Transformations between Nonlocal and Local Integrable Equations

Recently, a number of nonlocal integrable equations, such as the ‐symmetric nonlinear Schrödinger (NLS) equation and ‐symmetric Davey–Stewartson equations, were proposed and studied. Here, we show that many of such nonlocal integrable equations can be converted to local integrable equations through simple variable transformations. Examples include these nonlocal NLS and Davey–Stewartson equations, a nonlocal derivative NLS equation, the reverse space‐time complex‐modified Korteweg–de Vries (CMKdV) equation, and many others. These transformations not only establish immediately the integrability of these nonlocal equations, but also allow us to construct their Lax pairs and analytical solutions from those of the local equations. These transformations can also be used to derive new nonlocal integrable equations. As applications of these transformations, we use them to derive rogue wave solutions for the partially ‐symmetric Davey–Stewartson equations and the nonlocal derivative NLS equation. In addition, we use them to derive multisoliton and quasi‐periodic solutions in the reverse space‐time CMKdV equation. Furthermore, we use them to construct many new nonlocal integrable equations such as nonlocal short pulse equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa–Satsuma equations.     

Asymptotics beyond All Orders for a Certain Type of Nonlinear Oscillators

This paper shows that a special class of smooth nonlinear oscillators, called bisuperlinear, has a family of adiabatically symmetric solutions. This was motivated by a problem studied in sloshing water waves. A potential application of the work is to compute the nontrivial leading order term of the adiabatic invariants for a certain type of nonlinear nearly periodic Hamiltonian systems.     

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 .     

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 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.     

Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation

Complex analytical structure of Stokes wave for two‐dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave prop agating with the constant velocity. Simulations with the quadruple (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with radians angle on the crest. A conformal map is used that maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half‐plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half‐plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase in the numerical precision and subsequent increase in the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one square‐root branch point per horizontal spatial period λ of Stokes wave located at the distance from the real line. The increase in the scaled wave height from the linear limit to the critical value marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave (also called the Stokes wave of the greatest height). Here, H is the wave height from the crest to the trough in physical variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least 10^?26. The number of poles in tables increases from a few for near‐linear Stokes wave up to about hundred poles to highly nonlinear Stokes wave with      

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.     

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.     

Solutions of Nonlocal Equations Reduced from the AKNS Hierarchy

In this paper, nonlocal reductions of the Ablowitz–Kaup–Newell–Suger (AKNS) hierarchy are collected, including the nonlocal nonlinear Schrödinger hierarchy, nonlocal modified Korteweg‐de Vries hierarchy, and nonlocal versions of the sine‐Gordon equation in nonpotential form. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics, we illustrate new interaction of two‐soliton solutions of the reverse‐t nonlinear Schrödinger equation. Although as a single soliton, it is stationary that two solitons travel along completely symmetric trajectories in plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal 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.     

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.     

On the Generalizations of a Theorem of Beppo Levi

In this paper we give two generalizations of a theorem of Beppo Levi ([1], p. 347, Formula (12)). This theorem affirms that, under certain conditions, the following assertion is true: where φ(x) is a function that verifies φ(0) > 0; f(x) is defined and bounded in the interval (a, b) and continuous in the point 0 with f(0) ≠ 0; f(x) and φ(x) are integrable functions in the interval [a, b]; c >, 0 and υ > 1. This problem was studied by Laplace [2], Darboux [3], Stieltjes [4], Lebesgue [5], Romanovsky [6], and Fowler [7]. The first generalization (Section 1, Theorem 1.2, Formula (1.35)) says that, under certain conditions, the following formula is valid: where φ_n(x) is a sequence of functions and Bn(a) designates the n-dimentional ball of radius a and center in the origin. The extension follows by Romanovsky's method. The absolute maximum of φ(x) in the extremes of the interval of definition is treated in the second generalization of the Theorem of Beppo Levi (Section 2, Theorem 2.2, Formulas (2.1), (2.2)). We note that Beppo Levi proves this assertion in the interior of the interval.     

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.     

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 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.     

Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment,from Finite n to Double Scaling

In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely, the probability that the interval is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by , , and . We find that each one satisfies a second‐order differential equation. We show that after a double scaling, the large second‐order differential equation in the variable a with n as parameter satisfied by can be reduced to the Jimbo–Miwa–Okamoto σ form of the Painlevé V equation.     

Asymptotic Expansions for Second-Order Linear Difference Equations,II

Infinite asymptotic expansions are derived for the solutions to the second-order linear difference equation where p and q are integers, a(n) and b(n) have power series expansions of the form for large values of n, and a₀ ≠ 0, b₀ ≠ 0. Recurrence relations are also given for the coefficients in the asymptotic solutions. Our proof is based on the method of successive approximations. This paper is a continuation of an earlier one, in which only the special case p ≤ 0 and q = 0 is considered.     

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.     

Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

We consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we rederive the formulas for the classical periodic traveling waves, while for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. In both cases, we show spectral stability, for all values of the parameters. This is achieved by studying the nonstandard eigenvalue problems in the form , where is a Hill operator.