Square root of the monodromy map for the equation of RSJ model of Josephson junction
aa r X i v : . [ m a t h . C A ] J a n SQUARE ROOT OF THE MONODROMY MAP FOR THEEQUATION OF RSJ MODEL OF JOSEPHSON JUNCTION
SERGEY I. TERTYCHNIY
Abstract.
Several noteworthy properties of the differential equation utilizedfor the modeling of Josephson junctions are considered. The explicit represen-tation of the monodromy transform of the space of its solutions is given. Incase of positive integer order, the transformation interpreted as the square rootof the monodromy transformation noted is derived making use of a symmetrythe associated linear second order differential equation possesses.
The present notes are devoted to discussion of some noteworthy properties ofthe differential equation(1) ϕ ` sin ϕ “ B ` A cos ωt, in which ϕ “ ϕ p t q is the unknown function, the symbols A, B, ω stand for somereal constants, and t is a free real variable, the dot denoting derivation withrespect to t . Eq. (1) and its generalizations are of interest, in particular, in viewof their application in a number of models in physics, mechanics, and geometry[1, 2]. Most widely Eq. (1) is known as the equation utilized in the so calledRSJ model of Josephson junction [3, 4, 5, 6, 7] which applies if the effect ofthe junction electric capacitance is negligible (the case of so called overdampedJosephson junctions).Eq. (1) is equivalent to the Riccati equation Φ “ p ω z q ´ p ´ Φ q ` p ℓ z ´ ` µ p ` z ´ qq Φ , (2)where Φ “ Φ p z q is a holomorphic function of the free complex variable z , theprime denoting the derivative with respect to the latter, and ℓ, µ, ω are the con-stant parameters. Indeed, the formal substitutions(3) z Ø e i ωt , Φ p z q Ø e i ϕ p t q convert Eq. (2) to Eq. (1) get with the parameters related to the parametersinvolved in the former equation by the transformation(4) A “ ωµ, B “ ωℓ. The nonlinear equation (2) has the only singular point z “ but its solutionsmay diverge at of some other values of argument. Nevertheless, one can claimthe following [13]: Supported in part by RFBR grant N 17-01-00192.
Proposition 1.
Let the constants ℓ, µ , and ω ą be real and let Φ p z q be asolution to Eq. (2) holomorphic at z “ such that | Φ p q| “ . Then Φ p z q isalso holomorphic in some vicinity of the curve | z | “ ; moreover, if | z | “ then | Φ p z q| “ . Indeed, any solution to Eq. (1) is real analytic and can be extended to thewhole real axis R . Carrying out analytic continuation of ϕ p t q from R to someits vicinity in C , one obtains the function Φ “ exp p i ϕ p t qq which is holomorphicin t and which can also be considered as a holomorphic function of the variable z “ exp p i ωt q varying in some open set embodying the curve | z | “ . The function Φ p z q possesses the properties asserted above, obviously.It is also obvious that if we are given the function Φ p z q obeying Eq. (2) andunimodular on the curve | z | “ , then the real-valued smooth function ϕ p t q suchthat e i ϕ p t q “ Φ p e i ωt q can be constructed. It verifies Eq. (1), evidently.It is in order now to comment on the term “the curve | z | “ ” used aboveinstead of something like “the unit circle S Ă C ” which one could argue to bemore customary. The point is that, strictly speaking, apart of the very specialconditions, a solution to Eq. (2) can not be holomorphic on S . The rationaleis here fairly simple: indeed, there is no reason why a generic solution ϕ p t q toEq. (1) should obey the constraint ϕ p T q “ ϕ p - T q p mod 2 π q , where T “ πω ´ is the period of the right hand side expression in Eq. (1). Accordingly, followingthe way of constructing of the function Φ via the function ϕ utilized above, oneobtains Φ p´ ` q ´ Φ p´ ´ q “ e i ϕ p T q ´ e i ϕ p - T q “ , meaning that Φ , whenconsidered on S , proves to be not continuous at ´ .There is, definitely, no singularity of Φ at ´ and the deficiency in the aboveconstruction originates in the improper selection of the Φ domain. Generallyspeaking, it can not be the complex plane or any its subset; instead, the universalcover C of the punctured complex plane C ˚ “ C K (with the subset of the isolatedsingular points of Φ removed) has to be utilized. In this setting, the image of the“ t -axis”, produced by the map extending the transformation (3) to C , is not S but the non-compact curve covering S . It is this curve which, admitting someabuse of notations, was referred to as “the curve | z | “ ”.Here we shall not, however, consider solutions to Eq. (2) on their whole domainsbut only on the sub-domain C which is in bijective correspondence with (projectsto) the subset C ˚ of C ˚ obtained by removal of the ray of negative real numbers, C ˚ “ C K R ď , (and removal, for each Φ , its singular points, if any). In mostcases, C can be (and will be) considered to be undistinguished from C ˚ . However,there are two boundaries of C , which projects to the same (removed) ray R ă ,on which the values of Φ do not coincide. To mirror such a difference, we mayidentify these boundaries with the two edges of the corresponding cut in C ˚ . ONODROMY MAP 3
It is then convenient to refer to the points of the cut edge contacting the half-plane t z P C ˚ , ℑ z ą u by the symbol ρ e i π , where ρ stands for a positive realnumber, ρ P R ą , and by the symbol ρ e ´ i π for a point belonging to the cut edgecontacting the half-plane t z P C ˚ , ℑ z ă u . If ρ “ the factor ρ is omitted. Insuch a framework, the portion of S coming to be in C ˚ is “the punctured circle”(5) S “ t z P C , | z | “ , z “ ´ u . It approaches near the ends the (distinct) boundary points denoted e i π and e ´ i π .Each solution to Eq. (2) is holomorphic and non-zero in some vicinity of S .Besides, at the boundary points of S , it holds(6) Φ p e i π q “ e i ϕ p T q p“q Φ p e ´ i π q “ e i ϕ p - T q . The following statement is an obvious consequence of the periodicity of theright hand side of Eq. (1):
Proposition 2.
Let the function ϕ p t q verify Eq. (1) . Then the function ϕ M p t q : “ ϕ p t ` T q is also a solution to Eq. (1) . Let the solution Φ p z q to Eq. (2) be constructed in accordance with the algo-rithm specified above. The above statement and definitions imply the following. Proposition 3.
There exists the solution Φ M p z q to Eq. (2) such that (7) Φ M p e ´ i π q “ Φ p e i π q . Moreover, if (7) holds true for some solutions Φ , Φ M then (8) Φ M p ρ e ´ i π q “ Φ p ρ e i π q for all ρ ą excluding ones at which Φ p ρ e i π q is not analytic (i.e. is undefined). Indeed, the function Φ M p z q wanted can be constructed from the function ϕ M p t q in the same way as the function Φ p z q is constructed from ϕ p t q . The function ϕ M p t q is actually some “portion” of the maximally extended solution ϕ p t q get from thesegment p - T ` T, T ` T q and “put down” to the segment p - T, T q consideredas the common domain with ϕ p t q ; similarly, the function Φ M is, essentially, thefunction Φ get on the sub-domain adjacent via the common boundary with C andconsidered on the same domain with Φ trough their projections to the commonarea C ˚ which can be considered equivalent to C .Alternatively, the function Φ M can also be defined as the result of point-wiseanalytic continuations in C ˚ of the function Φ defined on C ˚ along the full circleswith centers situated at zero which are passed in the counter-clockwise direction(or along the curves avoiding Φ singularities and homotopic to such circles). Thelatter interpretation allows one to refer to transformation M : Φ ÞÑ Φ M as the monodromy map which acts on the space of solutions to Eq. (2).We are now ready to formulate the first non-evident result the present notesare devoted to. SERGEY I. TERTYCHNIY
Theorem 4.
Let a solution Φ to Eq. (2) holomorphic in some vicinity of S be given. Let also Ψ “ Ψ p z q be a solution of the linear homogeneous first orderordinary differential equation (9) ωz Ψ “ p Φ ` Φ ´ q Ψ . Let, finally, (10) Ψ p q “ and | Φ p q| “ . Then the formula (11) Φ M p z q “ ´ e P p T q cos ϕ p T q ¨ Ψ p z q Φ p z q ` i e P p - T q sin p ϕ p T q ´ ϕ p - T qq ¨ Ψ p { z q Φ p { z q - ¯ ˆ ´ e P p T q cos ϕ p T q ¨ Ψ p z q Φ p z q - ´ i e P p - T q sin p ϕ p T q ´ ϕ p - T qq ¨ Ψ p { z q Φ p { z q ¯ ´ in which the continuous (and then necessarily real analytic) function ϕ is deter-mined by the equation Φ p e i ωt q “ e i ϕ p t q , t P p´ T, T q , yields the explicit represen-tation of the result of the monodromy transformation of the function Φ . The above assertion means that any solution to Eq. (2) can be extended fromits sub-domain with the closure equal to C to the whole domain with the closureequal to C by means of certain algebraic transformations (provided the function Ψ had once only been computed on C ).Before proving these, it is worth commenting on the existence of Ψ . Given Φ ,it is determined on the base of the equality Ψ p e i ωt q “ e P p t q , t P p´ T, T q , where P p t q “ ş t cos ϕ p ˜ t q d ˜ t, reducing, therefore, to the quadrature and subsequent analytic continuation ofits result from an arc of the curve | z | “ . Notice also that for such Ψ the squareroot Ψ { is uniquelly defined via analytic continuation of e P p t q{ . The functions Φ ˘ { are endowed with unique values in a similar way.The proof of the formula (11) splits into two steps. First, its right hand sideis evaluated for the argument z “ e ´ i π . Performing substitutions in accord withdefinitions, one obtains Φ M p e ´ i π q “ e ϕ p T q “ Φ p e i π q . Second, the expression (11)is substituted into Eq. (2). Then, upon elimination of the derivatives Φ and Ψ with the help of Eq. (2) and Eq. (9), respectively, the identical equality follows.Thus, the expression (11) verifies the first order differential equation (2) andobeys the initial condition (7) which distinguishes the solution representing themonodromy transformation of Φ . The identical coincidence follows and we aredone.It will be further assumed throughout that the parameter ℓ is a positive integer, ℓ P N . ONODROMY MAP 5
We set up the following definition [9].
Definition 1.
Let the four sequences p k , q k , r k , s k , k “ , , . . . of functionsof the complex variable z and the constant parameters ℓ, µ, λ “ p ω q ´ ´ µ bedefined by means of the following recurrent scheme p “ , q “ , r “ z ´ , s “ ´ µ ; (12) $’’’’&’’’’% p k “ p ´ ℓ q z p k ´ ` q k ´ ` z p k ´ ,q k “ z p´ λ ` p ℓ ` q µz q p k ´ ` µ ` ´ z ˘ q k ´ ` z q k ´ ; r k “ p k ´ q z r k ´ ´ s k ´ ´ z r k ´ ,s k “ z p λ ´ p ℓ ` q µz q r k ´ ` ` p k ´ ℓ ´ q z ` µ ` z ´ ˘˘ s k ´ ´ z s k ´ , (13) We pick up their “diagonal” representatives denoting them p , q , r , s , i.e. define (14) p “ p ℓ , q “ q ℓ , r “ r ℓ , s “ s ℓ . It can be shown that the functions p , q , r , s are the polynomials in z of thedegrees ℓ ´ , ℓ, ℓ ´ , ℓ , respectively [9]; they are polynomial in the parameters λ and µ as well.We define now the following four holomorphic functions Φ B , Ψ B , Θ B , ˜Θ B rep-resenting them in terms of the two other holomorphic functions Φ and Ψ of acomplex variable and the functions ϕ, P of a real variable, the latter pair beingevaluated for several fixed values of their argument alone:(15) Φ B p z q “ ´ ´ e P p T q p D ` w ´ u ´ ` D ´ w ` u ` q ¨ Ψ p z q Φ p z q ` ` ´ D ` w ´ p e P p T q u ´ ` e P p - T q v ´ q` D ´ w ` p e P p T q u ` ` e P p - T q v ` q ˘ ¨ Ψ p { z q Φ p { z q - ¯ ˆ ´ ´ e P p T q p D ` w ´ u ´ ` D ´ w ` u ` q ¨ Ψ p z q Φ p z q - ` ` ´ D ` w ´ p e P p T q u ´ ` e P p - T q v ´ q` D ´ w ` p e P p T q u ` ` e P p - T q v ` q ˘ ¨ Ψ p { z q Φ p { z q ¯ ´ , (16) Ψ B p z q “ p q ´ p Θ B p z q ´ ˜Θ B p z qq , where SERGEY I. TERTYCHNIY (17) Θ B p z q “p cos ϕ p qq ´ ˆ ´ ´ i ` D ´ w ` pp ϕ p q ´ q e P p T q u ` ´ e P p - T q v ` q` D ` w ´ pp ϕ p q ` q e P p T q u ´ ` e P p - T q v ´ q ˘ ¨ Ψ p z q Φ p z q - ` ` ´ D ´ w ` pp sin ϕ p q ´ q e P p T q u ` ` sin ϕ p q e P p - T q v ` q` D ` w ´ pp sin ϕ p q ` q e P p T q u ´ ` sin ϕ p q e P p - T q v ´ q ˘ ˆ Ψ p { z q Φ p { z q ¯ ˆ ´ ´ e P p T q p D ` w ´ u ´ ` D ´ w ` u ` q ¨ Ψ p z q Φ p z q - ` ` ´ D ` w ´ p e P p T q u ´ ` e P p - T q v ´ q` D ´ w ` p e P p T q u ` ` e P p - T q v ` q ˘ ¨ Ψ p { z q Φ p { z q ¯ ´ , (18) ˜Θ B p z q “p cos ϕ p qq ´ ˆ ´ i ` ´ D ´ w ` p´p ϕ p q ´ q e P p T q u ` ` e P p - T q v ` q` D ` w ´ pp ϕ p q ` q e P p T q u ´ ` e P p - T q v ´ q ˘ ¨ Ψ p z q Φ p z q ` ` ´ D ´ w ` pp sin ϕ p q ´ q e P p T q u ` ` sin ϕ p q e P p - T q v ` q` D ` w ´ pp sin ϕ p q ` q e P p T q u ´ ` sin ϕ p q e P p - T q v ´ q ˘ ˆ Ψ p { z q Φ p { z q - ¯ ˆ ´ e P p T q p D ` w ´ u ´ ` D ´ w ` u ` q ¨ Ψ p z q Φ p z q ` ` ´ D ` w ´ p e P p T q u ´ ` e P p - T q v ´ q` D ´ w ` p e P p T q u ` ` e P p - T q v ` q ˘ ¨ Ψ p { z q Φ p { z q - ¯ ´ . Above, the following coefficient shortcuts u ˘ : “ p´ q ℓ e i2 ϕ p T q ˘ i e - i2 ϕ p T q v ˘ : “ e i2 ϕ p - T q ˘ i p´ q ℓ e - i2 ϕ p - T q ,w ˘ : “ e i2 ϕ p q ˘ i e - i2 ϕ p q , D ˘ : “ p p q ˘ ω r p q , (19)are utilized.It is worth noting that the involvement of the functions ϕ, P in Eq.s (15)-(18) is not obligatory. Their values utilized there can be expressed in terms ofthe functions Φ , Ψ alone, provided the following identifications are taken into ONODROMY MAP 7 account ( cf Eq.s (6)): e i ϕ p T q “ Φ p e i π q , e i ϕ p q “ Φ p q , e i ϕ p - T q “ Φ p e ´ i π q ; (20) e P p T q “ Ψ p e i π q , e P p - T q “ Ψ p e ´ i π q . (21)The (second) non-obvious result to be here reported is as follows: Theorem 5.
Let the functions Φ and Ψ verify the equations (2) and (9) , respec-tively, obeying also the constraints (10) . Then ‚ the functions Φ B (15) and Ψ B (16) verify the same equations and con-straints as Φ and Ψ , respectively; ‚ the transformation B : p Φ , Ψ q ÞÑ p Φ B , Ψ B q repeated twice coincides withthe monodromy transformation M . Thus, B can be considered as a square root of M .The first assertion is proven by straightforward computation. With regard tothe second one, we replace here its formal proof with outline of derivation of thevery formulas (15)-(18) demonstrating how they had been arisen. Besides, alongthe way, a profound relationship of the equation (2) (and (1)) with another familyof differential equations is demonstrated.To that end, let us consider the two holomorphic functions E t˘u p z q definedthrough the functions Φ , Ψ as follows:(22) E t˘u p z q : “ ´ e µ p z ` { z ´ q{ z ´ ℓ { ˆ $’% ˘ i ? p Ψ p z q Φ p z qq { ` ¯ i ? p Ψ p { z q{ Φ p { z qq { ,/- . Straightforward calculation proves the following equalities(23) E p z q “ ˘p ω q ´ z ´ ℓ ´ E t˘u p { z q ` µE t˘u p z q , taking place provided the functions Φ , Ψ obey the equations (2) and (9), respec-tively. Eq.s (23) imply, in turn, the fulfillment of the equation(24) z E p z q ` ` p ℓ ` q z ` µ p ´ z q ˘ E p z q ` ` ´ µ p ℓ ` q z ` λ ˘ E p z q “ by the both functions E “ E t`u and E “ E t´u .The equations of the form (24) with arbitrary constant parameters ℓ, λ, µ con-stitute a subfamily of the family of so called double confluent Heun equations,see Refs. [10, 11, 12].Eq. (24) is a linear homogeneous differential equation with coefficients holo-morphic everywhere except zero. Hence their solutions, including E t˘u , are holo-morphic everywhere except zero including the points of divergence and roots ofthe solution Φ of the non-linear equation (9) connected with E t˘u via Eq.s (22).At the same time, the common singular point z “ for all the functions E t`u , E t´u and Φ , Ψ is actually the branching point of their common domain, the universalcover C of the punctured complex plane C ˚ (for Φ and Ψ , with their singularpoints removed). Being defined on C , the functions E t˘u behave like multi-valued SERGEY I. TERTYCHNIY functions on C ˚ and may thus undergone the monodromy transformation withoutviolation of fulfillment of Eq. (24). Similarly to the case of solutions to Eq. (2),the monodromy transformation of E t˘u can be understood as point-wise analyticcontinuations along the arcs projected to full circles with centers situated at zerowhich are passed in the counter-clockwise direction (as opposed to the case of Φ ,no singular points can now be encountered on such arcs).Substituting z “ into (22), one gets(25) E t˘u p q “ ¯ sin p p ϕ p q ¯ π qq . Thus, if ϕ p q “ π p mod π q (i.e. if(26) Φ p q “ ´ , see Eq.s (20)) then E t`u p z q ı ı E t´u p z q and the functions E t`u and E t´u arelinear independent. Moreover, since the linear space of solutions to Eq. (24)is two-dimensional, the functions E t˘u constitute its basis and any solution toEq. (24) can be represented as their linear combination with constant coefficients.Thus, the two formulas (22) ensure, in fact, the explicit representation of all thesolutions to Eq. (24) in terms of any generic solution Φ to Eq. (2) and somerelated quadrature (the function Ψ ).Conversely, the formula Φ p α q p z q : “ ´ i z l cos p α q E t`u p z q ` i sin p α q E t´u p z q cos p α q E t`u p { z q ´ i sin p α q E t´u p { z q (27)in which α stands for an arbitrary real number, represent a solution to Eq. (2)obeying the constraint | Φ p e i ωt q| “ , provided the functions E t˘u obey Eq.s (23)and ℑ E t˘u p q “ . The composition of the transformations Eq. (22) and Eq. (27)takes a solution to Eq. (2) to the function verifying the same equation. If α “ π then this map of the space of solutions to Eq. (2) into itself reduces to the identicalmap.On the other hand, in case of integer ℓ , there exist two additional (as comparedto the case of generic ℓ ) transformations preserving the space of solutions toEq. (24) [9]. One of them, which we denote L B , can be represented as follows:(28) L B : E p z q ÞÑ L B r E sp z q : “p´ q ℓ ω z ´ ℓ ` e µ p z ` z ´ q ` z r p´ z q E p´ z q ` s p´ z q E p´ z q ˘ . The invariance of the space of solutions to Eq. (24) with respect to L B can beestablished by straightforward computations, provided the following property ofthe polynomials p , q , r , s (29) p p´ z q “ p´ q ℓ ` p λ ` µ q ´ ` µz r p z q ` s p z q ˘ , q p´ z q “ µz p p z q ` q p z q ` p´ q ℓ p λ ` µ q ´ µz ` µz r p z q ` s p z q ˘ , r p´ z q “ r p z q , s p´ z q “ p´ q ℓ ` p λ ` µ q p p z q ´ µz r p z q ; ONODROMY MAP 9 and the differential equations(30) z p “ ` µ ` p ℓ ´ q z ˘ p ´ q ` p´ q ℓ z r , q “ ` λ ´ p ℓ ` q µz ˘ p ` µ q ` p´ q ℓ s ,z r “p´ q ℓ ` ` λ ` µ ˘ p ` z ` p ℓ ´ q ´ µz ˘ r ´ s ,z s “p´ q ℓ ` ` λ ` µ ˘ q ` z ` λ ´ p ℓ ` q µz ˘ r ` ` p ℓ ´ q z ´ µ ˘ s which they obey [9] are taken into account.Moreover, applying the operator L B twice and utilizing the same reductionsensured by Eq.s (29) and Eq.s (30), one finds that on solutions to Eq. (24),the function-argument E is finally restored up to some constant factor and upto modification of its argument which undergoes, ultimately, the transformationlooking like a full revolution around zero yielding no ultimate effect in projectionto C ˚ but identical to the monodromy transformation on the actual domain C of E . This result can be captured by means of the following equality: L B ˝ L B “ D ¨ M . (31)In computation of the operator composition L B ˝ L B , the factor D appears orig-inally in the following form(32) D “ z p ´ l q ` p p z q s p z q ´ q p z q r p z q ˘ . However, a straightforward computation shows that D is the first integral ofthe system of differential equations (30) which the polynomials involved in itsdefinition obey. Thus D does not actually depend on the variable z and can bedetermined setting any value of the latter. Substituting, in particular, z “ , oneobtains(33) D “ p ω q ´ D ` D ´ , where the factors on the right are defined in (19).The equality (31) and formulas (19) now say us the following. Proposition 6. If (34) D ` “ “ D ´ or, equivalently, p p q “ p ω q r p q then the linear operator L B (28) determines the automorphism of the space ofsolutions to Eq. (24) . The violation of the condition (34) would impose severe restrictions on theconstant parameters involved in Eq. (24). We assume to consider a generic caseclaiming (34) to be fulfilled throughout.The linear operator L B acting on the two-dimensional linear space of solutionsto Eq. (24) can be presented with respect to any basis of this space as some ˆ matrix. In particular, it can be shown that in the basis p E t`u , E t´u q introducedabove the matrix form of the operator L B reads(35) B “ i ℓ p ω q ´ e - P p q ˆ diag ´ ´ e ´ i4 π D ` cos ` p ϕ p q ´ π q ˘ ´ , e i4 π D ´ cos ` p ϕ p q ` π q ˘ ´ ¯ ˆ ˜ e P p T q u ´ ´ e P p - T q v ´ i p e P p T q u ´ ` e P p - T q v ´ q i p e P p T q u ` ` e P p - T q v ` q ´ e P p T q u ` ` e P p - T q v ` ¸ . Let us note now that the numerator of the fraction in Eq. (27) is a solution toEq. (24) and the denominator is also a solution get with the modified argument( { z substituted in place of z ), the former and the later being mutually conjugatedsince E t˘u p ¯ z q “ E t˘u p z q under the conditions assumed.If one replaces in (27), formally, the functions E t˘u in the numerator with thefunctions L B E t˘u expanding them further as linear combinations of the original E t˘u derived making use of the matrix (35), and carry out the correspondingtransformation of the denominator preserving its complex conjugacy with thenumerator on the unit circle, then the formula similar to (27) but with distinctparameter α results. Repeating such a transformation twice, one comes, in view of(31), to the original functions E t˘u with the original coefficients cos p α q , sin p α q (times the constant factor of D which cancels out) but with arguments undergonethe monodromy transformation. In other words, the transformation of (27) in-duced by the map (28), repeated twice, results in the monodromy transformationof the function Φ p α q . Setting α “ π , performing such a transformation of Φ p α q once, and eliminating the functions E t˘u by means of their expansions (22), theformula (15) results.The functions Θ and ˜Θ (see Eq.s (17), (18)) which have been used for deter-mination of the function Ψ alone (see Eq. (16)) are actually of notable interestin their own rights. Such functions are closely related to solutions to (2). Theycan be defined as solutions to the linear differential equations(36) ω z Θ “ ´ Φ p Θ ´ ˜Θ q , ω z ˜Θ “ Φ ´ p Θ ´ ˜Θ q obeying the initial conditions(37) Θ p q “ i , ˜Θ p q “ ´ i . If these are fulfilled, then the difference Ψ “ p q ´ p Θ ´ ˜Θ q (see Eq. (16))obeys Eq. (9) and represents the analytic continuation of the function e P p t q “ exp ş t cos ϕ p ˜ t q d ˜ t , where ϕ p t q “ Φ p e i ωt q ` Φ p e i ωt q ´ , from an arc of the unitcircle to its vicinity in C ˚ .Concerning the very formulas (17), (18), the fulfillment of Eq.s (36), (37) bythe functions the former define for the corresponding right hand side factors Φ ˘ B defined by Eq. (15) is verified by straightforward computations. ONODROMY MAP 11
In conclusion, it should be emphasized that the transformation (17), taking so-lutions to the nonlinear equation (2) to solutions of the same equation (and, then,determining the associated transformation on the space of solutions to Eq. (1)),arises here as a byproduct of the specific symmetry of the space of solutions tothe linear equation (24). Such a symmetry has been shown to exists in case ofinteger value of the parameter ℓ (sometimes called the order). The existence ofanalogue of the above B -transformation under less restrictive conditions remainsan open problem. References [1] R. L. Foote.
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