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.     

A Solvable Nonlinear Wave Equation

It is pointed out that the nonlinear wave equation can be solved by quadratures. Here a and c are constants, A(y) and B(y) (arbitrary) functions; a t-dependence of all these quantities can also be accommodated. This wave equation can also be rewritten in the (purely differential) form via the substitutions .     

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.     

Conservation Laws for the Fully Nonlinear Long Wave Equations

The fully nonlinear long wave equations describe the motion over a flat bottom of a two-dimensional inviscid fluid with a free surface in a gravitational field in the long wave approximation. These equations are shown to possess an infinite number of conservation laws (in two space dimensions) in the form The conserved densities T and the fluxes ?X and ?Y are polynomials in the height h and the horizontal and vertical components of velocity, u and v, and also in integrals of powers of u. The method of proof is a modification of the method recently devised by D. J. Benney to prove that these same equations possess an infinite number of conservation laws (in one space dimension) in the form where T and X are polynomials in the height h and integrals of powers of u. Conservation laws which explicitly contain x and t are also given.     

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.     

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.     

Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes an electron trapped in its own hole. The interesting mathematical aspect of the problem is that is not convex, and usual methods to show existence and uniqueness of the minimum do not apply. By using symmetrie decreasing re arrangement inequalities we are able to prove existence and uniqueness (modulo translations) of a minimizing ?. To prove uniqueness a strict form of the inequality, which we believe is new, is employed.     

The Evolution of Finite Amplitude Solitary Rossby Waves on a Weak Shear

The evolution equation is derived for finite amplitude, long Rossby waves on a weak shear generalizing an earlier version given by Benney [1].     

Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight

We study the Hankel determinants associated with the weight where , , , is analytic in a domain containing [ ? 1, 1] and for . In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as and . We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.     

The Asymptotic Solution of a Connection Problem of a Second Order Ordinary Differential Equation

The solutions of the equation are discussed in the limit ρ → 0. The solutions which oscillate about ? |t| as t → ∞ have asymptotic expansions whose leading terms are where Ã₊, , Ã_?, and are constants. The connection problem is to determine the asymptotic expansion at + ∞. In other words, we wish to find (Ã₊, ) as functions of Ã_? and The nonlinear solutions with Ã_± not small are analyzed by using the method of averaging. It is shown that this method breaks down for small amplitudes. In this case a solution can be obtained on [0, ∞) as a small amplitude perturbation about the exact nonoscillating solution W(t) whose asymptotic expansion is This is a solution of (1) which corresponds to Ã₊ ≡ 0 in (2). A quantity which determines the scale of the small amplitude response is ?W'(0). This quantity is found to be exponentially small. The determination of this constant is shown to reduce to a solution of the equation for the first Painlevé transcendent. The asymptotic behavior of the required solution is determined by solving an integral equation.     

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.     

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 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.     

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.     

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.     

Asymptotics beyond All Orders in a Model of Crystal Growth

In its simplest form, the geometric model of crystal growth is a third-order, nonlinear, ordinary differential equation for θ(s, ε): A needle crystal is a solution that satisfies boundary conditions The geometric model admits a needle-crystal solution for ε = 0; for small ε, it admits an asymptotic expansion that is valid to all orders for such a solution. Even so, we prove that the geometric model in this form admits no needle crystal for any small, nonzero ε, a fact that lies beyond all orders of the asymptotic expansion. A more complicated version of the geometric model is where α represents crystalline anisitropy. We show that for 0 < α < 1, the geometric model admits needle crystals for a discrete set of values of α. The number of such values of α increases like ε^?1 as ε → 0.     

Solitary-Wave Interactions in Elastic Rods

The propagation of longitudinal deformation waves in an elastic rod is modelled by the nonlinear partial differential equation with p = 3 or 5. This equation is first derived under a range of possible constraints. We then show that this equation and even certain generalizations do not pass the Painlevé test, and hence are probably not completely integrable. Finally, we study the head-on collision of two equal solitary waves numerically and also asymptotically for small and large amplitude.     

On Mean Convergence of Fourier-Bessel Series of Negative Order

The expansion of f ∈ L^p(0, 1) Fourier series of Bessel functions of order converges to f in L^p whenever Let be the space of p-integrable functions with respect to the measure t dt and where {s_n}, n = 1, 2, …, is the set of positive zeros of J_v. Then, the expansion of in a Fourier series of functions ψ_n, ?1 < ν < ?½, converges to in whenever