Viscous Critical-Layer Analysis of Vortex Normal Modes

The linear stability properties of an incompressible axisymmetrical vortex of axial velocity   W ₀( r )  and angular velocity  Ω₀( r )  are considered in the limit of large Reynolds number. Inviscid approximations and viscous WKBJ approximations for three-dimensional linear normal modes are first constructed. They are then shown to be singular at the critical points r_c satisfying  ω= m Ω₀( r_c ) + kW ₀( r_c )  , where ω is the frequency, k and m the axial and azimuthal wavenumbers. The goal of this paper is to resolve these singularities. We show that a viscous critical-layer analysis is analytically tractable. It leads to a single sixth-order equation for the perturbation pressure. This equation is identical to the one obtained in stratified shear flows for a Prandtl number equal to 1. Integral expressions for typical solutions of this equation are provided and matched to the outer inviscid and viscous approximations in the complex plane around r_c . As for planar flows, it is proved that the critical layer solution with a balanced behavior matches a non-viscous approximation in a  4π/3  sector of the complex-plane. As a consequence, the Frobenius expansions of a non-viscous mode on each side of a critical point r_c differ by a π phase jump.     

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.     

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.     

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.     

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 .     

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     .     

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}   .     

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.     

A Hierarchical Cluster System Based on HortonStrahler Rules for River Networks

We consider a cluster system in which each cluster is characterized by two parameters: an "order" i , following HortonStrahler rules, and a "mass" j following the usual additive rule. Denoting by c _i,j ( t ) the concentration of clusters of order i and mass j at time t , we derive a coagulation-like ordinary differential system for the time dynamics of these clusters. Results about the existence and the behavior of solutions as   t   are obtained; in particular, we prove that   c _i,j ( t ) 0  and   N _i ( c ( t )) 0  as   t ,  where the functional   N _i (·)  measures the total amount of clusters of a given fixed order i . Exact and approximate equations for the time evolution of these functionals are derived. We also present numerical results that suggest the existence of self-similar solutions to these approximate equations and discuss their possible relevance for an interpretation of Horton's law of river numbers.     

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.     

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.     

Asymmetric Spotty Patterns for the Gray–Scott Model in R2

In this paper, we rigorously prove the existence and stability of asymmetric spotty patterns for the Gray–Scott model in a bounded two-dimensional domain. We show that given any two positive integers   k ₁, k ₂  , there are asymmetric solutions with   k ₁  large spots (type A) and   k ₂  small spots (type B). We also give conditions for their location and calculate their heights. Most of these asymmetric solutions are shown to be unstable. However, in a narrow range of parameters, asymmetric solutions may be stable.     

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.     

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.     

On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs),   S_mn ( c , η)  , and their eigenvalues,  λ _mn   , for arbitrary complex size parameter c in the asymptotic regime of large  | c |  , m and n fixed. The ability to calculate these SWFs for large and complex size parameters is important for many applications in mathematics, engineering, and physics. For arbitrary  arg( c )  , the PSWFs and their eigenvalues are accurately expressed by established prolate -type or oblate -type asymptotic expansions. However, determining the proper expansion type is dependent upon finding spheroidal branch points,   c ^mn _{○; r}   , in the complex c -plane where the PSWF alternates expansion type due to analytic continuation. We implement a numerical search method for tabulating these branch points as a function of spheroidal parameters m , n , and  arg( c )  . The resulting table allows rapid determination of the appropriate asymptotic expansion type of the SWFs. Normalizations, which are dependent on c , are derived for both the prolate - and oblate -type asymptotic expansions and for both  ( n − m )  even and odd. The ordering for these expansions is different from the original ordering of the SWFs and is dictated by the location of   c ^mn _{○; r}   . We document this ordering for the specific case of  arg( c ) =π/4  , which occurs for the diffusion equation in spheroidal coordinates. Some representative values of  λ _mn   and   S_mn ( c , η)  for large, complex c are also given.     

The Existence and Stability of Asymmetric Spike Patterns for the Schnakenberg Model

Asymmetric spike patterns are constructed for the two-component Schnakenburg reaction–diffusion system in the singularly perturbed limit of a small diffusivity of one of the components. For a pattern with k spikes, the construction yields   k ₁  spikes that have a common small amplitude and   k ₂= k − k ₁  spikes that have a common large amplitude. A k -spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. Explicit conditions for the existence and linear stability of these asymmetric spike patterns are determined using a combination of asymptotic techniques and spectral properties associated with a certain nonlocal eigenvalue problem. These asymmetric solutions are found to bifurcate from symmetric spike patterns at certain critical values of the parameters. Two interesting conclusions are that asymmetric patterns can exist for a reaction–diffusion system with spatially homogeneous coefficients under Neumann boundary conditions and that these solutions can be linearly stable on an O (1) time scale.     

Large-Reynolds-Number Asymptotics of the Berman Problem

Similarity flow of a viscous fluid in a channel is considered, driven by uniform withdrawal of the fluid through the channel walls. The nonlinear ordinary differential boundary value problem that results has several branches of solutions; those of Types III, III₁, and I₁ are investigated here, in the limit of large wall-suction Reynolds number. This paper gives a markedly more accurate Type III asymptotic solution than previously available, and describes the true asymptotics of the other branches for the first time. The asymptotic structure of the Type III₁ solution is particularly subtle, requiring matching between seven different layers. Numerical solutions of the boundary value problem provide support for the asymptotic solutions obtained.     

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.     

On the Number of Solutions to Carrier's Problem

Let N _ɛ denote the maximum number of spikes that a solution to Carrier's problem can have, where ɛ is a small positive parameter. We show that N _ɛ is asymptotically equal to [ K /ɛ], where   K = 0.4725⋯  , and the square brackets represent the greatest integer less than or equal to the quantity inside. If n (ɛ) stands for the number of solutions to this problem, then it is also shown that 4 N _ɛ− 3 ≤ n (ɛ) ≤ 4 N _ɛ. Our approach is based on the shooting method used by Ou and Wong ( Stud. Appl. Math. 111 (2003)) and on the construction of an envelope function for the minimum values of the solutions as ɛ approaches zero.     

Spectral Properties in Modon Stability Theory

Spectral properties of an invariant functional, denoted by H, for the linear stability equation associated with the modon , or solitary drift vortex, solutions of the quasi-geostrophic equivalent barotropic potential vorticity, or Charney–Hasegawa–Mima (CHM), equation are investigated. It is shown that H, which is the only known quadratic invariant in modon stability theory, is identical in form to the second variation of a "Benjamin-like" variational principle for solitary vortices. However, such a principle does not exist for the modon. The discrete spectrum of the "form operator" in H contains two simple negative eigenvalues and the simple zero eigenvalue. For the leftward-traveling solution there are only a finite number of positive eigenvalues. For the rightward-traveling solution, there are a countable infinity of positive eigenvalues. A sharp lower bound on the spectrum, for both the rightward- and leftward-traveling solutions, and a sharp upper bound for the leftward traveling solution, is determined. For the leftward-traveling solutions, the eigenfunctions span a finite-dimensional vector space and are orthogonal with respect to an inner product which is valid for all of L ₂. For the rightward-traveling solutions, the eigenfunctions span an infinite-dimensional Hilbert space, but are orthogonal with respect to an inner product, which is not valid for all of L ₂.