The Large‐Time Solution of Burgers' Equation with Time‐Dependent Coefficients. I. The Coefficients Are Exponential Functions

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.     

Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line

Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.     

The Kidder Equation:

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.     

The Large‐Time Solution of Burgers' Equation with Time‐ Dependent Coefficients. II. Algebraic Coefficients

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, while , are given continuous functions of t ( > 0). In particular, we consider the case when the initial data has algebraic decay as , with as and as . The constant states and are problem parameters. We focus attention on the case when (with ) and . The method of matched asymptotic coordinate expansions is used to obtain the large‐t asymptotic structure of the solution to the initial‐value problem over all parameter values.     

Multiple Homoclinic Solutions for Second‐Order Perturbed Hamiltonian Systems

In this paper, we study the second‐order perturbed Hamiltonian systems where is a parameter, is positive definite for all but unnecessarily uniformly positive definite for , and W is either asymptotically quadratic or superquadratic in x as . Based on variational methods, we prove the existence of at least two nontrivial homoclinic solutions for the above system when small enough.     

The Cartesian Vector Solutions for the N‐Dimensional Compressible Euler Equations

In this paper, based on matrix and curve integration theory, we theoretically show the existence of Cartesian vector solutions for the general N‐dimensional compressible Euler equations. Such solutions are global and can be explicitly expressed by an appropriate formulae. One merit of this approach is to transform analytically solving the Euler equations into algebraically constructing an appropriate matrix . Once the required matrix is chosen, the solution is directly obtained. Especially, we find an important solvable relation between the dimension of equations and pressure parameter, which avoid additional independent constraints on the dimension N in existing literatures. Special cases of our results also include some interesting conclusions: (1) If the velocity field is a linear transformation on , then the pressure p is a relevant quadratic form. (2) The compressible Euler equations admit the Cartesian solutions if is an antisymmetric matrix. (3) The pressure p possesses radial symmetric form if is an antisymmetrically orthogonal matrix.     

The Analytical Solutions for the N‐Dimensional Damped Compressible Euler 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.     

Nonlocality and the Inverse Scattering Transform for the Pavlov Equation

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is nonlocal, and the proper choice of integration constants should be the one dictated by the associated inverse scattering transform (IST). Using the recently made rigorous IST for vector fields associated with the so‐called Pavlov equation , in this paper we establish the following. 1. The nonlocal term arising from its evolutionary form corresponds to the asymmetric integral . 2. Smooth and well‐localized initial data evolve in time developing, for , the constraint , where . 3. Because no smooth and well‐localized initial data can satisfy such constraint at , the initial () dynamics of the Pavlov equation cannot be smooth, although, because it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results should be successfully used in the study of the nonlocality of other basic examples of integrable dispersionless PDEs in multidimensions.     

Asymptotics of the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomials are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points , N being a fixed positive integer. By using a double integral representation, we have recently obtained asymptotic expansions for in the double scaling limit, namely, and , where and ; see [8]. In this paper, we continue to investigate the behavior of these polynomials when the parameter b approaches the endpoints of the interval (0, 1). While the case is relatively simple (because it is very much like the case when b is fixed), the case is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities n, x, and , and different special functions have been used as approximants, including Airy, Bessel, and Kummer functions.     

Localized Analytical Solutions and Parameters Analysis in the Nonlinear Dispersive Gross–Pitaevskii Mean‐Field GP (m,n) Model with Space‐Modulated Nonlinearity and Potential

The novel nonlinear dispersive Gross–Pitaevskii (GP) mean‐field model with the space‐modulated nonlinearity and potential (called GP equation) is investigated in this paper. By using self‐similar transformations and some powerful methods, we obtain some families of novel envelope compacton‐like solutions spikon‐like solutions to the GP equation. These solutions possess abundant localized structures because of infinite choices of the self‐similar function . In particular, we choose as the Jacobi amplitude function and the combination of linear and trigonometric functions of space x so that the novel localized structures of the GP(2, 2) equation are illustrated, which are much different from the usual compacton and spikon solutions reported. Moreover, it is shown that GP(m, 1) equation with linear dispersion also admits the compacton‐like solutions for the case and spikon‐like solutions for the case .     

On Airy Solutions of the Second Painlevé Equation

In this paper, we discuss Airy solutions of the second Painlevé equation (P_II) and two related equations, the Painlevé XXXIV equation () and the Jimbo–Miwa–Okamoto σ form of P_II (S_II), are discussed. It is shown that solutions that depend only on the Airy function have a completely different structure to those that involve a linear combination of the Airy functions and . For all three equations, the special solutions that depend only on are tronquée solutions, i.e., they have no poles in a sector of the complex plane. Further, for both and S_II, it is shown that among these tronquée solutions there is a family of solutions that have no poles on the real axis.     

Initial‐Boundary Value Problem for Integrable Nonlinear Evolution Equation with 3 × 3 Lax Pairs on the Interval

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.     

A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV‐Like Eigenvalue Problems

The Hamiltonian–Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem , where is skew‐symmetric and is self‐adjoint. If has a bounded inverse the index is well established, and it is given by the number of negative eigenvalues of the operator constrained to act on some finite‐codimensional subspace. There is an important class of problems—namely, those of KdV‐type—for which does not have a bounded inverse. In this paper, we overcome this difficulty and derive the index for eigenvalue problems of KdV‐type. We use the index to discuss the spectral stability of homoclinic traveling waves for KdV‐like problems and Benjamin—Bona—Mahony‐type problems.     

Ground State and Geometrically Distinct Solitons of Discrete Nonlinear Schrödinger Equations with Saturable Nonlinearities

We study the discrete nonlinear equation where (the spectrum of L) and is asymptotically linear as for all . We obtain the existence of ground state solitons and the existence of infinitely many pairs of geometrically distinct solitons of this equation. Our method is based on the generalized Nehari manifold method developed recently by Szulkin and Weth. To the best of our knowledge, this technique has not been used for discrete equations with saturable nonlinearities.     

Solutions to ABS Lattice Equations via Generalized Cauchy Matrix Approach

The usual Cauchy matrix approach starts from a known plain wave factor vector and known dressed Cauchy matrix . In this paper, we start from a determining matrix equation set with undetermined and . From the determining equation set we can build shift relations for some defined scalar functions and then derive lattice equations. The determining equation set admits more choices for and and in the paper we give explicit formulae for all possible and . As applications, we get more solutions than usual multisoliton solutions for many lattice equations including the lattice potential KdV equation, the lattice potential modified KdV equation, the lattice Schwarzian KdV equation, NQC equation, and some lattice equations in ABS list.     

Singular Values of Products of Ginibre Random Matrices

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.     

The Kontorovich–Lebedev Transform as a Map between d‐Orthogonal Polynomials

A slight modification of the Kontorovich–Lebedev transform is an auto‐morphism on the vector space of polynomials. The action of this ‐transform over certain polynomial sequences will be under discussion, and a special attention will be given to the d‐orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the ‐transform of a 2‐orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose ‐transform is a d‐orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and d is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.     

Bifurcation of Soliton Families from Linear Modes in Non‐‐Symmetric Complex Potentials

Continuous families of solitons in the nonlinear Schrödinger equation with non‐‐symmetric complex potentials and general forms of nonlinearity are studied analytically. Under a weak assumption, it is shown that stationary equations for solitons admit a constant of motion if and only if the complex potential is of a special form , where is an arbitrary real function. Using this constant of motion, the second‐order complex soliton equation is reduced to a new second‐order real equation for the amplitude of the soliton. From this real soliton equation, a novel perturbation technique is employed to show that continuous families of solitons bifurcate out from linear discrete modes in these non‐‐symmetric complex potentials. All analytical results are corroborated by numerical examples.     

Dynamics in ‐Symmetric Honeycomb Lattices with Nonlinearity

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.