A new perspective on the Ermakov-Pinney and scalar wave equations
aa r X i v : . [ m a t h . G M ] J un A new perspective on the Ermakov-Pinney and scalar waveequations
Giampiero Esposito ORCID: 0000-0001-5930-8366 ∗ Istituto Nazionale di Fisica Nucleare, Sezione di Napoli,Complesso Universitario di Monte S. Angelo,Via Cintia Edificio 6, 80126 Napoli, Italy
Marica Minucci ORCID: 0000-0002-7095-5115 † Dipartimento di Fisica “Ettore Pancini”, Universit`a Federico II,Complesso Universitario di Monte S. Angelo,Via Cintia Edificio 6, 80126 Napoli, Italy (Dated: June 19, 2019)
Abstract
The first part of the paper proves that a subset of the general set of Ermakov-Pinney equationscan be obtained by differentiation of a first-order non-linear differential equation. The second partof the paper proves that, similarly, the equation for the amplitude function for the parametrix ofthe scalar wave equation can be obtained by covariant differentiation of a first-order non-linearequation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary1-forms ( ψ, ρ ). The 1-form ψ satisfies the divergenceless condition div( ψ ) = 0, whereas the 1-form ρ fulfills the non-linear equation div( ρ ) + ρ = 0. The auxiliary 1-forms ( ψ, ρ ) are evaluated explicitlyin Kasner space-time, and hence also amplitude and phase function in the parametrix are obtained.Thus, the novel method developed in this paper can be used with profit in physical applications. PACS numbers: 02.30.Hq, 02.30.Jr ∗ E-mail: [email protected] † E-mail: [email protected] . INTRODUCTION Although the modern theoretical description of gravitational interactions [1] has com-pletely superseded Newtonian gravity, the investigation of ordinary differential equationsprovides an invaluable tool in the analysis of chaotic dynamical systems [2] and in studyingthe interplay between linear and non-linear differential equations [3].In particular, in our paper we are interested inn the Ermakov-Pinney [4, 5] non-lineardifferential equation y ′′ + py = qy − , (1.1)which has found, along the years, many applications in theoretical physics, including quan-tum mechanics [6, 7] and relativistic cosmology [8]. The first aim of our work is to provideyet another perspective on the way of arriving at equations of type (1.1). For this purpose,section 2 provides a concise summary of well-established results on the canonical form ofsecond-order linear differential equations. Section 3 applies an ansatz based on amplitudeand phase functions, and proves eventually equivalence between Eqs. of type (1.1) with p = 0 and our Eq. (3.9), which is a nonlinear equation with only first derivatives of thedesired solution. Section 4 studies the correspondence between sections 2 and 3 on the onehand, and the parametrix construction for scalar wave equation on the other hand. Section5 obtains a first-order non-linear equation for the amplitude function occurring in such aparametrix. Section 6 evaluates in Kasner space-time the auxiliary 1-forms that are neededfor a successful application of our method. Section 7 solves the first-order equations foramplitude and phase function in Kasner space-time. Concluding remarks are then made insection 8. II. CANONICAL FORM OF SECOND-ORDER LINEAR DIFFERENTIALEQUATIONS
In the theory of ordinary differential equations, it is well-known that every linear second-order equation (cid:20) d dx + P ( x ) ddx + Q ( x ) (cid:21) u ( x ) = 0 (2.1)can be solved by expressing the unknown function u in the form of a product u ( x ) = ϕ ( x ) χ ( x ) , (2.2)2here [9, 10] ϕ ( x ) = exp (cid:18) − Z P ( x ) dx (cid:19) , (2.3)while χ solves the linear equation (cid:20) d dx + J ( x ) (cid:21) χ ( x ) = 0 , (2.4)having defined J ( x ) ≡ Q ( x ) − P ( x ) − P ′ ( x ) . (2.5)All complications arising from the variable nature of coefficient functions P and Q in Eq.(2.1) are encoded into the potential term J ( x ) of Eq. (2.4) defined in Eq. (2.5). One cantherefore hope to gain insight by the familiar solution of linear second-order equations solvedby sin( x ) , cos( x ) or real-valued exponentials. More precisely, a theorem [11] guarantees that,if J ( x ) is continuous on the closed interval [ a, b ], and if there exist real constants ω, Ω suchthat 0 < ω < J ( x ) < Ω , (2.6)one can compare the zeros of solutions of Eq. (2.4) with the zeros of solutions of the equations η ′′ + µ η = 0 , µ = ω or Ω . (2.7)Equations (2.7) are solved by periodic functions sin( µ ( x − x )) which have zeros at x + kπµ , k being an integer and µ being equal to ω or Ω as in Eq. (2.7). One can then prove that thedifference δ between two adjacent zeros of a solution of Eq. (2.4) lies in between π Ω and πω [11]. Equation (2.4) is therefore regarded as the canonical form of every linear second-orderdifferential equation [9]. III. AN ANSATZ IN TERMS OF AMPLITUDE AND PHASE FUNCTIONS
The work in Ref. [12] has shown a long ago that, on looking for solutions of Eq. (2.4),one can use with profit the ansatz χ ( x ) = u ( x )exp (cid:18) i Z π ( x ) u − λ ( x ) dx (cid:19) . (3.1)By doing so, we find χ ′′ ( x ) = (cid:2) u ′′ ( x ) + iπ ′ ( x ) u − λ ( x ) + i (2 − λ ) π ( x ) u ′ ( x ) u − λ ( x ) − π ( x ) u − λ ( x ) (cid:3) × exp (cid:18) i Z π ( x ) u − λ ( x ) dx (cid:19) . (3.2)3f the potential term J ( x ) vanishes in Eq. (2.4), we therefore find ( A and B being integrationconstants) χ ( x ) = A + Bx. (3.3)On the other hand, by virtue of Eq. (3.3), Eq. (3.2) yields u ′′ ( x ) + iπ ′ ( x ) u − λ ( x ) + i (2 − λ ) π ( x ) u ′ ( x ) u − λ ( x ) − π ( x ) u − λ ( x ) = 0 . (3.4)Equation (3.4) suggests setting λ = 2, and hence we find u ( x ) u ′′ ( x ) − π ( x ) = − iπ ′ ( x ) u ( x ) . (3.5)Thus, if π ( x ) = constant = τ , we obtain a particular case of the Ermakov-Pinney equation(1.1) with p = 0 and q = τ therein, i.e. u ( x ) u ′′ ( x ) = τ . (3.6)Furthermore, we can write that χ ( x ) = A + Bx = u ( x )exp (cid:18) iτ Z dxu ( x ) (cid:19) , (3.7)which implies log( A + Bx ) = log( u ( x )) + iτ Z dxu ( x ) . (3.8)By differentiation, this yields eventually the non-linear equation u ′ ( x ) u ( x ) + i τu ( x ) = B ( A + Bx ) . (3.9)In other words, the Ermakov-Pinney equations with p = 0 in Eq. (1.1) are equivalent to thenon-linear equation (3.9), provided that u is a function of class C . On the other hand, Eq.(3.9) allows for solutions for u which are just of class C . An useful check of our calculationis obtained by differentiating with respect to x both sides of Eq. (3.9) when u is of class C , and then using (3.9) in order to re-express the square of u ′ ( x ) u ( x ) . One then recovers theErmakov-Pinney Eq. (3.6), which therefore originates from Eq. (3.9), and in turn from Eq.(3.8). 4 V. AMPLITUDE-PHASE ANSATZ FOR THE PARAMETRIX OF THE SCALARWAVE EQUATION
The work in Ref. [13] has studied the parametrix for the scalar wave equation in curvedspacetime. The topic is relevant both for the mathematical theory of hyperbolic equations onmanifolds [10] and for the modern trends in mathematical relativity [14]. For our purposes,we can limit ourselves to the following outline.In a pseudo-Riemannian manifold (
M, g ) endowed with a Levi-Civita connection ∇ , thewave operator ≡ X µ,ν =1 ( g − ) µν ∇ ν ∇ µ (4.1)is a variable-coefficient operator, and the homogeneous wave equation φ = 0, for givenCauchy data φ ( x, t = 0) = ζ ( x ) , ∂φ∂t ( x, t = 0) = ζ ( x ) , (4.2)can be solved in the form φ ( x, t ) = X j =0 E j ( t ) ζ j ( x ) , (4.3)where E j are the Fourier-Maslov integral operators [13, 15] E j ( t ) ζ j ( x ) = X k =1 (2 π ) − Z e iϕ k ( x,t,ξ ) α jk ( x, t, ξ ) ˜ ζ j ( ξ ) d ξ + R j ( t ) ζ j ( x ) , (4.4)the R j ( t ) being regularizing operators which smooth out the singularities upon which theyact [15]. In the simplest possible terms, the meaning of Eq. (4.4) is that the integraloperators which generalize the Fourier transform to pseudo-Riemannian manifolds involveagain an integrand proportional to amplitude × eponential of ( i times a phase function),but, unlike flat space-time, the amplitude function depends explicitly on all cotangent bundlecoordinates, while the phase function is no longer linear in these variables [15].The amplitude and phase functions, denoted by α (of class C ) and ϕ (of class C ),respectively, can be obtained by solving the coupled equations [13] X γ,β =1 ( g − ) γβ ∇ β (cid:16) α ∇ γ ϕ (cid:17) = 0 , (4.5) X γ,β =1 ( g − ) γβ ( ∇ β ϕ )( ∇ γ ϕ ) = αα . (4.6)5hese equations lead in turn to the following recipe [13]. First, find a divergenceless covector ψ γ , i.e. div ψ = X γ =1 ∇ γ ψ γ = X γ,β =1 ( g − ) γβ ∇ β ψ γ = 0 , (4.7)then solve the non-linear equation α α = X γ =1 ψ γ ψ γ = X γ,β =1 ( g − ) γβ ψ β ψ γ , (4.8)and eventually obtain the phase from the equation ∇ γ ϕ = α − ψ γ . (4.9)Interestingly, upon defining q ≡ X γ =1 ψ γ ψ γ , (4.10)Eq. (4.8) becomes of the type (1.1) with p = 0. Thus, bearing in mind our finding inSec. III, we remark that a simple but non-trivial correspondence exists between a subset ofthe general set of Ermakov-Pinney equations and their tensor-calculus counterpart for theanalysis of the scalar wave equation, expressed by the following recipes: π ( x ) = τ = constant in Eq . (3 . ↔ Eq . (4 . , Eq . (3 . ↔ Eq . (4 .
8) with constant value q of the right − hand side ,u ′′ in Eq . (3 . ↔ α in Eq . (4 . . Moreover, we know that Eq. (3.6) is solved by a function solving the possibly simplerequation (3.9). This implies in turn that Eq. (4.8) for the amplitude α must be obtainablefrom the as yet unknown solution U of an unknown nonlinear equation involving at mostfirst-order derivatives of U . This is the topic of next section. V. A FIRST-ORDER NON-LINEAR EQUATION FOR THE AMPLITUDE INTHE PARAMETRIX
We are now going to prove that not only does our approach shed new light on theErmakov-Pinney equation as resulting from differentiation of the non-linear equation (3.9),6hich is therefore more fundamental (allowing also for solutions which are only of class C ,but not C ), but that also the second-order equation for the amplitude α in the parametrixcan be replaced by a first-order equation. For this purpose, since α should be the coun-terpart of u ′′ ( x ), and the divergenceless condition, Eq. (4.7), the counterpart of π ′ = 0 inSection 3, we are led to consider the first-order non-linear equation( ∇ γ α ) α + i ψ γ α = ρ γ , (5.1)where ρ γ are the components of a suitable covector that should generalize the behaviour of R ( x ) = B ( A + Bx ) on the right-hand side of Eq. (3.9). At this stage, inspired by Section 3, weperform covariant differentiation ∇ γ of both sides of Eq. (5.1), finding first the equation − X γ,β =1 ( g − ) γβ ( ∇ β α ) α ( ∇ γ α ) α + αα − i ψ γ α ( ∇ γ α ) α = X γ,β =1 ( g − ) γβ ∇ β ρ γ , (5.2)because the divergenceless condition (4.7) holds by assumption. Next, we exploit Eq. (5.1)by re-expressing all first covariant derivatives of α in Eq. (5.2) in the form( ∇ β α ) α = ρ β − i ψ β α , hence finding − X γ,β =1 ( g − ) γβ (cid:18) ρ β − i ψ β α (cid:19) (cid:18) ρ γ − i ψ γ α (cid:19) + αα − i X γ =1 ψ γ α (cid:18) ρ γ − i ψ γ α (cid:19) = X γ,β =1 ( g − ) γβ ∇ β ρ γ . (5.3)In this equation, the terms proportional to P γ =1 ρ γ ψ γ add up to 0, and hence we obtain αα − X γ =1 ψ γ ψ γ α = X γ,β =1 ( g − ) γβ (cid:16) ∇ β ρ γ + ρ β ρ γ (cid:17) . (5.4)Thus, provided that X γ,β =1 ( g − ) γβ (cid:16) ∇ β ρ γ + ρ β ρ γ (cid:17) = 0 , (5.5) Note that, strictly speaking, since α , ϕ and ψ γ are real-valued, we are dealing with a complex-valuedvector field P γ =1 ρ γ ∂∂x γ [16], with the associated dual concept of complex-valued 1-form field.
7e obtain eventually the second-order equation (4.8) for the amplitude α in the parametrixfor the scalar wave equation. Remarkably, Eq. (5.5) is precisely the tensorial generalizationof the differential equation obeyed by the right-hand side R ( x ) = B ( A + Bx ) of Eq. (3.9),because ddx R ( x ) + R ( x ) = − B ( A + Bx ) − + B ( A + Bx ) − = 0 . VI. EVALUATION OF THE AUXILIARY -FORMS ψ AND ρ So far, the critical reader might think that our method, despite being elegant and correct,does not offer any concrete advantage with respect to the direct investigation of the coupledequations (4.5) and (4.6), or (4.8) and (4.9). The aim of the present section is therefore toprove that the 1-forms ψ and ρ fulfilling Eqs. (4.7) and (5.5) are explicitly computable in anon-trivial case of physical interest.For this purpose, inspired again by our Ref. [13], we consider Kasner spacetime, whosemetric in c = 1 units reads as [1] g = − dt ⊗ dt + t p dx ⊗ dx + t p dy ⊗ dy + t p dz ⊗ dz, (6.1)where the real numbers p , p , p satisfy the condition X k =1 p k = 1 , (6.2)as well as the unit 2-sphere condition X k =1 ( p k ) = 1 . (6.3)Let us assume for simplicity that the only non-vanishing component of the desired 1-form ψ is ψ = ψ ( t ). Hence we find (since the Christoffel coefficients Γ kk = p k t p k − , ∀ k = 1 , , ψ ) = X µ,ν =1 ( g − ) µν ∇ ν ψ µ = − ∂ ψ + ψ Γ − X k =1 t − p k Γ kk ! = − dψ dt − ψ t X k =1 p k = − dψ dt − ψ t . (6.4)The vanishing divergence condition (4.7) is therefore satisfied by ψ = κt , κ = constant . (6.5)8imilarly, assuming that also the auxiliary 1-form ρ has only one non-vanishing component ρ ( t ), one findsdiv( ρ ) + ρ = − dρ dt − ρ t + ( g − ) ( ρ ) = − dρ dt − ρ t − ( ρ ) . (6.6)But we know from section 5 that ρ should be complex-valued, hence we set ρ ( t ) = β ( t ) + iβ ( t ) , (6.7) β and β being the real and imaginary part of ρ , respectively. Thus, by virtue of theidentity (6.6), Eq. (5.5) leads to the non-linear coupled system dβ dt + β ( t ) t + ( β ( t )) − ( β ( t )) = 0 , (6.8) dβ dt + β ( t ) t + 2 β ( t ) β ( t ) = 0 . (6.9)Equations (6.8) and (6.9) suggest re-expressing them in terms of the unknown function B ( t ) ≡ β ( t ) β ( t ) . (6.10)This leads to the equivalent system dBdt β ( t ) + B ( t ) (cid:18) dβ dt + β ( t ) t (cid:19) + ( B ( t ) − β ( t )) = 0 , (6.11) dβ dt + β ( t ) t + 2 B ( t )( β ( t )) = 0 . (6.12)By insertion of Eq. (6.12) into Eq. (6.11), we find β ( t ) = B ′ ( t )(1 + B ( t )) , (6.13) β ( t ) = B ( t ) β ( t ) = B ( t ) B ′ ( t )(1 + B ( t )) , (6.14)and hence Eq. (6.12) yields for B ( t ) the equation B ′′ ( t )(1 + B ( t )) − B ( t )( B ′ ( t )) (1 + B ( t )) + 1 t B ′ ( t )(1 + B ( t )) + 2 B ( t )( B ′ ( t )) (1 + B ( t )) = 0 , (6.15)which is equivalent to the linear differential equation B ′′ ( t ) + 1 t B ′ ( t ) = 0 . (6.16) Note also that (1 + B ( t )) in Eq. (6.15) can never vanish, bearing in mind the real nature of B ( t ) fromthe definition (6.10). B ′ ( t ) is proportional to t , and hence, upon introducing thereal parameter σ , one can write that ( κ being the same parameter used in (6.5)) dBdt = σκ t , (6.17)and hence B ( t ) = B ( T ) + σκ log (cid:18) tT (cid:19) . (6.18)Hereafter we set B ( T ) = 0 for simplicity. By virtue of (6.13), (6.14) and (6.18) we obtaineventually, upon defining D σ ( t ) ≡ κ σ + σ log (cid:18) tT (cid:19) , (6.19)the exact formulae β ( t ) = σt log (cid:0) tT (cid:1) D σ ( t ) , (6.20) β ( t ) = κt D σ ( t ) . (6.21)Remarkably, the exact solution of the non-linear equations (6.8) and (6.9) has been obtainedfrom the general solution of the linear equation (6.16). VII. AMPLITUDE AND PHASE FUNCTIONS IN KASNER SPACE-TIME
In light of (5.1), we can now evaluate the amplitude function α from the equation ∇ γ (log( α )) + i ψ γ α = ρ γ , (7.1)and eventually the phase function ϕ from Eq. (4.9), which reads in our case dϕdt = κtα . (7.2)From Eq. (7.1) we obtain, in Kasner space-time, the ordinary differential equation ddt log( α ( t )) + i κtα ( t ) = β ( t ) + iβ ( t ) , (7.3)i.e., upon separating real and imaginary part, the pair of equations ddt log( α ( t )) = β ( t ) , (7.4) The work in Ref. [17] arrives instead at Eq. (6.19) by solving directly for the amplitude α , withoutmaking any use of the auxiliary 1-form ρ and of our Eq. (5.1). tα ( t ) = β ( t ) = ϕ ( t ) . (7.5)Hence we find in Kasner space-time the amplitude function α ( t ) = α ( T )exp Z tT στ log (cid:0) τT (cid:1) D σ ( τ ) dτ = s κ σ + σ log (cid:18) tT (cid:19) , (7.6)for which α ( T ) = q κ σ , as well as the phase function ϕ ( t ) = ϕ ( T ) + κ Z tT dττ D σ ( τ ) = ϕ ( T ) + arctan (cid:18) σκ log (cid:18) tT (cid:19)(cid:19) , (7.7)which holds for all positive values of the real ratio T ( t − T ) . It should be stressed that, ina generic space-time without any symmetry, the amplitude and phase, if computable, willdepend on all cotangent bundle local coordinates [15] (see further comments in Section 8). VIII. CONCLUDING REMARKS
In our paper, starting from well known properties in the theory of linear differentialequations, we have first proved that the Ermakov-Pinney equations with p = 0 in Eq.(1.1) result from differentiation of the more fundamental equation (3.9), provided that thefunction u solving (3.9) is taken to be at least of class C .By comparison with the construction of amplitude and phase in the scalar parametrix,we have then proved that finding the amplitude α for which Eq. (4.8) holds with a diver-genceless covector ψ γ , is equivalent to finding also a covector ρ γ for which Eq. (5.5) holds.One can then obtain the amplitude α from the first-order non-linear equation (7.1). Oursuccessful calculations of sections 6 and 7, where we have evaluated the auxiliary 1-formswith components ψ µ ( t ) = (cid:16) κt , , , (cid:17) , (8.1) ρ µ ( t ) = (cid:18) tD σ ( t ) (cid:18) σ log (cid:18) tT (cid:19) + iκ (cid:19) , , , (cid:19) , (8.2)prove that our original method leads to a powerful tool for studying the scalar wave equationwith the associated parametrix. This will be of concrete interest in applied mathematicsand in the theoretical physics of fundamental interactions.Note also that, in principle, there might exist solutions of Eq. (7.1) which are of class C but not C . Thus, the consideration of Eq. (5.1) is closer to the modern emphasis on finding11ew solutions of partial differential equations under weaker differentiability properties. Ofcourse, the corresponding physical interpretation is a relevant open problem. Acknowledgments
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