On the Ackerberg–O'Malley Resonance

In this paper, we continue our study of the boundary value problem where A , B are prescribed constants and  0 < ɛ≪ 1  is a small positive parameter. We assume that the coefficients   a ( x )  and   b ( x )  are sufficiently smooth functions with the behavior given by   a ( x ) ∼α x   and   b ( x ) ∼β  as   x → 0  . In our previous work, the problem has been studied for both  α > 0  and  α < 0  except for the cases  β/α= 1,2,3,…  when  α > 0  and  β/α= 0,−1,−2,…  when  α < 0  . In the present paper, we study these exceptional cases and obtain, by rigorous analysis, uniformly valid asymptotic solutions of the problem. From these solutions, we also show that the conditions in these exceptional cases are exactly the ones which are necessary and sufficient for the Ackerberg–O'Malley resonance.     

Uniform Asymptotics for Orthogonal Polynomials with Exponential Weights—the Riemann–Hilbert Approach

Let   Q ( x ) = q _{2 m} x ^{2 m} + q _{2 m −1} x ^{2 m −1}+⋯  be a polynomial of degree 2 m with   q _{2 m} > 0  , and let  {π _n ( x )} _{n ≥1}  be the sequence of monic polynomials orthogonal with respect to the weight   w ( x ) = e ^{− Q ( x )}  on     . Furthermore, let  α _n   and  β _n   denote the Mhaskar–Rakhmanov–Saff (MRS) numbers associated with Q ( x ). By using the Riemann–Hilbert approach, an asymptotic expansion is constructed for  π _n ( c_nz + d_n )  , which holds uniformly for all z bounded away from  (−∞, −1)  , where     and     .     

Shallow Water Wave Systems

In this article we study various systems that represent the shallow water wave equation
v_xxt + αvv_t − βv_x∂_x^-1 ( v_t ) − v_t − v_x = 0,
where (∂ _x ⁻¹ f )( x )=∫ _x ^∞ f ( y ) d y , and α and β are arbitrary, nonzero, constants. The classical method of Lie, the nonclassical method of Bluman and Cole [ J. Math. Mech. 18:1025 (1969)], and the direct method of Clarkson and Kruskal [ J. Math. Phys. 30:2201 (1989)] are each applied to these systems to obtain their symmetry reductions. It is shown that for both the nonclassical and direct methods unusual phenomena can occur, which leads us to question the relationship between these methods for systems of equations. In particular an example is exhibited in which the direct method obtains a reduction that the nonclassical method does not.     

Justification of a Perturbation Approximation of the Klein–Gordon Equation

Consider the nonlinear wave equation
u_tt − γ ² u_xx + f(u) = 0
with the initial conditions
u ( x ,0) = εφ ( x ), u _t( x ,0) = εψ ( x ),
where f ( u ) is either of the form f ( u )= c ² u −σ u ^{2 s +1}, s =1, 2,…, or an odd smooth function with f '(0)>0 and | f '( u )|≤ C ₀².The initial data φ( x )∈ C ² and ψ( x )∈ C ¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u ( x ,  t ; ɛ), and prove its boundedness in x ∈ R and t >0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u ( x ,  t ; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ³ uniformly for x ∈ R and 0≤ t ≤ T /ɛ², as long as ɛ is sufficiently small, T being an arbitrary positive number.     

Toward the Classification of Scalar Nonpolynomial Evolution Equations: Polynomiality in Top Three Derivatives

We prove that arbitrary (nonpolynomial) scalar evolution equations of order    m  ≥ 7  , that are integrable in the sense of admitting the canonical conserved densities   ρ⁽¹⁾, ρ⁽²⁾  , and   ρ⁽³⁾   introduced in [ 1 ], are polynomial in the derivatives    u _{m −  i}    for  i  = 0, 1, 2. We also introduce a grading in the algebra of polynomials in     u _k     with     k  ≥  m  − 2    over the ring of functions in     x ,  t ,  u , … ,  u _{m −3}    and show that integrable equations are scale homogeneous with respect to this grading .     

Movable Singularities of a Class of Nonlinear Ordinary Differential Equations of Arbitrary Order

In this paper we consider nonlinear ordinary differential equations   y ^{( n )}= F ( y ', y , x )  of arbitrary order   n ≥ 3  , where F is algebraic in   y , y '  and locally analytic in x . We prove that for   n > 3  these equations always admit movable branch points. In the case   n = 3  these equations admit movable branch points unless they are of the known class   y '= a ( x )( y ')²+ ( b ₂( x ) y ²+ b ₁( x ) y + b ₀( x )) y '+ ( c ₄( x ) y ⁴+ c ₃( x ) y ³+ c ₂( x ) y ²+ c ₁( x ) y + c ₀( x ))  , where   a ,  b_j ,  c_j   are locally analytic in x .     

Recursion Operators for a Class of Integrable Third-Order Evolution Equations

We consider   u_t = u ^α u_xxx + n ( u ) u_xu_xx + m ( u ) u ³ _x + r ( u ) u_xx + p ( u ) u ² _x + q ( u ) u_x + s ( u )  with  α= 0  and  α= 3  , for those functional forms of   m , n , p , q , r , s   for which the equation is integrable in the sense of an infinite number of Lie-Bäcklund symmetries. Recursion operators which are x - and t -independent that generate these infinite sets of (local) symmetries are obtained for the equations. A combination of potential forms, hodograph transformations, and x -generalized hodograph transformations are applied to the obtained equations.     

On Interfaces and Oscillatory Solutions of Higher-Order Semilinear Parabolic Equations with Non-Lipschitz Nonlinearities

We study local properties of solutions and their asymptotic extinction behavior for the fourth-order semilinear parabolic equation of diffusion–absorption type where p < 1, so that the absorption term is not Lipschitz continuous at u = 0. The Cauchy problem with bounded compactly supported initial data possesses solutions with finite interfaces, and we describe their oscillatory, sign changing properties for     . For p ∈ (0, 1), we also study positive solutions of the free-boundary problem with zero contact angle and zero-flux conditions. Finally, we describe families { f_k } of similarity extinction patterns   u_S ( x , t ) = ( T − t )^{1/(1− p )} f ( y )  , where   y = x /( T − t )^1/4  , that vanish in finite time, as   t → T ⁻∈ (0, ∞)  . Similar local and asymptotic properties are indicated for the sixth-order equation with source      

Asymptotic Analysis of a Perturbed Periodic Solution for the KdV Equation

We consider the solution of the Korteweg–de Vries (KdV) equation with periodic initial value where C , A , k , μ, and β are constants. The solution is shown to be uniformly bounded for all small ɛ, and a formal expansion is constructed for the solution via the method of multiple scales. By using the energy method, we show that for any given number   T > 0  , the difference between the true solution v ( x , t ; ɛ) and the N th partial sum of the asymptotic series is bounded by  ɛ ^{N +1}  multiplied by a constant depending on T and N , for all  −∞ < x < ∞, 0 ≤ t ≤ T /ɛ  , and  0 ≤ɛ≤ɛ₀  .     

Conditional Lie Bäcklund Symmetries and Sign-Invariants to Quasi-Linear Diffusion Equations

Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term u _t =[ u ^m ( u _x ) ⁿ ] _x + P ( u ) u _x + Q ( u ) , where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bäcklund conditional symmetry with characteristic η= u _xx + H ( u ) u ² _x + G ( u )( u _x )^{2− n} + F ( u ) u ^{1− n} _x and the Hamilton–Jacobi sign-invariant J = u _t + A ( u ) u ^{n +1} _x + B ( u ) u _x + C ( u ) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.     

Rates of Convergence to Self-Similar Solutions of Burgers' Equation

We study the large-time behavior of solutions to Burgers' equation with localized initial conditions. Previous studies have demonstrated that these solutions converge to a self-similar asymptotic solution  Θ( x, t )  with an error whose   L_p   norm is of order   t ^{−1+1/2 p}   . Noting that the temporal and spatial translational invariance of the underlying equations leads to a two-parameter family of self-similar solutions  Θ( x − x _*, t + t _*)  , we demonstrate that the optimal choice of   x _*  and   t _*  reduces the   L_p   error to the order of   t ^{−2+1/2 p}   .     

Asymptotic Expansions for Two Singularly Perturbed Convection–Diffusion Problems with Discontinuous Data: The Quarter Plane and the Infinite Strip

We consider a singularly perturbed convection–diffusion equation,     , defined on two domains: a quarter plane,  ( x , y ) ∈ (0, ∞) × (0, ∞)  , and an infinite strip,  ( x , y ) ∈ (−∞, ∞) × (0, 1)  . We consider for both problems discontinuous Dirichlet boundary conditions:   u ( x , 0) = 0  and   u (0, y ) = 1  for the first one and   u ( x , 0) =χ_{[ a , b ]}( x )  and   u ( x , 1) = 0  for the second. For each problem, asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) when the singular parameter  ε→ 0⁺  (with fixed distance r to the discontinuity points of the boundary condition) and (b) when that distance   r → 0⁺  (with fixed ε). It is shown that in both problems, the first term of the expansion at  ε= 0  is an error function or a combination of error functions. This term characterizes the effect of the discontinuities on the ε-behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the discontinuities of the boundary condition, the solution u ( x , y ) of both problems is approximated by a linear function of the polar angle at the discontinuity points.     

Nonexistence Results for the Korteweg–de Vries and Kadomtsev–Petviashvili Equations

We study characteristic Cauchy problems for the Korteweg–de Vries (KdV) equation u_t = uu_x + u_xxx , and the Kadomtsev–Petviashvili (KP) equation u_yy =( u_xxx + uu_x + u_t ) _x with holomorphic initial data possessing non-negative Taylor coefficients around the origin. For the KdV equation with initial value u (0,  x )= u ₀( x ), we show that there is no solution holomorphic in any neighborhood of ( t ,  x )=(0, 0) in C² unless u ₀( x )= a ₀+ a ₁ x . This also furnishes a nonexistence result for a class of y -independent solutions of the KP equation. We extend this to y -dependent cases by considering initial values given at y =0, u ( t ,  x , 0)= u ₀( x ,  t ), u_y ( t ,  x , 0)= u ₁( x ,  t ), where the Taylor coefficients of u ₀ and u ₁ around t =0, x =0 are assumed non-negative. We prove that there is no holomorphic solution around the origin in C³, unless u ₀ and u ₁ are polynomials of degree 2 or lower. MSC 2000: 35Q53, 35B30, 35C10.     

Author Index to Volumes 100 and 101: 1998

Through both analytical and numerical methods, we solve the eigenproblem u_zz >+(1/ z −λ−( z −1/ε)²) u =0 on the unbounded interval z ∈[−∞, ∞], where λ is the eigenvalue and u ( z )→0 as | z |→∞. This models an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere. It is the usual parabolic cylinder equation with Hermite functions as the eigenfunctions except for the addition of an extra term, which is a simple pole. The pole, which is on the interior of the interval, is interpreted as the limit δ→0 of 1/( z − i δ). The eigenfunction has a branch point of the form z  log( z ) at z =0, where the branch cut is on the upper imaginary axis. The eigenvalue is complex valued with an imaginary part, which we show, through matched asymptotics, to be approximately √ π exp(−1/ε²){1−2ε log ε+ε log 2+γε}. Because T ( λ ) is transcendentally small in the small parameter ε, it lies "beyond all orders" in the usual Rayleigh–Schrödinger power series in ε. Nonetheless, we develop special numerical algorithms that are effective in computing T ( λ ) for ε as small as 1/100.     

A Ray Approximation to a PdE that Arises in the Study of Tandem Queues

We use singular perturbation methods to analyze a diffusion equation that arose in studying two tandem queues. Denoting by p ( n ₁,  n ₂) the probability that there are n ₁ customers in the first queue and n ₂ customers in the second queue, we obtain the approximation p ( n ₁,  n ₂)∼ɛ² P ( X ,  Y )=ɛ² P (ɛ n ₁, ɛ n ₂), where ɛ is a small parameter. The diffusion approximation P satisfies an elliptic PDE with a nondiagonal diffusion matrix and boundary conditions that involve both normal and tangential derivatives. We analyze the boundary value problem using the ray method of geometrical optics and other singular perturbation techniques. This yields the asymptotic behavior of P ( X ,  Y ) for X and/or Y large.     

Absolute and Convective Instability for Evolution PDEs on the Half-Line

We analyze evolution PDEs exhibiting absolute (temporal) as well as convective (spatial) instability. Let  ω( k )  be the associated symbol, i.e., let  exp[ ikx −ω( k ) t ]  be a solution of the PDE. We first study the problem on the infinite line with an arbitrary initial condition   q ₀( x )  , where   q ₀( x )  decays as  | x | → ∞  . By making use of a certain transformation in the complex k -plane, which leaves  ω( k )  invariant, we show that this problem can be analyzed in an elementary manner. We then study the problem on the half-line, a problem physically more realistic but mathematically more difficult. By making use of the above transformation, as well as by employing a general method recently introduced for the solution of initial-boundary value problems, we show that this problem can also be analyzed in a straightforward manner. The analysis is presented for a general PDE and is illustrated for two physically significant evolution PDEs with spatial derivatives up to second order and up to fourth order, respectively. The second-order equation is a linearized Ginzburg–Landau equation arising in Rayleigh–Bénard convection and in the stability of plane Poiseuille flow, while the fourth-order equation is a linearized Kuramoto–Sivashinsky equation, which includes dispersion and which models among other applications, interfacial phenomena in multifluid flows.     

Large Time Asymptotic Behaviors for Periodic Solutions of Generalized Burgers Equations With Spherical Symmetry or Linear Damping

Using the method of balancing arguments, large time asymptotic behaviors for the periodic solutions of generalized Burgers equations   u_t  +  u ³ u_x  +  ju /2 t  =δ/2 u_xx   and   u_t  +  u ³ u_x  +λ u  =δ/2 u_xx   subject to the periodic initial condition     and the vanishing boundary conditions   u (0,  t ) =  u ( l ,  t ) = 0,   t  ≥ 0   or    t ₀,  where   A ,  A ₁, δ, λ,  l ,  t ₀, ∈ R ⁺  and   j  = 1, 2  , are obtained.     

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight   z ^{− n +ν} e ^{− Nz} ( z − 1)^{2 b}   , in the limit where   n , N →∞  with   N / n → 1  and ν is a fixed number in     . With the effect of the factor (   z − 1)^{2 b}   , the local parametrix near the critical point z = 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.     

The Stokes and Krasovskii Conjectures for the Wave of Greatest Height

The integral equation:
φ_μ(s) = (1/3 π)∫^π ₀((sin φ_μ(t))/(μ ⁻¹+ ∫^t ₀sin φ_μ(u) d u )) (log((sin½( s + t ))/ (sin½( s − t )))d t
was derived by Nekrasov to describe waves of permanent form on the surface of a nonviscous, irrotational, infinitely deep flow, the function φ_μ giving the angle that the wave surface makes with the horizontal. The wave of greatest height is the singular case μ=∞, and it is shown that there exists a solution φ_∞ to the equation in this case and that it can be obtained as the limit (in a specified sense) as μ→∞ of solutions for finite μ. Stokes conjectured that φ_∞( s )→⅙π as s ↓0, so that the wave is sharply crested in the limit case; and Krasovskii conjectured that sup _{s ∈[0,π]}φ_μ( s )≤⅙π for all finite μ. Stokes' conjecture was finally proved by Amick, Fraenkel, and Toland, and the present article shows Krasovskii's conjecture to be false for sufficiently large μ, the angle exceeding ⅙π in what is a boundary layer.     

Shooting Method for Nonlinear Singularly Perturbed Boundary-Value Problems

Asymptotic formulas, as  ɛ→ 0⁺  , are derived for the solutions of the nonlinear differential equation  ɛ u" + Q ( u ) = 0  with boundary conditions   u (-1) = u (1) = 0  or   u '(-1) = u '(1) = 0  . The nonlinear term Q ( u ) behaves like a cubic; it vanishes at   s _-, 0, s ₊  and nowhere else in  [ s _-, s ₊]  , where   s _- < 0 < s ₊  . Furthermore,   Q '( s _±) < 0, Q '(0) > 0  and the integral of Q on the interval [ s _-, s ₊] is zero. Solutions to these boundary-value problems are shown to exhibit internal shock layers, and the error terms in the asymptotic approximations are demonstrated to be exponentially small. Estimates are obtained for the number of internal shocks that a solution can have, and the total numbers of solutions to these problems are also given. All results here are established rigorously in the mathematical sense.