Stokes Phenomenon Research Articles

Stokes' phenomenon at second-order pole and reflectionless potentials

Some classes of analytic potentials with second-order poles and corresponding asymptotic solutions of the Schrodinger equation are studied. The main goal of the analysis is to answer the question as to whether a Stokes phenomenon and therefore a reflection takes place for those equations. It is shown that one can notice the Stokes phenomenon not only in the leading term of an asymptotic expansion but also in higher-order ones. Therefore for only a restricted set of potentials proposed, the reflection coefficient is exactly equal to zero although it looks like that for other potentials in the first approximation. This letter gives rise to a further extension of the original approach of Berry and Howls (1990).

Fake Airy functions and the asymptotics of reflectionlessness

Two classes of analytic refractive-index profile P2(z, in ), whose reflection coefficients r are zero for all values of a parameter in , are studied as in to 0. The aim is to understand why r=0 rather than r varies as exp(-1/ in ) as for generic profiles. The authors find that reflectionlessness is a consequence of the fact that transition points of P2 (zeros or poles in the complex z plane) form tight clusters (whose size vanishes with in ) which can be regarded neither as coalesced nor well separated. Expansion near a cluster yields the local wave not as the usual Airy function, whose Stokes phenomenon generates reflection, but as Bessel functions of half-integer order (fake Airy functions) which are exactly trigonometric functions with no Stokes phenomenon and so no reflection.

Waves near Stokes lines

The large- k asymptotics of d 2 u ( z )/d z 2 = k 2 R 2 ( z ) u ( z ) are studied near a Stokes line ( ω ≡ ∫ z z 0 R d z real, where z 0 is a zero of R 2 ( z ), of any order), on which there is greatest disparity between the dominant and subdominant exponential waves in the phase-integral (WKB) approximations. The aim is to establish precisely how the multiplier b _ of the subdominant wave varies across the Stokes line. Although b _ always has a total change proportional to i times the multiplier of the dominant wave (the Stokes phenomenon), the form of the change depends on the convention used to define the two waves. The optimal convention, for which the variation is maximally compact and smooth, is to define them by the phase-integral approximation truncated at its least term, whose order is proportional to k and therefore large (‘asymptotics of asymptotics’). Then the variation of b _ is proportional to the error function of the natural Stokes-crossing variable Im ω √( k /Re ω ). This result is obtained without resumming divergent series (thereby avoiding ‘asymptotics of asymptotics of asymptotics’). An application is given, to the birth of exponentially weak reflected waves in media with smoothly varying refractive index.

Quantum behavior at a cusp, or how wave mechanics avoids a catastrophe

The quantum mechanical system whose potential is the cusp catastrophe function is canonical for the uniform asymptotic analysis of the Stokes phenomenon for four transition points. It is shown that across the singularity which marks the bifurcation from a single to two modes, quantum eigenlevels vary smoothly. Also, their derivatives and nonadiabatic coupling matrix elements are continuous with respect to control parameters. These matrix elements, which are of interest for the three-body rearrangement problem, clearly exhibit the ridge effect, i.e. a peaking astride potential maxima.

A Simple Explanation of the Stokes Phenomenon

The Stokes Phenomenon is known to be a pervasive feature of asymptotics, but its explanation in the literature is obscured by intricate and lengthy technicalities. This article presents a simpler approach to its understanding and treatment as a natural aspect of a well-motivated characterization of functions by approximands of different multivaluedness

Comment on 'Stokes phenomena and the monodromy deformation problem for the nonlinear Schrodinger equation'

For original paper see ibid., vol.19, p.3741 (1986). A comment is presented on a recent paper by Chowdhury and Naskar (1986) which describes a method that is claimed to be successful in calculating the Stokes parameters of the linear differential equation to which the Painleve IV equation defines isomonodromic deformations. However, the author shows that the simplification made by Chowdhury and Naskar is not uniform and the resulting formulae for Stokes parameters are not invariant under the deformations satisfying the Painleve IV equation.

Stokes phenomena and the monodromy deformation problem for the non-linear Schrodinger equation

Following Flaschka and Newell (1979) the authors have formulated the inverse problem for Painleve IV, with the help of similarity variables. The Painleve IV arises as the eliminant of the two second-order differential equations originating from the non-linear Schrodinger equation. They have obtained the asymptotic expansions near the singularities at 0 and infinity of the complex eigenvalue plane. The corresponding analysis then displays Stokes phenomena. The monodromy matrices connecting the solution Yj in the sector Sj to that in Sj+1 are fixed in structure by the imposition of certain conditions. They then show that a deformation keeping the monodromy data fixed leads to the non-linear Schrodinger equation. At this point they may mention that, while Flaschka and Newell did not make any absolute determination of the Stokes parameter, the approach yields the values of the Stokes parameter in an explicit way, which in turn can determine the matrix connecting the solutions near 0 and infinity . Such a realisation was not possible in the approach of Flashchka and Newell. Lastly they show that the integral equation originating from the analyticity and asymptotic nature leads to the similarity solution previously determined by Boiti and Pempinelli (1979).

Stokes phenomena and monodromy deformation problem for nonlinear Schr�dinger equation

Following Flaschka and Newell, the inverse problem for Painleve IV is formulated with the help of similarity variables. The Painleve IV arises as the eliminant of the two second-order ordinary differential equations originating from the nonlinear Schrodinger equation. Asymptotic expansions are obtained near the singularities at zero and infinity of the complex eigenvalue plane. The corresponding analysis then displays the Stokes phenomena. The monodromy matrices connecting the solutionY j , in the sectorS j to that inS j +1 are fixed in structure by the imposition of certain conditions. It is then shown that a deformation keeping the monodromy data fixed leads to the nonlinear Schrodinger equation. While Flaschka and Newell did not make any absolute determination of the Stokes parameters, the present approach yields the values of the Stokes parameters in an explicit way, which in turn can determine the matrix connecting the solutions near zero and infinity. Finally, it is shown that the integral equation originating from the analyticity and asymptotic nature of the problem leads to the similarity solution previously determined by Boiti and Pampinelli.

On the occurrence of multiple spectra of eigenvalues in the one-dimensional complex scaled Schrödinger equation

Numerical experiments performed on the complex scaled one-dimensional radial Schrodinger equation with an exponential wall show that different rotation angles may produce different spectra. An explanation is given from a study of the analytic solutions and the relation with the Stokes phenomenon is pointed out. However, the spectra predicted by this analysis are not always obtained numerically. This is shown to be due to a self-correction effect which erases the memory of the boundary conditions.

Asymptotic Approximations and Error Bounds

The purpose of this paper is to demonstrate that well-constructed error bounds for asymptotic approximations can provide useful analytical insight into the nature and reliability of the approximations, enable somewhat unsatisfactory concepts such as multiple asymptotic expansions and generalized asymptotic expansions to be avoided, and lead to significant extensions of asymptotic results. Also included are numerical values of error terms in the asymptotic expansion of Macdonald’s modified aesse)function $K_\nu (z)$, and some new results pertaining to the Stokes phenomenon for this expansion.

Admissible Domains for Higher Order Differential Equations

This paper analyzes the geometric structure of certain domains in the complex plane which arise in the asymptotic theory of linear ordinary differential equations containing a parameter. These domains, called admissible, are domains in which an asymptotic representation of the solution of the differential equation may be found and across whose boundaries these representations may undergo a rapid change of asymptotic behavior (the Stokes phenomenon). A knowledge of the disposition of those domains associated with a particular differential equation is necessary for a satisfactory asymptotic theory of the equation. The main analysis gives necessary and sufficient conditions for identifying admissible domains and gives a procedure for obtaining particular admissible subdomains of a given domain. Sufficient conditions are established to determine the maximality of admissible domains. A section of examples is included to highlight the salient features of this theory. In all of the results, criteria involving only purely local properties of the boundary are needed to determine the global properties of admissibility and maximal admissibility .

Solutions of one-dimensional Schr�dinger equation with hardy-type potential. The S-matrix formalism

A general method is developed for investigating the solutions of the scattering problem for the one-dimensional SchrSdinger equation with Hardy potential. The method uses the formalism of stationary scattering theory. In the quasiclassical and wave (anticlassical) approximations, calculations are made of the amplitude of reflection of particles from a potential with Hardy function describing Bragg scattering of shortwavelength radiation on static elastic distortions of dislocation type in a crystal lattice. In the theory of Bragg scattering of x rays, fast electrons and short-wavelength radiation generally in crystals with static fields of elastic distortions one sometimes encounters the problem of finding and analyzing asymptotic solutions of a one-dimensional scattering problem on a potential of Hardy type with U(x) =2-'[--0 '2 (x) +i0" (x) ], where 0 (x) is a function of the field of elastic displacements and U (x) is the one-dimensional potential of the SchrSdinger equation. The general principles for constructing quasiclassical solutions of the one-dimensional Schrt~dinger equation were formulated by Pokrovskii and Khalatnflcov [1]. Their approach is based on an investigation of the analytic properties of the wave function ,I,(x) in the complex x plane with allowance for the Stokes phenomenon - the rules for going round the singular points and turning points of the wave vector k(x) = [2(E--U(z)) ]'" (E is the energy of the particles; here and in what follows m=h=i); their method made it possible to analyze completely the problem of the scattering of particles by a quasiclassical potential [2-4]. In the general case of diffraction scattering of radiation in crystals with elastic displacement fields, the quasiclassical and wave (anticlassical) asymptotic behaviors of the wave function of the problem [5-6] have physical interest. For the one-dimensional SchrSdinger equation with potential of Hardy type, the construction and investigation of special quasiclassical asymptotic solutions for the wave function were first discussed apparently by Jeffreys [7] (Jeffreys solutions) in connection with the well-known problem of the free gravitational oscillations of water in an elliptical lake; however, the question of the correctness of his proposed modification to the quasiclassical approximation remained open (in [7] there is an error in the calculations; for more detail, see [8.9]). In the present paper, we investigate the problem of "above-barrier reflection" of particles on a potential of Hardy type when 0o' (x) =b-'+2-'an (x2+a 2)-' (n= l, 2, 3 .... ). Physically, this function 0~(x) corresponds to Bragg scattering of short-wavelength radiation in a crystal with a field of elastic displacements of dislocation type [5]. It is important that in this case the potential function U (x) has a second-order pole on

A complex ray tracing study of ion cyclotron whistlers in the inosphere

The complex ray theory of Budden & Terry (1971) has been used to study ion cyclotron whistlers in the topside ionosphere. A study of mode coupling processes by the use of ray tracing is impossible using standard real ray theory, but they can sometimes be handled by using complex rays and complex space. Losses by coupling have a similar effect to losses by collisional damping, and large coupling losses produce large complex lateral deviation in the ray path, which in turn requires the use of angles of incidence with an appreciable imaginary part to ensure that the ray path ends at a real point. The importance of understanding the branch points of the Booker quartic equation, and the Stokes phenomenon at these points is stressed. Results are presented for calculations in a model ionosphere, which agree quite well with experimental observations.

Numerical Investigations into the Stokes Phenomenon. I

Numerical Investigations into the Stokes Phenomenon. I

Numerical Investigations into the Stokes Phenomenon. II

The Stokes phenomenon associated with the differential equations W “ = WZ (z2−a2) and W” = w(z2 −a2)(x2−b2)is considered. As an application to the method introduced in paper I, some numerical and analytical results concerning the Stokes constants of these equations are presented.

Deductions from some artificial refractive index profiles based on the elementary functions

AbstractThe familiar W.K.B.J. method expresses in terms of elementary functions approximate solutions of second order differential equations in normal form that otherwise have no analytical solutions expressible in terms of the standard elementary or transcendental functions. Proofs of the connexion formulae required to trace these approximate solutions across transition points are usually derived by comparison with known functions such as the Airy integral or Bessel functions, or by an appeal to the Stokes phenomenon in the complex plane. A new device is developed for the synthesis of refractive index profiles with transition points, and yet yielding solutions in terms of the elementary functions. These are then used to derive the W.K.B.J. connexion formula by means of a novel limiting process based on solutions considered only along the real z-axis. The method is obviously more complicated than the more usual approach, but contains special features throwing new light on the connexion formula across a single transition point.

2.—Barriers bounded by Two Transition Points of Arbitrary Odd Order

SynopsisA generalisation is considered of the potential barrier problem beyond the familiar case in which the barrier is bounded by transition points of order one. Here, the two transition points involved are of arbitrary odd order. The approximate method employed, though formal in character, avoids certain pitfalls often made in the past whereby certain exponentially small terms within the barrier are confounded with inherent error terms. This confusion is avoided in the treatment given here by tracing uniformly approximate solutions round the transition points in the complex plane by means of the Stokes phenomenon, the method not requiring the dubious concept of a subdominant term existing in the presence of a dominant term on a Stokes line. At the same time, the solutions to which the reflection and transmission coefficients may be attached are carefully discussed, so that the appearance of a small exponential term may be seen to be genuine when taken in conjunction with inherent error terms. The resulting formula for the modulus of the reflection coefficient generalizes the more elementary formula.

Further transformable nth order differential equations with transition points of particular order m

Further consideration is given to properties of the Stokes phenomenon associated with a class of nth order differential equations with transition points of order m (a rational fraction) at the origin. The systematic vanishing of the Stokes multipliers leads to the discovery of sets of equations that are transformable from certain specific values of m to particular lower values of m. A series of theorems in the theory of congruences is first proved, from which the main transformation theorem is then deduced. The theory is ifiustrated by various numerical examples.

The Stokes phenomenon and generalized contiguous transformations of some generalized hypergeometric functions

AbstractPrevious investigations by the author into the Stokes phenomenon pertaining to solutions of the differential equation dnu/dzn = (–1)nzmu are extended in order to find when different equations have the same set of Stokes multipliers, with perhaps a series of zeros being additionally allowed. The reason for periodic cycles to exist (with n fixed and m varying), with the same Stokes multipliers regained after a complete cycle, is traced to certain transformation properties of the equations. Within the first cycle (with n fixed and m varying) further remarkable identities exist between the Stokes multipliers, and this also is traced to special transformations between the equations. Relations are found toexist when values of m are chosen so that the highest common factors of the two integers n and n/(n + m) are identical. Finally, a transformation of the independent variable is deduced whereby the set of Stokes multipliers for an equation of order n is identical (apart from the additional zeros) to that for an equation of lower order. A hierarchy of equations is thrown up, whereby certain basic equations are transformed to yield more advanced equations of higher order and different m.

Stokes multipliers for the Orr-Sommerfeld equation

The comparison equation method is used to study the outer expansions of the solutions of the Orr-Sommerfeld equation. All but one of these expansions are multiple-valued and must therefore exhibit the Stokes phenomenon. One of the major aims of the present paper is to obtain first approximations to the Stokes multipliers which describe the continuation of these expansions on crossing a Stokes line in the complex plane. By restricting the domains of validity of these expansions appropriately we can insure that all of the expansions are ‘complete ’ in the sense of Olver and this is an essential feature of the work. The resulting approximations show that, in some sectors, a sharp distinction can no longer be made between approximations of inviscid and viscous type. A consistent first-order approximation to the characteristic equation in the complete sense is derived and compared with the more usual second-order approximation of Poincare type. Calculations of the curve of neutral stability for plane Poiseuille flow clearly show that a first approximation in the complete sense provides a substantially better approximation to the neutral curve than a second approximation in the Poincare sense.