lattice, the Krein signature of the internal mode relative to that of the wave continuum may change depending on the period of the multi-site breather state. For the periods for which the Krein signatures of the internal mode and the wave continuum coincide, multi-site breather states are observed to be nonlinearly stable.

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight

with parameters and , and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation. Further we derive a second-order linear ordinary differential equation and a differential-difference equation satisfied by the generalized Freud polynomials.

]]>We compute the scattering data of the Benjamin–Ono equation for arbitrary rational initial conditions with simple poles. Specifically, we obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient.

]]>We revisit in this paper the strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with a free surface proposed by Choi and Camassa and Barros et al. . Its solitary-wave solutions were the object of the work by Barros and Gavrilyuk , who proved that such solutions are governed by a Hamiltonian system with two degrees of freedom. A detailed analysis of the critical points of the system is presented here, leading to some new results. It is shown that conjugate states for the long wave model are the same as those predicted by the fully nonlinear Euler equations. Some emphasis will be given to the baroclinic mode, where interfacial waves are known to change polarity according to different values of density and depth ratios. A critical depth ratio separates these two regimes and its analytical expression is derived directly from the model. In addition, we prove that such waves cannot exist throughout the whole range of speeds.

]]>The purpose of this paper is to construct the inverse scattering transform for the focusing Ablowitz-Ladik equation with nonzero boundary conditions at infinity. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann-Hilbert problem on a doubly connected curve in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz-Ladik potential.

]]>We study solutions of the Cauchy problem for nonlinear Schrödinger system in with nonlinear coupling and linear coupling modeling synthetic magnetic field in spin-orbit coupled Bose–Einstein condensates. Three main results are presented: a proof of the local existence, a proof of the sufficient condition for the blowup result in finite time for some solutions, and the persistence of the nonlinear dynamics in the limit where the spin-orbit coupling converges to zero.

]]>A technique to manufacture *solvable* variants of the “goldfish” many-body problem is introduced, and several many-body problems yielded by it are identified and discussed, including cases featuring *multiperiodic* or *isochronous* dynamics.

By introducing an elliptic vortex ansatz, the 2+1-dimensional two-layer fluid system is reduced to a finite-dimensional nonlinear dynamical system. Time-modulated variables are then introduced and multicomponent Ermakov systems are isolated. The latter is shown to be also Hamiltonian, thereby admitting general solutions in terms of an elliptic integral representation. In particular, a subclass of vortex solutions is obtained and their behaviors are simulated. Such solutions have recently found applications in oceanic and atmospheric dynamics. Moreover, it is proved that the Hamiltonian system is equivalent to the stationary nonlinear cubic Schrödinger equations coupled with a Steen-Ermakov-Pinney equation.

]]>In this paper, we consider an initial-value problem for Burgers' equation with variable coefficients

where *x* and *t* represent dimensionless distance and time, respectively, and , are given functions of *t*. In particular, we consider the case when the initial data have algebraic decay as , with as and as . The constant states and are problem parameters. Two specific initial-value problems are considered. In initial-value problem 1 we consider the case when and , while in initial-value problem 2 we consider the case when and . The method of matched asymptotic coordinate expansions is used to obtain the large-*t* asymptotic structure of the solution to both initial-value problems over all parameter values.

Chebyshev and Legendre polynomial spectral methods are bedeviled by highly nonuniform grids. The separation between nearest neighbors of an *N*-point grid at the center of the interval is larger than the spacing of a uniform grid with the same number of points. Quasi-Uniform Spectral Schemes (QUSS) redistribute grid points and choose basis functions in order to recover this factor of as nearly as possible while retaining a high density of points near the endpoints to avoid the horrors of the Gibbs or Runge Phenomenon. Here, we introduce a systematic approach, dubbed “mapped cosine bases,” that embraces the widely used Kosloff/Tal-Ezer functions as a special case. The mapped cosine approach uses grid points that are the images of a uniform grid under the coordinate mapping . Here, we show how to generalize the well-known graphical construction of the Chebyshev grid using a circle to QUSS mappings using a generalized ellipse. This provides a way to visualize the maps and grids and the subtle differences between different mappings of the mapped cosine family. We illustrate and compare the Kosloff/Tal-Ezer map with two new maps that use elliptic integrals and Jacobian theta functions, respectively. We show that the elliptic integral grid is an asymptotic approximation to the usual grid for prolate spheroidal functions. This suggests the conjecture that one can obtain the benefits of a prolate basis without the complications of prolate functions by using mapped polynomials instead.

In the coastal ocean, the interaction of barotropic tidal currents with topographic features such as the continental shelf, sills in narrow straits, and bottom ridges are often observed to generate large amplitude, horizontally propagating internal solitary waves. These are long nonlinear waves and hence can be modeled by equations of the Korteweg–de Vries type. Typically they occur in regions of variable bottom topography, with the consequence that the appropriate nonlinear evolution equation has variable coefficients. Further, as these waves can be long-lived it is necessary to take account of the effects of the Earth's background rotation. We review this family of model evolution equations and some of their pertinent solutions, obtained both asymptotically and numerically.

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