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.     

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 .     

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.     

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     .     

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.     

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.     

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  .     

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.     

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.     

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 .     

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.     

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.     

Asymptotic Analysis of a Loss Model with Trunk Reservation II: Trunks Reserved for Slow Traffic

We consider a model for a single link in a circuit switched network. The link has C circuits, and the input consists of offered calls of two types, that we call primary and secondary traffic. Of the C links R are reserved for primary traffic. We assume that both traffic types arrive as Poisson arrival streams. Assuming that C is large and   R = O (1)  , the arrival rate of secondary traffic is   O ( C )  while that of primary traffic is smaller, of the order     . The holding times of the secondary calls are assumed exponentially distributed with unit mean. Those of the primary calls are exponentially distributed with a large mean, that is     . The loads for both traffic types are thus comparable  ( O ( C ))  and we assume that the system is "critically loaded," i.e., the system's capacity is approximately equal to the total load. We analyze asymptotically the steady state probability that   n ₂  (resp.   n ₁  ) circuits are occupied by primary (resp. secondary) calls. In particular, we obtain two-term asymptotic approximations to the blocking probabilities for both traffic types. This work complements part I, where we assumed that the secondary traffic had an arrival rate that was     and a service rate that was     . Thus in part I the R trunks were reserved for the "fast traffic," whose arrival and service rates were   O ( C )  and   O (1)  .     

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.     

The Existence and Stability of Spike Equilibria in the One-Dimensional Gray–Scott Model: The Low Feed-Rate Regime

In a singularly perturbed limit of small diffusivity ɛ of one of the two chemical species, equilibrium spike solutions to the Gray–Scott (GS) model on a bounded one-dimensional domain are constructed asymptotically using the method of matched asymptotic expansions. The equilibria that are constructed are symmetric k -spike patterns where the spikes have equal heights. Two distinguished limits in terms of a dimensionless parameter in the reaction-diffusion system are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime, the solution branches of k -spike equilibria are found to have a saddle-node bifurcation structure. The stability properties of these branches of solutions are analyzed with respect to the large eigenvalues λ in the spectrum of the linearization. These eigenvalues, which have the property that  λ= O (1)  as  ɛ→ 0  , govern the stability of the solution on an O (1) time scale. Precise conditions, in terms of the nondimensional parameters, for the stability of symmetric k -spike equilibrium solutions with respect to this class of eigenvalues are obtained. In the low feed-rate regime, it is shown that a large eigenvalue instability leads either to a competition instability, whereby certain spikes in a sequence are annihilated, or to an oscillatory instability (typically synchronous) of the spike amplitudes as a result of a Hopf bifurcation. In the intermediate regime, it is shown that only oscillatory instabilities are possible, and a scaling-law determining the onset of such instabilities is derived. Detailed numerical simulations are performed to confirm the results of the stability theory. It is also shown that there is an equivalence principle between spectral properties of the GS model in the low feed-rate regime and the Gierer–Meinhardt model of morphogenesis. Finally, our results are compared with previous analytical work on the GS model.     

Rational ODEs with Movable Algebraic Singularities

A class of second-order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite-length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations   P_II – P   _VI   are poles.     

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.     

On the Inverse Problem for Scattering of Electromagnetic Radiation by a Periodic Structure

We consider a smooth perturbation  δε( x , y , z )  of a constant background permittivity  ε=ε₀  that varies periodically with x , does not depend on y , and is supported on a finite-length interval in z . We investigate the theoretical and numerical determination of such perturbation from (several) fixed frequency y -invariant electromagnetic waves.
By varying the direction and frequency of the probing radiation a scattering matrix is defined. By using an invariant-imbedding technique we derive an operator Riccati equation for such scattering matrix. We obtain a theoretical uniqueness result for the problem of determining the perturbation from the scattering matrix.
We also investigate a numerical method for performing such reconstruction using multi-frequency information of the truncated scattering matrix. This relies on ideas of regularization and recursive linearization. Numerical experiments are presented validating such approach.