A new perspective on the Ermakov-Pinney and scalar wave equations

TL;DR

This paper introduces a novel method linking Ermakov-Pinney and scalar wave equations to first-order non-linear equations using auxiliary 1-forms, with explicit applications in Kasner space-time.

Contribution

It demonstrates that certain Ermakov-Pinney and scalar wave equations can be derived from first-order non-linear equations involving auxiliary 1-forms, providing new tools for physical applications.

Findings

Explicit construction of auxiliary 1-forms in Kasner space-time

Derivation of amplitude and phase functions for the scalar wave parametrix

Potential applications in physical models using the new method

Abstract

The first part of the paper proves that a subset of the general set of Ermakov-Pinney equations can be obtained by differentiation of a first-order non-linear differential equation. The second part of the paper proves that, similarly, the equation for the amplitude function for the parametrix of the scalar wave equation can be obtained by covariant differentiation of a first-order non-linear equation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary 1-forms (psi,rho). The 1-form psi satisfies the divergenceless condition div(psi)=0, whereas the 1-form rho fulfills the non-linear equation div(rho)+rho**2=0. The auxiliary 1-forms (psi,rho) are evaluated explicitly in 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.

Equations

$$y^{\prime \prime }+py=q{y}^{-3},$$

$$\left[\frac{d^2}{d{x}^2}+P(x)\frac{d}{dx}+Q(x)\right]u(x)=0$$

$$u(x)=\phi (x)\chi (x),$$

$$\phi (x)=\exp \left(-\frac{1}{2}\int P(x)dx\right),$$

$$\left[\frac{d^2}{d{x}^2}+J(x)\right]\chi (x)=0,$$

$$J(x)\equiv Q(x)-\frac{1}{4}{P}^2(x)-\frac{1}{2}{P}^{\prime }(x).$$

$$0<{\omega }^2<J(x)<{\Omega }^2,$$

$${\eta }^{\prime \prime }+{\mu }^2\eta =0,\mu =\omega \mathrm{or}\Omega .$$

$$\chi (x)=u(x)\exp \left(i\int \pi (x)u^{-\lambda }(x)dx\right).$$

$${\chi }^{\prime \prime }(x)$$

$$\chi (x)=A+Bx.$$

$$u^{\prime \prime }(x)+i{\pi }^{\prime }(x)u^{1-\lambda }(x)+i(2-\lambda )\pi (x)u^{\prime }(x)u^{-\lambda }(x)-{\pi }^2(x)u^{1-2\lambda }(x)=0.$$

$$u^3(x)u^{\prime \prime }(x)-{\pi }^2(x)=-i{\pi }^{\prime }(x)u^2(x).$$

$$u^3(x)u^{\prime \prime }(x)={\tau }^2.$$

$$\chi (x)=A+Bx=u(x)\exp \left(i\tau \int \frac{dx}{u^2(x)}\right),$$

$$\log (A+Bx)=\log (u(x))+i\tau \int \frac{dx}{u^2(x)}.$$

$$\frac{u^{\prime }(x)}{u(x)}+i\frac{\tau }{u^2(x)}=\frac{B}{(A+Bx)}.$$

$$\text{leavevmode}\text{vrule}height=6.0pt\text{vtop}\text{vbox}\text{hrule}\text{hrule}\text{vrule}height=6.0pt\equiv \sum _{\mu ,\nu =1}^4(g^{-1}{)}^{\mu \nu }{\nabla }_{\nu }{\nabla }_{\mu }$$

$$\varphi (x,t=0)={\zeta }_0(x),\frac{\partial \varphi }{\partial t}(x,t=0)={\zeta }_1(x),$$

$$\varphi (x,t)=\sum _{j=0}^1{E}_j(t){\zeta }_j(x),$$

$$E_j(t){\zeta }_j(x)=\sum _{k=1}^2(2\pi {)}^{-3}\int e^{i{\phi }_k(x,t,\xi )}{\alpha }_{jk}(x,t,\xi ){\tilde{\zeta }}_j(\xi )d^3\xi +R_j(t){\zeta }_j(x),$$

$$\sum_{\gamma,\beta=1}^{4}(g^{-1})^{\gamma\beta}\nabla_{\beta}\Bigr{(}\alpha^{2}\nabla_{\gamma}\varphi\Bigr{)}=0,$$

$$\sum _{\gamma ,\beta =1}^4(g^{-1}{)}^{\gamma \beta }({\nabla }_{\beta }\phi )({\nabla }_{\gamma }\phi )=\frac{\text{leavevmode}\text{vrule}height=6.0pt\text{vtop}\text{vbox}\text{hrule}\text{hrule}\text{vrule}height=6.0pt\alpha }{\alpha }.$$

$$\mathrm{div}\psi =\sum _{\gamma =1}^4{\nabla }^{\gamma }{\psi }_{\gamma }=\sum _{\gamma ,\beta =1}^4(g^{-1}{)}^{\gamma \beta }{\nabla }_{\beta }{\psi }_{\gamma }=0,$$

$${\alpha }^3\text{leavevmode}\text{vrule}height=6.0pt\text{vtop}\text{vbox}\text{hrule}\text{hrule}\text{vrule}height=6.0pt\alpha =\sum _{\gamma =1}^4{\psi }_{\gamma }{\psi }^{\gamma }=\sum _{\gamma ,\beta =1}^4(g^{-1}{)}^{\gamma \beta }{\psi }_{\beta }{\psi }_{\gamma },$$

$${\nabla }_{\gamma }\phi ={\alpha }^{-2}{\psi }_{\gamma }.$$

$$q\equiv \sum _{\gamma =1}^4{\psi }_{\gamma }{\psi }^{\gamma },$$

$$\pi (x)=\tau =\mathrm{constant}\mathrm{in}Eq.(3.5)\leftrightarrow Eq.(4.7),$$

$$Eq.(3.6)\leftrightarrow Eq.(4.8)\mathrm{with}\mathrm{constant}\mathrm{value}q\mathrm{of}\mathrm{the}\mathrm{right}-\mathrm{hand}\mathrm{side},$$

$$u^{\prime \prime }\mathrm{in}Eq.(3.6)\leftrightarrow \text{leavevmode}\text{vrule}height=6.0pt\text{vtop}\text{vbox}\text{hrule}\text{hrule}\text{vrule}height=6.0pt\alpha \mathrm{in}Eq.(4.8).$$

Full text

I Introduction

Although the modern theoretical description of gravitational interactions YCB has completely superseded Newtonian gravity, the investigation of ordinary differential equations provides an invaluable tool in the analysis of chaotic dynamical systems Poincare and in studying the interplay between linear and non-linear differential equations CIMM .

In particular, in our paper we are interested inn the Ermakov-Pinney EP1 ; EP2 non-linear differential equation

[TABLE]

which has found, along the years, many applications in theoretical physics, including quantum mechanics EP3 ; EP4 and relativistic cosmology EP5 . The first aim of our work is to provide yet 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 of second-order linear differential equations. Section $3$ applies an ansatz based on amplitude and 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 the desired solution. Section $4$ studies the correspondence between sections $2$ and $3$ on the one hand, and the parametrix construction for scalar wave equation on the other hand. Section $5$ obtains a first-order non-linear equation for the amplitude function occurring in such a parametrix. Section $6$ evaluates in Kasner space-time the auxiliary $1$-forms that are needed for a successful application of our method. Section $7$ solves the first-order equations for amplitude and phase function in Kasner space-time. Concluding remarks are then made in section $8$.

II Canonical form of second-order linear differential equations

In the theory of ordinary differential equations, it is well-known that every linear second-order equation

[TABLE]

can be solved by expressing the unknown function $u$ in the form of a product

[TABLE]

where WW ; E

[TABLE]

while $\chi$ solves the linear equation

[TABLE]

having defined

[TABLE]

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 can therefore hope to gain insight by the familiar solution of linear second-order equations solved by $\sin(x),\cos(x)$ or real-valued exponentials. More precisely, a theorem Valiron guarantees that, if $J(x)$ is continuous on the closed interval $[a,b]$, and if there exist real constants $\omega,\Omega$ such that

[TABLE]

one can compare the zeros of solutions of Eq. (2.4) with the zeros of solutions of the equations

[TABLE]

Equations (2.7) are solved by periodic functions $\sin(\mu(x-x_{0}))$ which have zeros at $x_{0}+{k\pi\over\mu}$, $k$ being an integer and $\mu$ being equal to $\omega$ or $\Omega$ as in Eq. (2.7). One can then prove that the difference $\delta$ between two adjacent zeros of a solution of Eq. (2.4) lies in between ${\pi\over\Omega}$ and ${\pi\over\omega}$ Valiron . Equation (2.4) is therefore regarded as the canonical form of every linear second-order differential equation WW .

III An ansatz in terms of amplitude and phase functions

The work in Ref. Majorana has shown a long ago that, on looking for solutions of Eq. (2.4), one can use with profit the ansatz

[TABLE]

By doing so, we find

[TABLE]

If the potential term $J(x)$ vanishes in Eq. (2.4), we therefore find ($A$ and $B$ being integration constants)

[TABLE]

On the other hand, by virtue of Eq. (3.3), Eq. (3.2) yields

[TABLE]

Equation (3.4) suggests setting $\lambda=2$, and hence we find

[TABLE]

Thus, if $\pi(x)={\rm constant}=\tau$, we obtain a particular case of the Ermakov-Pinney equation (1.1) with $p=0$ and $q=\tau^{2}$ therein, i.e.

[TABLE]

Furthermore, we can write that

[TABLE]

which implies

[TABLE]

By differentiation, this yields eventually the non-linear equation

[TABLE]

In other words, the Ermakov-Pinney equations with $p=0$ in Eq. (1.1) are equivalent to the non-linear equation (3.9), provided that $u$ is a function of class $C^{2}$. On the other hand, Eq. (3.9) allows for solutions for $u$ which are just of class $C^{1}$. An useful check of our calculation is obtained by differentiating with respect to $x$ both sides of Eq. (3.9) when $u$ is of class $C^{2}$, and then using (3.9) in order to re-express the square of ${u^{\prime}(x)\over u(x)}$. One then recovers the Ermakov-Pinney Eq. (3.6), which therefore originates from Eq. (3.9), and in turn from Eq. (3.8).

IV Amplitude-phase ansatz for the parametrix of the scalar wave equation

The work in Ref. EBD has studied the parametrix for the scalar wave equation in curved spacetime. The topic is relevant both for the mathematical theory of hyperbolic equations on manifolds E and for the modern trends in mathematical relativity M . For our purposes, we can limit ourselves to the following outline.

In a pseudo-Riemannian manifold $(M,g)$ endowed with a Levi-Civita connection $\nabla$, the wave operator

[TABLE]

is a variable-coefficient operator, and the homogeneous wave equation $\leavevmode\thinspace\hbox{\vrule height=6.0pt\vtop{\vbox{\hrule\kern-2.0pt\hbox{\vphantom{\tt/}\thinspace{\tt\ }\thinspace}} \kern 1.0pt\hrule}\vrule height=6.0pt}\thinspace\phi=0$, for given Cauchy data

[TABLE]

can be solved in the form

[TABLE]

where $E_{j}$ are the Fourier-Maslov integral operators EBD ; T

[TABLE]

the $R_{j}(t)$ being regularizing operators which smooth out the singularities upon which they act T . In the simplest possible terms, the meaning of Eq. (4.4) is that the integral operators which generalize the Fourier transform to pseudo-Riemannian manifolds involve again an integrand proportional to amplitude $\times$ eponential of ($i$ times a phase function), but, unlike flat space-time, the amplitude function depends explicitly on all cotangent bundle coordinates, while the phase function is no longer linear in these variables T .

The amplitude and phase functions, denoted by $\alpha$ (of class $C^{2}$) and $\varphi$ (of class $C^{1}$), respectively, can be obtained by solving the coupled equations EBD

[TABLE]

[TABLE]

These equations lead in turn to the following recipe EBD . First, find a divergenceless covector $\psi_{\gamma}$, i.e.

[TABLE]

then solve the non-linear equation

[TABLE]

and eventually obtain the phase from the equation

[TABLE]

Interestingly, upon defining

[TABLE]

Eq. (4.8) becomes of the type (1.1) with $p=0$. Thus, bearing in mind our finding in Sec. III, we remark that a simple but non-trivial correspondence exists between a subset of the general set of Ermakov-Pinney equations and their tensor-calculus counterpart for the analysis of the scalar wave equation, expressed by the following recipes:

[TABLE]

[TABLE]

[TABLE]

Moreover, we know that Eq. (3.6) is solved by a function solving the possibly simpler equation (3.9). This implies in turn that Eq. (4.8) for the amplitude $\alpha$ must be obtainable from the as yet unknown solution ${\cal U}$ of an unknown nonlinear equation involving at most first-order derivatives of ${\cal U}$. This is the topic of next section.

V A first-order non-linear equation for the amplitude in the parametrix

We are now going to prove that not only does our approach shed new light on the Ermakov-Pinney equation as resulting from differentiation of the non-linear equation (3.9), which is therefore more fundamental (allowing also for solutions which are only of class $C^{1}$, but not $C^{2}$), but that also the second-order equation for the amplitude $\alpha$ in the parametrix can be replaced by a first-order equation. For this purpose, since $\leavevmode\thinspace\hbox{\vrule height=6.0pt\vtop{\vbox{\hrule\kern-2.0pt\hbox{\vphantom{\tt/}\thinspace{\tt\ }\thinspace}} \kern 1.0pt\hrule}\vrule height=6.0pt}\thinspace\alpha$ should be the counterpart of $u^{\prime\prime}(x)$, and the divergenceless condition, Eq. (4.7), the counterpart of $\pi^{\prime}=0$ in Section $3$, we are led to consider the first-order non-linear equation

[TABLE]

where $\rho_{\gamma}$ are the components of a suitable covector that should generalize the behaviour of $R(x)={B\over(A+Bx)}$ on the right-hand side of Eq. (3.9).111Note that, strictly speaking, since $\alpha$, $\varphi$ and $\psi_{\gamma}$ are real-valued, we are dealing with a complex-valued vector field $\sum_{\gamma=1}^{4}\rho^{\gamma}{\partial\over\partial x^{\gamma}}$ OS , with the associated dual concept of complex-valued $1$-form field. At this stage, inspired by Section $3$, we perform covariant differentiation $\nabla^{\gamma}$ of both sides of Eq. (5.1), finding first the equation

[TABLE]

because the divergenceless condition (4.7) holds by assumption. Next, we exploit Eq. (5.1) by re-expressing all first covariant derivatives of $\alpha$ in Eq. (5.2) in the form

[TABLE]

hence finding

[TABLE]

In this equation, the terms proportional to $\sum_{\gamma=1}^{4}\rho^{\gamma}\psi_{\gamma}$ add up to [math], and hence we obtain

[TABLE]

Thus, provided that

[TABLE]

we obtain eventually the second-order equation (4.8) for the amplitude $\alpha$ in the parametrix for the scalar wave equation. Remarkably, Eq. (5.5) is precisely the tensorial generalization of the differential equation obeyed by the right-hand side $R(x)={B\over(A+Bx)}$ of Eq. (3.9), because

[TABLE]

VI Evaluation of the auxiliary $1$-forms $\psi$ and $\rho$

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 coupled equations (4.5) and (4.6), or (4.8) and (4.9). The aim of the present section is therefore to prove that the $1$-forms $\psi$ and $\rho$ fulfilling Eqs. (4.7) and (5.5) are explicitly computable in a non-trivial case of physical interest.

For this purpose, inspired again by our Ref. EBD , we consider Kasner spacetime, whose metric in $c=1$ units reads as YCB

[TABLE]

where the real numbers $p_{1},p_{2},p_{3}$ satisfy the condition

[TABLE]

as well as the unit $2$-sphere condition

[TABLE]

Let us assume for simplicity that the only non-vanishing component of the desired $1$-form $\psi$ is $\psi_{1}=\psi_{0}(t)$. Hence we find (since the Christoffel coefficients $\Gamma_{kk}^{0}=p_{k}t^{2p_{k}-1}$, $\forall k=1,2,3$)

[TABLE]

The vanishing divergence condition (4.7) is therefore satisfied by

[TABLE]

Similarly, assuming that also the auxiliary $1$-form $\rho$ has only one non-vanishing component $\rho_{0}(t)$, one finds

[TABLE]

But we know from section $5$ that $\rho$ should be complex-valued, hence we set

[TABLE]

$\beta_{1}$ and $\beta_{2}$ being the real and imaginary part of $\rho_{0}$, respectively. Thus, by virtue of the identity (6.6), Eq. (5.5) leads to the non-linear coupled system

[TABLE]

[TABLE]

Equations (6.8) and (6.9) suggest re-expressing them in terms of the unknown function

[TABLE]

This leads to the equivalent system

[TABLE]

[TABLE]

By insertion of Eq. (6.12) into Eq. (6.11), we find

[TABLE]

[TABLE]

and hence Eq. (6.12) yields for $B(t)$ the equation

[TABLE]

which is equivalent to the linear differential equation222Note also that $(1+B^{2}(t))$ in Eq. (6.15) can never vanish, bearing in mind the real nature of $B(t)$ from the definition (6.10).

[TABLE]

Equation (6.16) implies that $B^{\prime}(t)$ is proportional to ${1\over t}$, and hence, upon introducing the real parameter $\sigma$, one can write that ($\kappa$ being the same parameter used in (6.5))

[TABLE]

and hence

[TABLE]

Hereafter we set $B(T)=0$ for simplicity. By virtue of (6.13), (6.14) and (6.18) we obtain eventually, upon defining333The work in Ref. BE19 arrives instead at Eq. (6.19) by solving directly for the amplitude $\alpha$, without making any use of the auxiliary $1$-form $\rho$ and of our Eq. (5.1).

[TABLE]

the exact formulae

[TABLE]

[TABLE]

Remarkably, the exact solution of the non-linear equations (6.8) and (6.9) has been obtained from 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 $\alpha$ from the equation

[TABLE]

and eventually the phase function $\varphi$ from Eq. (4.9), which reads in our case

[TABLE]

From Eq. (7.1) we obtain, in Kasner space-time, the ordinary differential equation

[TABLE]

i.e., upon separating real and imaginary part, the pair of equations

[TABLE]

[TABLE]

Hence we find in Kasner space-time the amplitude function

[TABLE]

for which $\alpha(T)=\sqrt{{\kappa^{2}\over\sigma}}$, as well as the phase function

[TABLE]

which holds for all positive values of the real ratio ${T\over(t-T)}$. It should be stressed that, in a generic space-time without any symmetry, the amplitude and phase, if computable, will depend on all cotangent bundle local coordinates T (see further comments in Section $8$).

VIII Concluding remarks

In our paper, starting from well known properties in the theory of linear differential equations, 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 the function $u$ solving (3.9) is taken to be at least of class $C^{2}$.

By comparison with the construction of amplitude and phase in the scalar parametrix, we have then proved that finding the amplitude $\alpha$ for which Eq. (4.8) holds with a divergenceless covector $\psi_{\gamma}$, is equivalent to finding also a covector $\rho_{\gamma}$ for which Eq. (5.5) holds. One can then obtain the amplitude $\alpha$ from the first-order non-linear equation (7.1). Our successful calculations of sections $6$ and $7$, where we have evaluated the auxiliary $1$-forms with components

[TABLE]

[TABLE]

prove that our original method leads to a powerful tool for studying the scalar wave equation with the associated parametrix. This will be of concrete interest in applied mathematics and in the theoretical physics of fundamental interactions.

Note also that, in principle, there might exist solutions of Eq. (7.1) which are of class $C^{1}$ but not $C^{2}$. Thus, the consideration of Eq. (5.1) is closer to the modern emphasis on finding new solutions of partial differential equations under weaker differentiability properties. Of course, the corresponding physical interpretation is a relevant open problem.

Acknowledgements.

G. E. is grateful to Dipartimento di Fisica “Ettore Pancini” for hospitality and support, and to D. Bini for a careful reading of a draft version, and for encouragement.