Curvilinear parallelogram identity and mean-value property for

a semilinear hyperbolic equation of second-order

V. I. Korzyuk, J. V. Rudzko

Key words: hyperbolic equation, characteristics parallelogram, mean-value
theorem, mean-value property.

AMS Mathematics Subject Classification: 35B05 (Primary), 35L10, 35L71, 35C15
(Secondary).

Abstract. In this paper, we discuss some of the important qualitative
properties of solutions of second-order hyperbolic equations, whose
coefficients of the terms involving the second-order derivatives are
independent of the desired function and its derivatives. Solutions of these
equations have a special property called the curvilinear parallelogram
identity (or the mean-value property), which can be used to solve some
initial-boundary value problems.

##  1 Introduction

The terms ‘‘mean value theorem’’, ‘‘mean value property’’, ‘‘mean formula’’,
and ‘‘mean value’’ are quite common in mathematics (e. g., real and complex
analysis, probability theory, partial differential equations) and physics. But
they may pertain to diverse phenomena.

In the theory of partial differential equations, mean value theorems for
harmonic functions and solutions of various elliptic equations are best known.
They include the classical mean value property for harmonic functions [12] and
the results obtained in works [9, 7, 8, 27] for more general elliptic
equations and elliptic operators. Similar theorems are formulated for
(hypoelliptic) parabolic equations [16, 17, 18].

Such facts can be established not only for elliptic and parabolic equations
but also for hyperbolic ones. First of all, we should mention the classical
Asgeirsson’s mean value theorem [3, 6] the ultrahyperbolic differential
equation, and the mean value theorem of Bitsadze and Nakhushev for the wave
equation [2]. Spherical means can be used to solve initial-value problems as
it’s done in the work [10] for the wave equation and the Darboux equation.
Using a symbolic approach [28] several results [24, 22, 23, 30, 25, 31, 29,
26, 33, 32] connected with mean values of solutions of various differential
equations were obtained in works of Polovinkin and Meshkov et al. It should
also be said that in these works, the parallelogram identity (parallelogram
rule) for the wave equation (which the authors call ‘difference mean-value
formula’) was generalized to the following cases: a (nonstrictly) hyperbolic
equation with constant coefficients of third-order [24], fourth-order [22],
higher-order [32], an equation with constant coefficients and with the
operator represented by the product of the first order hyperbolic operators
and the second-order elliptic operators [29]. These results can be used to
obtain analytical and numerical solutions to differential equations as it was
done in [12, 14, 11, 20, 21]. However, these results are given mainly for
equations with constant coefficients because of the methods used (Fourier
transform, search for accompanying distribution with compact support).

Moreover, the characteristic parallelogram of differential equations has some
applications in hydrodynamics [19].

In this paper, we derive the identity of a curvilinear characteristic
parallelogram for a general semilinear second-order hyperbolic equation using
the method of characteristics [12]. This identity can be considered as the
mean value theorem in some sense.

##  2 Semilinear hyperbolic equation

In the domain $\Omega\subseteq\mathbb{R}^{2}$ of two independent variables
$\mathbf{x}=(x_{1},x_{2})\in\Omega\subseteq\mathbb{R}^{2}$, consider the
semilinear hyperbolic equation of second-order

$Au(x_{1},x_{2})=f(x_{1},x_{2},u(x_{1},x_{2}),\partial_{x_{1}}u(x_{1},x_{2}),\partial_{x_{2}}u(x_{1},x_{2})),$
(2.1)

where the operator $A$ is defined as

$Au(x_{1},x_{2}):=a(x_{1},x_{2})\partial_{x_{1}}^{2}u(x_{1},x_{2})+2b(x_{1},x_{2})\partial_{x_{1}}\partial_{x_{2}}u(x_{1},x_{2})+c(x_{1},x_{2})\partial_{x_{2}}^{2}u(x_{1},x_{2}),$

and is hyperbolic (this means $b^{2}(\mathbf{x})-a(\mathbf{x})c(\mathbf{x})>0$
for any $x\in\Omega$).

Eq. (2.1) has two families of characteristics: $\gamma_{1}(x_{1},x_{2})$ and
$\gamma_{2}(x_{1},x_{2})$, which are the first integrals of the ordinary
differential equation [12]

$a(\mathbf{x})(\mathrm{d}x_{2})^{2}-2b(\mathbf{x})\mathrm{d}x_{1}\mathrm{d}x_{2}+c(\mathbf{x})(\mathrm{d}x_{1})^{2}=0,$
(2.2)

and solutions of the equation of characteristics [12]

$a\left(\frac{\partial\gamma_{i}}{\partial
x_{1}}\right)^{2}+2b\frac{\partial\gamma_{i}}{\partial
x_{1}}\frac{\partial\gamma_{i}}{\partial
x_{2}}+c\left(\frac{\partial\gamma_{i}}{\partial x_{2}}\right)^{2}=0,\quad
i=1,2.$ (2.3)

It is well known [12] that Eq. (2.2), in general, can be decomposed into two
equations

$a(\mathbf{x})\mathrm{d}x_{2}-(b(\mathbf{x})\pm\sqrt{b^{2}(\mathbf{x})-a(\mathbf{x})c(\mathbf{x})})\mathrm{d}x_{1}=0,\text{
if }a(\mathbf{x})\neq 0,$

or

$c(\mathbf{x})\mathrm{d}x_{1}-(b(\mathbf{x})\pm\sqrt{b^{2}(\mathbf{x})-a(\mathbf{x})c(\mathbf{x})})\mathrm{d}x_{2}=0,\text{
if }c(\mathbf{x})\neq 0,$

or

$\mathrm{d}x_{1}\mathrm{d}x_{2}=0,\text{ if }a(\mathbf{x})=c(\mathbf{x})=0.$

Therefore, we can assume that $\gamma_{1}$ and $\gamma_{2}$ are the first
integrals of different differential equations and they are functionally
independent since the Jacobian
$\left|\dfrac{\partial(\gamma_{1},\gamma_{2})}{\partial(x_{1},x_{2})}\right|$
is nonzero [12].

If the curves $\gamma_{i}$, $i=1,2$, have a parametric representation
$(x_{1}^{(i)}(t),x_{2}^{(i)}(t))$, where $x_{j}^{(i)}$, $j=1,2$, are some
twice continuously differentiable functions, then the equation holds [4]

$a\big{(}Dx_{2}^{(i)}\big{)}^{2}-2bDx_{1}^{(i)}Dx_{2}^{(i)}+c\big{(}Dx_{1}^{(i)}\big{)}^{2}=0,\quad
i=1,2,$

where $D$ is the operator of the ordinary derivative.

##  3 Curvilinear characteristic parallelogram

######  Definition 1.

Curvilinear characteristic parallelogram of the hyperbolic differential
equation (2.1) is a set
$\Pi=\\{\mathbf{x}\,|\,\gamma_{1}(\mathbf{x})\in[l_{1},l_{2}]\land\gamma_{2}(\mathbf{x})\in[r_{1},r_{2}]\\}$,
where $l_{1}$, $l_{2}$, $r_{1}$, $r_{2}$ are some real numbers and
$\gamma_{i}$, $i=1,2$ are two different functionally independent
characteristics.

######  Remark 1.

Definition 1 is correct. It is known [1] that any other first integral (2.2)
has the form $q\circ\gamma_{1}$, where $q$ is some continuously differentiable
function. If $\gamma_{1}(\mathbf{x})\in[l_{1},l_{2}]$, then, due to the
continuity of $q$, $q(\gamma_{1}(\mathbf{x}))\in
q([l_{1},l_{2}])=[\widetilde{l}_{1},\widetilde{l}_{2}]$. Thus, the curvilinear
characteristic parallelogram does not depend on the considered
characteristics.

Fig. 1. Curvilinear characteristic parallelogram

######  Definition 2.

Vertices of the curvilinear characteristic parallelogram
$\Pi=\\{\mathbf{x}\,|\,\gamma_{1}(\mathbf{x})\in[l_{1},l_{2}]\land\gamma_{2}(\mathbf{x})\in[r_{1},r_{2}]\\}$
are points $\mathbf{x}$ such that
$\gamma_{1}(x)=l_{i}\land\gamma_{2}(x)=r_{j}$,
$(i,j)\in\\{1,2\\}\times\\{1,2\\}$.

######  Remark 2.

Definition 2 is correct. We should show that $q\circ\gamma_{1}$, where $q$ is
some continuously differentiable function, maps $[l_{1},l_{2}]$ into
$[\widetilde{l}_{1},\widetilde{l}_{2}]$ and $\partial([l_{1},l_{2}])$ into
$\partial([\widetilde{l}_{1},\widetilde{l}_{2}])$. If the function $q$ is
increasing or decreasing, these mappings must be true. But if the function $q$
does not satisfy these conditions, then there exists at least one point
$l_{0}\in(l_{1},l_{2})$ such that $q^{\prime}(l_{0})=0$. Due to the continuity
of $q$, there exists a point $\mathbf{x}\in\Pi$ such that
$\gamma_{1}(\mathbf{x})=l_{0}\in(l_{1},l_{2})$ This implies

$\left|\frac{\partial(q\circ\gamma_{1},\gamma_{2})}{\partial(x_{1},x_{2})}\right|(\mathbf{x})=\begin{vmatrix}q^{\prime}(\gamma_{1}(\mathbf{x}))\partial_{x_{1}}\gamma_{1}(\mathbf{x})&q^{\prime}(\gamma_{1}(\mathbf{x}))\partial_{x_{2}}\gamma_{1}(\mathbf{x})\\\
\partial_{x_{1}}\gamma_{2}(\mathbf{x})&\partial_{x_{2}}\gamma_{2}(\mathbf{x})\end{vmatrix}=0\text{
when }\gamma_{1}(\mathbf{x})=l_{0}.$

But we only consider characteristics with nonzero Jacobian. The correctness is
proved.

######  Definition 3.

Opposite vertices of the curvilinear characteristic parallelogram
$\Pi=\\{\mathbf{x}\,|\,\gamma_{1}(\mathbf{x})\in[l_{1},l_{2}]\land\gamma_{2}(\mathbf{x})\in[r_{1},r_{2}]\\}$
are its vertices $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ such that
$\gamma_{1}(\mathbf{x}_{1})\neq\gamma_{1}(\mathbf{x}_{2})$ and
$\gamma_{2}(\mathbf{x}_{1})\neq\gamma_{2}(\mathbf{x}_{2})$.

Point transformation of variables of the form $y_{1}=\gamma_{1}(x_{1},x_{2})$,
$y_{1}=\gamma_{2}(x_{1},x_{2})$ is invertible [34], i.e., there is an inverse
change of variables $x_{1}=\gamma_{1}^{-1}(y_{1},y_{2})$,
$x_{2}=\gamma_{2}^{-1}(y_{1},y_{2})$.

######  Lemma 3.1.

Let
$\Pi=\\{\mathbf{x}\,|\,\gamma_{1}(\mathbf{x})\in[l_{1},l_{2}]\land\gamma_{2}(\mathbf{x})\in[r_{1},r_{2}]\\}$
is a curvilinear characteristic parallelogram and the conditions $a\in
C^{2}(\Pi)$, $b\in C^{2}(\Pi)$, $c\in C^{2}(\Pi)$, and $f\in
C^{1}(\Pi\times\mathbb{R}^{3})$ be satisfied. The function $u$ belongs to the
class $C^{2}(\Pi)$ and satisfies Eq. (2.1) if and only if it can be
represented as

$\displaystyle u(\mathbf{x})$
$\displaystyle=g_{1}\left(\gamma_{1}(\mathbf{x})\right)+g_{2}\left(\gamma_{2}(\mathbf{x})\right)+$
$\displaystyle+\int\limits_{l^{(0)}}^{\gamma_{1}(\mathbf{x})}dz_{1}\int\limits_{r^{(0)}}^{\gamma_{2}(\mathbf{x})}\frac{1}{2\left(a\partial_{x_{1}}\gamma_{1}\partial_{x_{1}}\gamma_{2}+b\left(\partial_{x_{2}}\gamma_{2}\partial_{x_{1}}\gamma_{1}+\partial_{x_{2}}\gamma_{1}\partial_{x_{1}}\gamma_{2}\right)+c\partial_{x_{2}}\gamma_{1}\partial_{x_{2}}\gamma_{2}\right)(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}))}\times$
$\displaystyle\times\left[f\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}),u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\right.\right.$
$\displaystyle\left.\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\right)-$
$\displaystyle-A\gamma_{1}\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\left(\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\partial_{y_{1}}\gamma_{1}^{-1}(\mathbf{z})\>+\right.$
$\displaystyle\left.+\>\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\partial_{y_{1}}\gamma_{2}^{-1}(\mathbf{z})\right)-$
$\displaystyle-A\gamma_{2}\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\left(\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\partial_{y_{2}}\gamma_{1}^{-1}(\mathbf{z})\>+\right.$
$\displaystyle\left.\left.+\>\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\partial_{y_{2}}\gamma_{2}^{-1}(\mathbf{z})\right)\right]dz_{2},$
(3.1)

where $l^{(0)}\in[l_{1},l_{2}]$, $r^{(0)}\in[r_{1},r_{2}]$, and the functions
$g_{1}$, $g_{2}$ belong to the classes $C^{2}(\mathfrak{D}(g_{1}))$,
$C^{2}(\mathfrak{D}(g_{2}))$ respectively.

###### Proof.

Let the function $u\in C^{2}(\Pi)$ satisfy Eq. (2.1). Making the nonlinear
nondegenerate change of independent variables $y_{1}=\gamma_{1}(x_{1},x_{2})$,
$y_{1}=\gamma_{2}(x_{1},x_{2})$ and denoting $u(x_{1},x_{2})=v(y_{1},y_{2})$
we obtain the new differential equation

$\displaystyle
2\left(a\partial_{x_{1}}\gamma_{1}\partial_{x_{1}}\gamma_{2}+b\left(\partial_{x_{2}}\gamma_{2}\partial_{x_{1}}\gamma_{1}+\partial_{x_{2}}\gamma_{1}\partial_{x_{1}}\gamma_{2}\right)+c\partial_{x_{2}}\gamma_{1}\partial_{x_{2}}\gamma_{2}\right)\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right)\times$
$\displaystyle\times\partial_{y_{1}}\partial_{y_{2}}v(\mathbf{y})+A\gamma_{1}\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right)\partial_{y_{1}}v(\mathbf{y})+A\gamma_{2}\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right)\partial_{y_{2}}v(\mathbf{y})=$
$\displaystyle=f\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y}),u\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right),\right.$
$\displaystyle\left.\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right)\right)=f\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y}),v(\mathbf{y}),\partial_{y_{1}}v(\mathbf{y})\partial_{x_{1}}\gamma_{1}\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right)\right.+$
$\displaystyle+\partial_{y_{2}}v(\mathbf{y})\partial_{x_{1}}\gamma_{2}\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right),\partial_{y_{1}}v(\mathbf{y})\partial_{x_{2}}\gamma_{1}\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right)$
$\displaystyle\left.+\>\partial_{y_{2}}v(\mathbf{y})\partial_{x_{2}}\gamma_{2}\left(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y})\right)\right)$

Let us integrate it twice to obtain the equation

$\displaystyle v(\mathbf{y})$
$\displaystyle=g_{1}\left(\mathbf{y}\right)+g_{2}\left(\mathbf{y}\right)+$
$\displaystyle+\int\limits_{l^{(0)}}^{y_{1}}dz_{1}\int\limits_{r^{(0)}}^{y_{2}}\frac{1}{2\left(a\partial_{x_{1}}\gamma_{1}\partial_{x_{1}}\gamma_{2}+b\left(\partial_{x_{2}}\gamma_{2}\partial_{x_{1}}\gamma_{1}+\partial_{x_{2}}\gamma_{1}\partial_{x_{1}}\gamma_{2}\right)+c\partial_{x_{2}}\gamma_{1}\partial_{x_{2}}\gamma_{2}\right)(\gamma_{1}^{-1}(\mathbf{y}),\gamma_{2}^{-1}(\mathbf{y}))}\times$
$\displaystyle\times\left[f\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}),u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\right.\right.$
$\displaystyle\left.\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\right)-$
$\displaystyle-\left.A\gamma_{1}(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}))\partial_{y_{1}}v(\mathbf{z})-A\gamma_{2}(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}))\partial_{y_{2}}v(\mathbf{z})\right]dz_{2},$

Returning to the variables $x_{1}$ and $x_{2}$, we obtain Eq. (3.1). This also
implies that the functions $g_{j}$ belong to the class
$C^{2}(\mathfrak{D}(g_{1}))$, $j=1,2$.

Substituting the representations (3.1) into Eq. (2.1), we verify that the
function $u$ satisfies this equation in $\Pi$. ∎

######  Remark 3.

Under some additional conditions on the functions $f$, $a$, $b$, $c$, $g_{1}$,
$g_{2}$, we can show the solvability of the integro-differential equation
(3.1) using the methods proposed in the works [5, 13, 35].

For the convenience of further presentation, we introduce the notation

$\displaystyle\beta=2\left(a\partial_{x_{1}}\gamma_{1}\partial_{x_{1}}\gamma_{2}+b\left(\partial_{x_{2}}\gamma_{2}\partial_{x_{1}}\gamma_{1}+\partial_{x_{2}}\gamma_{1}\partial_{x_{1}}\gamma_{2}\right)+c\partial_{x_{2}}\gamma_{1}\partial_{x_{2}}\gamma_{2}\right),$
$\displaystyle
K(\mathbf{z},p,q,r)=f(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}),p,q,r)\>-$
$\displaystyle-A\gamma_{1}(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}))(q\partial_{y_{1}}\gamma_{1}^{-1}(\mathbf{z})+r\partial_{y_{1}}\gamma_{2}^{-1}(\mathbf{z}))\>-$
$\displaystyle-A\gamma_{2}(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}))(q\partial_{y_{2}}\gamma_{1}^{-1}(\mathbf{z})+r\partial_{y_{2}}\gamma_{2}^{-1}(\mathbf{z})),$
$\displaystyle\widetilde{K}(\mathbf{z},p,q,r)=(\beta(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})))^{-1}K(\mathbf{z},p,q,r)$

##  4 Curvilinear parallelogram identity

######  Theorem 4.1.

Let the conditions $a\in C^{2}(\Omega)$, $b\in C^{2}(\Omega)$, $c\in
C^{2}(\Omega)$, $f\in C^{1}(\Omega\times\mathbb{R}^{3})$, and
$b^{2}(\mathbf{x})-a(\mathbf{x})c(\mathbf{x})>0$ be satisfied, and let the
function $u$ belong to the class $C^{2}(\Omega)$ and be a solution of the
hyperbolic equation (2.1). Then for any curvilinear characteristic
parallelogram
$\Pi=\\{\mathbf{x}\,|\,\gamma_{1}(\mathbf{x})\in[l_{1},l_{2}]\land\gamma_{2}(\mathbf{x})\in[r_{1},r_{2}]\\}\subseteq\Omega$
with vertices $A(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1}))$,
$B(\gamma_{1}^{-1}(l_{1},r_{2}),\gamma_{2}^{-1}(l_{1},r_{2}))$,
$C(\gamma_{1}^{-1}(l_{2},r_{2}),\gamma_{2}^{-1}(l_{2},r_{2}))$,
$(\gamma_{1}^{-1}(l_{2},r_{1}),\gamma_{2}^{-1}(l_{2},r_{1}))$, the equality
holds

$\displaystyle u(A)$ $\displaystyle-u(B)+u(C)-u(D)=$ (4.1)
$\displaystyle=\int\limits_{l_{1}}^{l_{2}}dz_{1}\int\limits_{r_{1}}^{r_{2}}\widetilde{K}\Big{(}\mathbf{z},u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!,$
$\displaystyle\quad\quad\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!\\!\Big{)}dz_{2}.$

###### Proof.

According to Lemma 1, the function $u$ can be represented in the form

$\displaystyle u(\mathbf{x})$
$\displaystyle=g_{1}(\gamma_{1}(\mathbf{x}))+g_{2}(\gamma_{2}(\mathbf{x}))\>+$
(4.2)
$\displaystyle+\int\limits_{l_{1}}^{\gamma_{1}(\mathbf{x})}dz_{1}\int\limits_{r_{1}}^{\gamma_{2}(\mathbf{x})}\widetilde{K}\Big{(}\mathbf{z},u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!,$
$\displaystyle\quad\quad\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!\\!\Big{)}dz_{2}.$

where $g_{i}\in C^{2}(\mathfrak{D}(g_{i}))$, $i=1,2$. Using the expression
(4.2), we compute

$\displaystyle
u(A)=g_{1}(l_{1})+g_{2}(r_{1}),u(B)=g_{1}(l_{1})+g_{2}(r_{2}),u(D)=g_{1}(l_{2})+g_{2}(r_{1}),$
$\displaystyle u(C)=g_{1}(l_{2})+g_{2}(r_{2})\>+$
$\displaystyle+\int\limits_{l_{1}}^{l_{2}}dz_{1}\int\limits_{r_{1}}^{r_{2}}\widetilde{K}\Big{(}\mathbf{z},u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!,$
$\displaystyle\quad\quad\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!\\!\Big{)}dz_{2}.$
(4.3)

Substituting (4.3) into (4.1) gives the correct equality. ∎

######  Theorem 4.2.

Let the conditions $u\in C^{2}(\Omega)$, $a\in C^{2}(\Omega)$, $b\in
C^{2}(\Omega)$, $c\in C^{2}(\Omega)$, $f\in
C^{1}(\Omega\times\mathbb{R}^{3})$, and
$b^{2}(\mathbf{x})-a(\mathbf{x})c(\mathbf{x})>0$ be satisfied. If for any
curvilinear characteristic parallelogram
$\Pi=\\{\mathbf{x}\,|\,\gamma_{1}(\mathbf{x})\in[l_{1},l_{2}]\land\gamma_{2}(\mathbf{x})\in[r_{1},r_{2}]\\}\subseteq\Omega$
with vertices $A(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1}))$,
$B(\gamma_{1}^{-1}(l_{1},r_{2}),\gamma_{2}^{-1}(l_{1},r_{2}))$,
$C(\gamma_{1}^{-1}(l_{2},r_{2}),\gamma_{2}^{-1}(l_{2},r_{2}))$,
$(\gamma_{1}^{-1}(l_{2},r_{1}),\gamma_{2}^{-1}(l_{2},r_{1}))$, where
$\gamma_{i}$, $i=1,2$ are solutions of Eqs. (2.2) and $\gamma_{i}^{-1}$ are
defined as before, the equality (4.1) is satisfied, then the function $u$ is a
solution of Eq. (2.1).

###### Proof.

Let $l_{2}=l+l_{1}$, $r_{2}=r+r_{1}$. So, we can write the coordinates of
points $A$, $B$, $C$ and $D$ in the form

$\displaystyle A(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1})),$
$\displaystyle
B(\gamma_{1}^{-1}(l_{1},r+r_{1}),\gamma_{2}^{-1}(l_{1},r+r_{1})),$
$\displaystyle
C(\gamma_{1}^{-1}(l+l_{1},r+r_{1}),\gamma_{2}^{-1}(l+l_{1},r+r_{1})),$
$\displaystyle
D(\gamma_{1}^{-1}(l+l_{1},r_{1}),\gamma_{2}^{-1}(l+l_{1},r_{1})).$

Let’s consider the expression

$\displaystyle\frac{u(A)-u(B)}{r}$
$\displaystyle=\frac{u(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1}))-u(\gamma_{1}^{-1}(l_{1},r+r_{1}),\gamma_{2}^{-1}(l_{1},r+r_{1}))}{r}\xrightarrow[r\to
0]{\>}$ $\displaystyle\xrightarrow[r\to
0]{\>}-\partial_{r}u(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1})).$

In the same way

$\frac{u(C)-u(D)}{r}\xrightarrow[r\to
0]{\>}\partial_{r}u(\gamma_{1}^{-1}(l_{1}+l,r_{1}),\gamma_{2}^{-1}(l_{1}+l,r_{1})).$

Now since

$\displaystyle\frac{\partial_{r}u(\gamma_{1}^{-1}(l_{1}+l,r_{1}),\gamma_{2}^{-1}(l_{1}+l,r_{1}))-\partial_{r}u(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1}))}{l}\xrightarrow[l\to
0]{\>}$ $\displaystyle\quad\quad\quad\quad\xrightarrow[l\to
0]{\>}\partial_{l}\partial_{r}u(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1})),$

we obtain
$\lim\limits_{(r,l)\to(0,0)}(lr)^{-1}(u(A)-u(B)+u(C)-u(D))=\partial_{l}\partial_{r}u(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1}))$.
Similarly, we get

$\displaystyle\lim\limits_{(r,l)\to(0,0)}\frac{1}{lr}\int\limits_{l_{1}}^{l+l_{1}}dz_{1}\int\limits_{r_{1}}^{r+r_{1}}\widetilde{K}\Big{(}\mathbf{z},u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!,$
$\displaystyle\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!\\!\Big{)}dz_{2}=$
$\displaystyle=\widetilde{K}\Big{(}\mathbf{z}=(l_{1},r_{1}),u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!\\!\Big{)}.$

Thus

$\displaystyle\lim\limits_{(r,l)\to(0,0)}\frac{1}{lr}\Bigg{(}u(A)-u(B)+u(C)-u(D)-$
$\displaystyle-\int\limits_{l_{1}}^{l+l_{1}}dz_{1}\int\limits_{r_{1}}^{r+r_{1}}\widetilde{K}\Big{(}\mathbf{z},u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!,\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z})\\!,\gamma_{2}^{-1}(\mathbf{z})\right)\\!\\!\Big{)}dz_{2}\Bigg{)}=$
$\displaystyle=\lim\limits_{(r,l)\to(0,0)}\frac{u(A)-u(B)+u(C)-u(D)}{lr}-$
$\displaystyle-\lim\limits_{(r,l)\to(0,0)}\frac{1}{lr}\int\limits_{l_{1}}^{l+l_{1}}dz_{1}\int\limits_{r_{1}}^{r+r_{1}}\widetilde{K}\Big{(}\mathbf{z},u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!,$
$\displaystyle\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z})\\!,\gamma_{2}^{-1}(\mathbf{z})\right)\\!\\!\Big{)}dz_{2}=\partial_{l}\partial_{r}u(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1}))-$
$\displaystyle-\>\frac{K\left(\mathbf{z}=(l_{1},r_{1}),u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\\!,\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z})\\!,\gamma_{2}^{-1}(\mathbf{z})\right)\right)}{\beta(\gamma_{1}^{-1}(l_{1},r_{1}),\gamma_{2}^{-1}(l_{1},r_{1}))}.$

This means that the function $u$ satisfies at the point
${(\gamma_{1}^{-1}(\mathbf{z}=(y_{1}=l_{1},y_{2}=r_{1})),\gamma_{2}^{-1}(\mathbf{z}))}$
the differential equation

$\displaystyle\beta(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}))\partial_{y_{1}}\partial_{y_{2}}u(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}))=$
(4.4)
$\displaystyle=f\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z}),u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\right.$
$\displaystyle\left.\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right),\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\right)-$
$\displaystyle-A\gamma_{1}\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\left(\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\partial_{y_{1}}\gamma_{1}^{-1}(\mathbf{z})\>+\right.$
$\displaystyle\left.+\>\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\partial_{y_{1}}\gamma_{2}^{-1}(\mathbf{z})\right)-$
$\displaystyle-A\gamma_{2}\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\left(\partial_{x_{1}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\partial_{y_{2}}\gamma_{1}^{-1}(\mathbf{z})\>+\right.$
$\displaystyle\left.+\>\partial_{x_{2}}u\left(\gamma_{1}^{-1}(\mathbf{z}),\gamma_{2}^{-1}(\mathbf{z})\right)\partial_{y_{2}}\gamma_{2}^{-1}(\mathbf{z})\right),$

where $x_{1}=\gamma_{1}^{-1}(y_{1},y_{2})$,
$x_{2}=\gamma_{2}^{-1}(y_{1},y_{2})$. By virtue of the arbitrariness of
$\Pi\subseteq\Omega$, Eq. (4.4) is true for any point
$(x_{1}=\gamma_{1}^{-1}(\mathbf{z}=(l_{1},r_{1})),x_{2}=\gamma_{2}^{-1}(\mathbf{z}=(l_{1},r_{1})))\in\Omega$.

Making the change of variables $x_{1}=\gamma_{1}^{-1}(y_{1},y_{2})$,
$x_{2}=\gamma_{2}^{-1}(y_{1},y_{2})$ in Eq. (4.4), we obtain Eq. (2.1). ∎

Note that formula (4.1) can be considered as a kind of mean value theorem.

##  5 Applications

###  5.1 Wave equation

Let’s consider
$Au(x_{1},x_{2})=\partial_{x_{1}}^{2}u(x_{1},x_{2})-a^{2}\partial_{x_{2}}^{2}u(x_{1},x_{2})$,
where $a>0$ (for definiteness). Then we have
$\gamma_{1}(x_{1},x_{2})=x_{2}-ax_{1}$,
$\gamma_{2}(x_{1},x_{2})=x_{2}+ax_{1}$,
$\gamma^{-1}_{1}(y_{1},y_{2})=(y_{2}-y_{1})/(2a)$,
$\gamma^{-1}_{2}(y_{1},y_{2})=(y_{1}+y_{2})/2$, $A\gamma_{1}\equiv 0$,
$A\gamma_{2}\equiv 0$.

####  5.1.1 Parallelogram identity

Let $f\equiv 0$. In this case, formula (4.1) transforms to

$\displaystyle
u\left(\frac{r_{1}-l_{1}}{2a},\frac{l_{1}+r_{1}}{2}\right)-u\left(\frac{r_{2}-l_{1}}{2a},\frac{l_{1}+r_{2}}{2}\right)+$
$\displaystyle+\>$ $\displaystyle
u\left(\frac{r_{2}-l_{2}}{2a},\frac{l_{2}+r_{2}}{2}\right)-u\left(\frac{r_{1}-l_{2}}{2a},\frac{l_{2}+r_{1}}{2}\right)=0,$
(5.1)

where $l_{1}$, $l_{2}$, $r_{1}$ and $r_{2}$ are some real numbers. Eq. (5.1)
is the well-known parallelogram identity for the wave equation.

####  5.1.2 Goursat problem

Let’s consider the Goursat problem [15]

$\left\\{\begin{array}[]{l}(\partial_{x_{1}}^{2}-a^{2}\partial_{x_{2}}^{2})u(\mathbf{x})=f(\mathbf{x}),\quad
0<x_{1},-ax_{1}<x_{2}<ax_{1},\\\ u(x_{1},x_{2}=ax_{1})=\phi^{(1)}(x_{1}),\quad
u(x_{1},x_{2}=-ax_{1})=\phi^{(1)}(x_{2}),\quad x_{1}>0,\end{array}\right.$
(5.2)

where $f\in C^{1}(\\{\mathbf{x}\,|\,0\leqslant x_{1},-ax_{1}\leqslant
x_{2}\leqslant ax_{1}\\})$, $\phi^{(1)}\in C^{2}([0,\infty))$, $\phi^{(2)}\in
C^{2}([0,\infty))$ and $\phi^{(1)}(0)=\phi^{(2)}(0)$. We can write the
classical solution of (5.2) using the formula (4.1). If we choose
$C(x_{1},x_{2})$,
$B\\!\left(\dfrac{ax_{1}+x_{2}}{2a},\dfrac{ax_{1}+x_{2}}{2}\right)$,
$D\\!\left(\dfrac{ax_{1}-x_{2}}{2a},\dfrac{x_{2}-ax_{1}}{2}\right)$, $A(0,0)$
and apply (4.1), we obtain

$\displaystyle u(x_{1},$ $\displaystyle
x_{2})=u(C)=\phi^{(1)}\\!\left(\frac{ax_{1}+x_{2}}{2a}\right)+\phi^{(2)}\\!\left(\frac{ax_{1}-x_{2}}{2a}\right)-\phi^{(1)}(0)\>-$
$\displaystyle-\frac{1}{4a^{2}}\int\limits_{0}^{x_{2}-ax_{1}}dy_{1}\int\limits_{0}^{x_{2}+ax_{1}}f\left(\frac{y_{2}-y_{1}}{2a},\frac{y_{1}+y_{2}}{2}\right)dy_{2},\quad
0<x_{1},-ax_{1}<x_{2}<ax_{1}.$

Fig. 2. To the Goursat problem (5.2).

####  5.1.3 Mixed problem

Let’s consider the first mixed problem [12]

$\left\\{\begin{array}[]{l}(\partial_{x_{1}}^{2}-a^{2}\partial_{x_{2}}^{2})u(\mathbf{x})=f(\mathbf{x}),\quad\mathbf{x}\in(0,\infty)\times(0,\infty),\\\
u(0,x_{2})=\phi(x_{1}),\quad\partial_{x_{1}}u(0,x_{2})=\psi(x_{2}),\quad
x_{1}>0,\\\ u(x_{1},0)=\mu(x_{1}),\quad x_{2}>0,\end{array}\right.$ (5.3)

where $f\in C^{1}([0,\infty)\times[0,\infty))$, $\phi\in C^{2}([0,\infty))$,
$\psi\in C^{1}([0,\infty))$, $\mu\in C^{2}([0,\infty))$.

If $x_{2}-ax_{1}>0$, then the solution of (5.3) at the point $(x_{1},x_{2})$
can be defined by d’Alembert formula

$\displaystyle u(x_{1},x_{2})$
$\displaystyle=\frac{\phi(x_{2}-ax_{1})+\phi(x_{2}+ax_{1})}{2}+\frac{1}{2a}\int\limits_{x_{2}-ax_{1}}^{x_{2}+ax_{1}}\psi(\xi)\,d\xi\>+$
(5.4)
$\displaystyle+\frac{1}{2a}\int\limits_{0}^{x_{1}}d\tau\int\limits_{x_{2}-a(x_{1}-\tau)}^{x_{2}+a(x_{1}-\tau)}f(\tau,\xi)d\xi,\quad
x_{2}-ax_{1}>0,\quad x_{1}>0,\quad x_{2}>0.$

If $x_{2}-ax_{1}<0$, then we can use the parallelogram identity (4.1) to
derive the solution of (5.3) at the point $(x_{1},x_{2})$. We can choose
$C(x_{1},x_{2})$, $B\\!\left(x_{1}-\dfrac{x_{2}}{a},0\right)$,
$D\\!\left(\dfrac{x_{2}}{a},ax_{1}\right)$, $A\left(0,ax_{1}-x_{2}\right)$,
apply (4.1) and obtain

$\displaystyle u(x_{1},x_{2})$
$\displaystyle=\mu\left(x_{1}-\frac{x_{2}}{a}\right)+\frac{\phi\left(ax_{1}+x_{2}\right)-\phi\left(ax_{1}-x_{2}\right)}{2}+\frac{1}{2a}\int\limits_{ax_{1}-x_{2}}^{ax_{1}+x_{2}}\psi(\xi)\,d\xi\>+$
(5.5)
$\displaystyle+\frac{1}{2a}\int\limits_{0}^{\frac{x_{2}}{a}}d\tau\int\limits_{ax_{1}-x_{2}+a\tau}^{ax_{1}+x_{2}-a\tau}f(\tau,\xi)d\xi-\frac{1}{4a^{2}}\int\limits_{ax_{1}-x_{2}}^{x_{2}-ax_{1}}dy_{1}\int\limits_{ax_{1}-x_{2}}^{ax_{1}+x_{2}}f\left(\frac{y_{2}-y_{1}}{2a},\frac{y_{2}+y_{1}}{2}\right)\,dy_{2},$
$\displaystyle x_{2}-ax_{1}<0,\quad x_{1}>0,\quad x_{2}>0.$

Fig. 3. To the first mixed problem (5.3).

Using the representations (5.4) and (5.5), we can easily derive the necessary
and sufficient matching conditions $\mu(0)=\phi(0)$, $\mu^{\prime}(0)=\psi(0)$
and $\mu^{\prime\prime}(0)=a^{2}\phi^{\prime\prime}(0)+f(0,0)$ under which the
solution $u$ of the first mixed problem (5.3) will be classical.

###  5.2 Nonlinear wave equation

For convenience, we will present equations in divergence form later in this
section. Let’s consider
$Au(x_{1},x_{2})=\partial_{x_{1}}\partial_{x_{2}}u(x_{1},x_{2})$. Then we have
$\gamma_{1}(x_{1},x_{2})=x_{1}$, $\gamma_{2}(x_{1},x_{2})=x_{2}$,
$\gamma^{-1}_{1}(y_{1},y_{2})=y_{1}$, $\gamma^{-1}_{2}(y_{1},y_{2})=y_{2}$,
$A\gamma_{1}\equiv 0$, $A\gamma_{2}\equiv 0$.

####  5.2.1 Darboux problem

Let’s consider the second Darboux problem for a nonlinear wave equation in
divergence form [11]

$\left\\{\begin{array}[]{l}\partial_{x_{1}}\partial_{x_{2}}u(\mathbf{x})+\lambda
g(\mathbf{x},u(\mathbf{x}))=f(\mathbf{x}),\quad 0<x_{1},\alpha
x_{1}<x_{2}<\beta x_{1},\\\ u(x_{1},x_{2}=\alpha x_{1})=u(x_{1},x_{2}=\beta
x_{1})=0,\quad x_{1}>0,\end{array}\right.$ (5.6)

where $\lambda\in\mathbb{R}$, $0<\alpha<1<\beta<\infty$, $f\in
C^{1}(\\{\mathbf{x}\,|\,0\leqslant x_{1}\land\alpha x_{1}\leqslant
x_{2}\leqslant\beta x_{1}\\})$, $g\in C^{1}(\\{\mathbf{x}\,|\,0\leqslant
x_{1}\land\alpha x_{1}\leqslant x_{2}\leqslant\beta
x_{1}\\}\times\mathbb{R})$, $|g(x_{1},x_{2},z)|\leqslant L_{1}+L_{2}|z|$,
$L_{1}\geqslant 0$, $L_{2}\geqslant 0$.

We want to obtain an expression for the classical solution $u$ of the problem
(5.6) at the point $P_{0}(x_{1},x_{2})$. Let us denote by
$P_{1}M_{0}P_{0}N_{0}$ the characteristic parallelogram, whose vertices
$N_{0}$ and $M_{0}$ lie, respectively, on the segments $x_{2}=\alpha x_{1}$
and $x_{2}=\beta x_{1}$, that is: $N_{0}:=(x_{1},\alpha x_{1})$,
$M_{0}:=\left(\beta^{-1}x_{2},x_{2}\right)$,
$P_{1}:=\left(\beta^{-1}x_{2},\alpha x_{1}\right)$. Since
$P_{1}\in\\{\mathbf{x}\,|\,0<x_{1}\land\alpha x_{1}<x_{2}<\beta x_{1}\\}$, we
construct analogously the characteristic parallelogram $P_{2}M_{1}P_{1}N_{1}$
whose vertices $N_{1}$ and $M_{1}$ lie, respectively, on the segments
$x_{2}=\alpha x_{1}$ and $x_{2}=\beta x_{1}$. Continuing this process, we
obtain the characteristic parallelogram $P_{i+1}M_{i}P_{i}N_{i}$ for which
$N_{i}\in\\{\mathbf{x}\,|\,x_{2}=\alpha x_{1}\\}$,
$M_{i}\in\\{\mathbf{x}\,|\,x_{2}=\beta x_{1}\\}$, and
$N_{i}:=\left(x_{1}^{(i)},\alpha x_{1}^{(i)}\right)$,
$M_{i}:=\left(\beta^{-1}x_{2}^{(i)},x_{2}\right)$,
$P_{i+1}:=\left(\beta^{-1}x_{2}^{(i)},\alpha x_{1}^{(i)}\right)$ if
$P_{i}:=\left(x_{1}^{(i)},x_{2}^{(i)}\right)$.

Fig. 4. To the second Darboux problem (5.6).

By (4.1) and (5.6), we have

$\displaystyle u(P_{i})=u(M_{i})+u(N_{i})-u(P_{i+1})$
$\displaystyle+\iint\limits_{P_{i+1}M_{i}P_{i}N_{i}}\left[f(\mathbf{z})-\lambda
g(\mathbf{z},u(\mathbf{z}))\right]d\mathbf{z}=$ $\displaystyle=-u(P_{i+1})$
$\displaystyle+\iint\limits_{P_{i+1}M_{i}P_{i}N_{i}}\left[f(\mathbf{z})-\lambda
g(\mathbf{z},u(\mathbf{z}))\right]d\mathbf{z},\>\>i\in\mathbb{N}\cup\\{0\\}.$

It follows that

$\displaystyle u(x_{1},x_{2})$
$\displaystyle=u(P_{0})=\iint\limits_{P_{1}M_{0}P_{0}N_{0}}\left[f(\mathbf{z})-\lambda
g(\mathbf{z},u(\mathbf{z}))\right]d\mathbf{z}-u(P_{1})=$
$\displaystyle=u(P_{2})+\iint\limits_{P_{1}M_{0}P_{0}N_{0}}\left[f(\mathbf{z})-\lambda
g(\mathbf{z},u(\mathbf{z}))\right]d\mathbf{z}-\iint\limits_{P_{2}M_{1}P_{1}N_{1}}\left[f(\mathbf{z})-\lambda
g(\mathbf{z},u(\mathbf{z}))\right]d\mathbf{z}=$
$\displaystyle=(-1)^{n}u(P_{n})+\sum\limits_{i=0}^{n-1}(-1)^{i}\iint\limits_{P_{i+1}M_{i}P_{i}N_{i}}\left[f(\mathbf{z})-\lambda
g(\mathbf{z},u(\mathbf{z}))\right]d\mathbf{z}.$

It is clear that
$\lim\limits_{n\rightarrow\infty}u(P_{n})=u\left(\lim\limits_{n\rightarrow\infty}P_{n}\right)=u(0,0)=0$.
Hence, passing to the limit, as $n\rightarrow\infty$, we obtain the following
integral representation

$u(x_{1},x_{2})=\sum\limits_{i=0}^{\infty}(-1)^{i}\iint\limits_{P_{i+1}M_{i}P_{i}N_{i}}\left[f(\mathbf{z})-\lambda
g(\mathbf{z},u(\mathbf{z}))\right]d\mathbf{z}.$ (5.7)

The further solution of the problem (5.6) is connected with the study of the
solvability of Eq. (5.7), and it is given in the work [11]. It turns out that
under the conditions given in the statement of the problem (5.6), it has a
unique classical solution. But we still notice that in the linear case, i.e.,
when $\lambda=0$, the formula (5.7) transforms into

$u(x_{1},x_{2})=\sum\limits_{i=0}^{\infty}(-1)^{i}\iint\limits_{P_{i+1}M_{i}P_{i}N_{i}}f(\mathbf{z})d\mathbf{z},$
(5.8)

The series in the right-hand side of equality (5.8) is uniformly and
absolutely convergent [11]. Thus, in the linear case, there exists the
solution $u$ of (5.6) written in the explicit analytic form (5.8).

###  5.3 Linear second-order hyperbolic equation

As in the previous subsection, we consider
$Au(x_{1},x_{2})=\partial_{x_{1}}\partial_{x_{2}}u(x_{1},x_{2})$. Then we have
$\gamma_{1}(x_{1},x_{2})=x_{1}$, $\gamma_{2}(x_{1},x_{2})=x_{2}$,
$\gamma^{-1}_{1}(y_{1},y_{2})=y_{1}$, $\gamma^{-1}_{2}(y_{1},y_{2})=y_{2}$,
$A\gamma_{1}\equiv 0$, $A\gamma_{2}\equiv 0$.

####  5.3.1 Goursat problem

Let’s consider the Goursat problem for a linear second-order hyperbolic
equation [12]

$\left\\{\begin{array}[]{l}\partial_{x_{1}}\partial_{x_{2}}u(\mathbf{x})+a(\mathbf{x})\partial_{x_{1}}u(\mathbf{x})+b(\mathbf{x})\partial_{x_{2}}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x}),\quad
x_{1}^{(0)}<x_{1},x_{2}^{(0)}<x_{2},\\\
u(x_{1}=x_{1}^{(0)},x_{2})=\phi(x_{2}),\quad x_{2}>x_{2}^{(0)},\\\
u(x_{1},x_{2}=x_{2}^{(0)})=\psi(x_{1}),\quad
x_{1}>x_{1}^{(0)},\end{array}\right.$ (5.9)

where $f\in C(\\{\mathbf{x}\,|\,x_{1}^{(0)}\leqslant x_{1}\land
x_{2}^{(0)}\leqslant x_{2}\\})$, $\phi\in C^{1}([x_{2}^{(0)},\infty))$,
$\psi\in C^{1}([x_{1}^{(0)},\infty))$ and
$\phi(x_{2}^{(0)})=\psi(x_{1}^{(0)})$. We can write the classical solution of
(5.9) using the formula (4.1). If we choose $C(x_{1},x_{2})$,
$B(x_{1}^{(0)},x_{2})$, $D(x_{1},x_{2}^{(0)})$, $A(x_{1}^{(0)},x_{2}^{(0)})$
and apply (4.1), we obtain

$\displaystyle u(\mathbf{x})$
$\displaystyle=u(C)=\phi(x_{2})+\psi(x_{1})-\psi(x_{2}^{(0)})\>+$
$\displaystyle+\int\limits_{x_{1}^{(0)}}^{x_{1}}dy_{1}\int\limits_{x_{2}^{(0)}}^{x_{2}}[f(\mathbf{y})-a(\mathbf{y})\partial_{x_{1}}u(\mathbf{y})-b(\mathbf{y})\partial_{x_{2}}u(\mathbf{y})-c(\mathbf{y})u(\mathbf{y})]dy_{2}.$
(5.10)

A representation of the solution in the form of the integro-differential
equation (5.10) is obtained. Under the conditions specified in the formulation
of the problem (5.9), Eq. (5.10) will be solvable [12], and the function $u$
will have the required smoothness. It proves the solvability of the problem
(5.9).

##  6 Conclusion

In this paper, we have generalized the parallelogram rule for the wave
equation to the case of a semilinear hyperbolic equation of the second order.
This identity connects not only the values of the points at the vertices of
the parallelogram but also the continuum of function values on the
parallelogram, in contrast to the linear cases with constant coefficients
considered earlier. We have shown how the obtained results can be used in
combination with other methods to solve various mixed problems.

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Viktor Korzyuk, Jan Rudzko

Department of Mathematical Cybernetics

Belarusian State University

4 Nezavisimosti Avenue,

Minsk, Belarus

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

Received: ??.??.20??


# The Fermionic Axion Interferometer

Nicolò Crescini<EMAIL_ADDRESS>Fondazione Bruno Kessler (FBK), I-38123,
Trento, Italy  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000
Grenoble, France

###### Abstract

The axion is an hypothetical beyond the Standard Model particle. Its
experimental search is an ongoing effort, and an expanding number of
techniques keep on narrowing its parameters space. Leveraging the interaction
between dark matter axions and spins, a fermionic interferometer is an
experiment which aims at detecting the axion-induced precession of a spin
resonance. We describe the detection scheme, outline the possible experimental
implementations, their sensitive axion-mass range and discovery potential.
Furthermore, the building and characterisation of an axion interferometer is
explained in details and the resulting setup is used to search for sub-neV
dark matter.

##  .1 Axions and spins

The main evidences of physics beyond the Standard Model [1] are the strong CP
problem, the asymmetry between matter and anti-matter, neutrino oscillations,
dark matter and dark energy [2]. The axion [3] is an hypothetical particle
which has the potential to solve two of these matters. Originally proposed to
accommodate for the absence of charge-parity symmetry violation in quantum
chromodynamics, i. e. the strong CP problem [4], it later became a compelling
dark matter candidate [5, 6]. This theoretical sparkle triggered the first
experimental searches that excluded early axion models, and which were
followed by new models and new experiments [7, 8, 9]. To this day, no signal
compatible with axions has been reported.

The first thing to consider when detecting a particle is its mass, and the
axion one is unknown. A virtually infinite mass range can be constrained by
means of models [10], astrophysics [11] and cosmology [12], suggesting a
preferred window for the axion mass $m_{a}$ between micro- and milli-
electronvolts. Still, many theoretical efforts and experimental searches are
directed towards lighter or heavier axions or axion-like particles [13].
Typically, the most sensitive experiments cover a narrow mass range, as in the
case of haloscopes [14, 15, 16, 17, 18, 19, 20, 21], while broadband
techniques cover a wide mass range at the expense of a reduced sensitivity
[22, 23, 24]. Among all of these are spin resonance axion searches can be
based on electrons [25, 26, 27], or nuclei, which can access low frequency
axions [28, 29, 30, 31].

Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) axions [32, 33] interact with
fermions with pseudoscalar interaction constant $g_{p}$ [34, 35] though the
Lagrangian

${\cal
L}_{f}=\frac{g_{p}}{2m_{f}}(\bar{\psi}\gamma_{\mu}\gamma_{5}\partial^{\mu}a\psi),$
(1)

where $\partial^{\mu}a$ is the axion field quadriderivative, $\gamma_{\mu}$
and $\gamma_{5}$ are Dirac matrices, and $\psi$ is the quantum field of a
fermion with mass $m_{f}$ and charge $e$. One can notice that ${\cal L}_{f}$
is analogous to the interaction Lagrangian of quantum electrodynamics

${\cal L}_{\mathrm{QED}}\supset-e(\bar{\psi}\gamma_{\mu}A^{\mu}\psi),$ (2)

minus a $\gamma_{5}$. Physically, this means that the axion field derivative
acts on fermionic spins as an effective magnetic field. In the non-
relativistic limit Eq. (1) can be recast in terms of the fermion’s spin
$s_{i}=\mu_{f}\sigma_{i}$, where $\mu_{f}=e/2m_{f}$ is the fermion magneton
and $\sigma_{i}$ is the Pauli vector. The remaining terms form a pseudo-
magnetic axionic field

$b_{i}=\frac{g_{p}}{2e}\partial_{i}a,$ (3)

which oscillates at the frequency of the axion mass $m_{a}$, such that
$b_{i}\propto\beta_{i}e^{-im_{a}t}$, where $\beta_{i}=(\beta,0,0)$ is the
axion speed relative to the speed of light $c$. If the spin $s_{i}$ is in a
static magnetic field $B_{i}$ and in an axion effective field, both parallel
to its direction, its equations of motion are Bloch equations [36]

$\dot{s_{i}}/\gamma=\epsilon_{ijk}s^{j}(B^{k}+b^{k}),$ (4)

where $\gamma$ is the fermion gyromagnetic ratio. This equation displays a
Larmor precession with an additional axion-induced spin precession [37, 38,
39], which is the core observable of the present work. Taking the non-
relativistic limit of Eq. (3), and treating the axion as a classical field are
justified by $\beta_{i}\ll 1$ and by the large axion occupation number
$n_{a}=\varrho_{\mathrm{dm}}/m_{a}$ [40, 27], respectively. An Earth-based
laboratory is immersed in the dark matter halo of the Milky Way, whose
estimated density is $\varrho_{\mathrm{dm}}=0.4\,\mathrm{GeV/cm^{3}}$, and
travels through it at a speed $\beta\simeq 10^{-3}$. The resulting axion
effective field is

$b_{a}=\frac{g_{p}}{2e}\beta n_{a}^{1/2}\simeq 3\times
10^{-18}\left(m_{a}/1\,\mathrm{eV}\right)\,\mathrm{T}.$ (5)

centered at the frequency corresponding to the axion mass, and with a quality
factor which is approximately $Q_{a}\simeq\beta^{-2}\simeq 10^{6}$ [8, 40].

The detection of an axion precession can be understood in a general
experimental scheme which can be realised at different energy scales—i.e.
axion masses—and therefore with different physical systems and technologies,
outlined in Table 1. Let’s first discuss the experimental scheme, outlined in
Fig. 1. The spin resonance under consideration is phase-modulated by the dark
matter axion field gradient which, being dependent on $\beta_{i}$, is a
directional effect. This suggests that two spin resonances with static
magnetic fields oriented perpendicularly to each other can be used as the two
arms of an interferometer. The east arm—(E)—is parallel to the axion field
gradient and carries the dark matter signal, while the north arm—(N)—acts as a
reference. In the east arm $B^{\mathrm{(E)}}_{i}=(B_{0},0,0)$ and
$b_{i}=(b_{a},0,0)$, so Eq. (4) describes a spin resonance whose resonant
frequency $\omega_{0}=\gamma B_{0}$ is periodically modulated by an axionic
field with amplitude $b_{a}$ and frequency $\omega_{a}$. The north arm has
$B^{\mathrm{(N)}}_{i}=(0,B_{0},0)$, giving the same resonance frequency and no
axionic modulation. One can therefore imagine to shine the same light beam to
both the resonances, and then let the outputs interfere with themselves,
providing an interferometric readout of a dark matter axion signal.

| $\omega_{0}/2\pi$ | $k_{0}/2\pi$ | $m_{a}$
---|---|---|---
Spin resonance | 10 MHz | 100 kHz |  $<0.4$ neV
Magnon polariton | 10 GHz | 1 MHz |  $<40$ neV
Atomic transition | 100 THz | 10 GHz |  $<40$ $\mu$eV
Table 1: Experimental platforms to implement a fermionic axion interferometer
at different energy scales.

This scheme is analogous to the one of gravitational wave observatories, where
the Fabry-Pérot cavities are substituted by the spin resonances, as shown in
Fig. 1. Although existing proposals use interferometric techniques to search
for axions [41, 42, 43, 44], the present one differs from them in both
detection principle and practical realisation. As is detailed in the
following, the advantages of this approach are that it leverages the
considerable experimental assets of interferometry, it is broadband in the
axion mass parameter space, and it can be engineered at different energy
scales.

Figure 1: Scheme illustrating the principle of a fermionic interferometer. Two
perpendicular spin resonances act as arms of an interferometer, which are
probed by splitted light beams. The probe signals are reported in pink while
the axion field $b_{a}$ is green, see text for further details.

##  .2 Detection scheme

As shown in the introduction, axionic dark matter produces coherent
oscillations in the frequency of a spin resonance. A suitable way to detect
this effect is based on phase modulation, and is detailed in the following. A
monochromatic input light on-resonance with the spin precession and with
amplitude $A_{0}$ is phase modulated by the oscillating resonance, yielding a
signal

$\xi(t)=A_{0}e^{-i\omega_{0}t}e^{-ix\sin(\omega_{a}t)},$ (6)

where $x=2\pi^{2}\gamma b_{a}/k_{0}$ is the modulation index, and $k_{0}$ is
the spin resonance linewidth. This effect is illustrated in Fig. 2 by using a
magnified effect of axions acting on a frequency
$\omega_{0}/2\pi=5\,\mathrm{MHz}$.

Figure 2: Effect of the axionic modulation on a monochromatic signal
interacting with a spin resonance. The axionic dark matter modulates the spin
resonance frequency, thereby inducing oscillations on the phase of the
monochromatic signal.

If the modulation index $x$ is small, $\xi(t)$ can be recast in terms of the
first kind Bessel functions

$\xi(t)=\sum_{n=0}^{\infty}\xi_{n}(x)=A_{0}e^{-i\omega_{0}t}\sum_{n=0}^{\infty}\frac{x^{n}}{2^{n}n!}e^{in\omega_{a}t}.$
(7)

At zero order Eq. (7) gives the monochromatic tone amplitude itself
$|\xi_{0}|=A_{0}$, while for $n=1$ it yields the first sideband amplitude

$|\xi_{1}|=\pi\gamma A_{0}b_{a}/2k_{0}$ (8)

which is the searched-for signal of the scheme. One can notice that the
sensitivity of phase-modulation schemes to first order does not depend on the
amount of material used, but rather on its coherence properties—i.e. the spin
resonance linewidth $k_{0}$—making them ideal table-top experiments [45].

Eq. (8) needs to be compared with a background to get the experimental
sensitivity. While the signal calculation is general enough to accommodate for
different experimental schemes, the treatment of the noise is more involved,
and is discussed hereafter in some details. Assuming that the sensitivity is
limited by a white noise with power spectral density $N^{2}_{0}$, we can
calculate the minimum detectable effective axion field as

$\displaystyle\begin{split}\sigma_{a}&=\frac{2k_{0}N_{0}}{\pi\gamma A_{0}}\\\
&=2.27\left(\frac{k_{0}}{10\,\mathrm{kHz}}\right)\left(\frac{N_{0}/A_{0}}{10^{-5}/\mathrm{\sqrt{Hz}}}\right)\,\frac{\mathrm{pT}}{\mathrm{Hz}^{1/2}},\end{split}$
(9)

where $\gamma$ is the electron gyromagnetic ratio, and the other quantities
are roughly the experimental parameters of the setup described in the next
part of this work. For instance, considering an axion signal at 1 kHz and
therefore an integration time of $10^{3}\,\mathrm{s}$—matching $Q_{a}$—we
obtain a sensitivity of about 72 fT.

The interferometer bandwidth—the range of axion masses probed by the
experiment—is in first approximation $k_{0}$, as when $\omega_{a}>k_{0}$ the
sideband $\xi_{1}$ falls out of resonance, and its amplitude decreases.
However, depending on the background origin, the signal-to-noise ratio could
be preserved, and the experimental bandwidth could be increased depending on
the measurement technique.

Let’s briefly discuss the three experimental cases of Table 1, which are
referred to as radiofrequency, microwave, and laser interferometer,
respectively. A radiofrequency interferometer can be realised with both nuclei
or electrons. The former resembles some configurations of the cosmic axion
spin precession experiment [46, 47], although not searching for axion-induced
oscillating dipole moments or magnetisation. However, one could use a
different detection configuration of the same setup to implement the scheme
described in this work. Electron spin based experiments work with low magnetic
fields, which is advantageous in terms of complexity but makes it particularly
sensitive to environmental fluctuations. A non-negligible advantage of
radiofrequency setups is that all the interferometry can be handled digitally
thanks to fast electronics, reducing the hardware requirements to a minimum.
At microwave frequencies nuclear spin resonance becomes inaccessible, although
some nuclei like Holmium show up to GHz precession [48, 49] thanks to a large
effective $\gamma$. Electron spin resonances on the other hand are widely
studied in this frequency range. A major problem occurring with spin
resonances at high frequencies is the broadening of the resonance linewidth,
often referred to as radiation damping [40], which is eliminated by embedding
the sample in a resonant cavity. The resulting photon-magnon hybrid systems
are applied e. g. to quantum technologies [50] and Dark Matter searches [27].

A laser axion interferometer has several intriguing features. Let us consider
as an example a magnetic dipole atomic transition in the infrared range. A
resonant laser beam transmitted through the sample will experience phase
modulation like the carrier signals treated beforehand, and with a quality
factor of order $10^{4}$ the bandwith of the interferometer would encompass
the whole QCD axion range [51]. On top of that, the experimental apparatus can
benefit from all the advantages of interferometers, similarly to gravitational
waves observatories. However, the effective modulation of a magnetic dipole
resonance should be estimated and would probably depend on the nature of the
transition itself 111The effective amplitude modulation and the noise
estimation are not trivial, therefore a thorough discussion of this setup will
be carried out in a forthcoming work.

##  .3 Experimental axion search

Figure 3: Circuit scheme of the halocope. The two resonators represent the
ferrimagnetic arms of the setup, which are probed with a coherent tone each.
The resulting signal is acquired with a data acquisition board and processed.

This section concerns the building and operation of a radiofrequency axion
interferometer sensitive to the axion-electron interaction at sub-megahertz
frequency [53], a scheme of the experimental setup is presented in Fig. 3.

Figure 4: Characterisation and operation of the fermionic interferometer.
Panel (a) shows the transmission spectrum of the north and east resonances
used to calculate the experiment properties and set the tones frequencies
$\omega_{1}$ and $\omega_{2}$. Panels (b) and (c) are an extract of the
interferometer data while in operation, and show the probing tones
superimposed to resonance phases.

The sensing material is a ferrimagnet, NiZn, shaped in two rods oriented
perpendicularly. The rods are the two arms of the interfetometer and are
labelled north (N) and east (E) from the direction they are pointing at. Each
rod is readout by coupling it to a coil, resulting in a resonance frequency of
about 5 MHz, and is then connected to the electronics (for more details see
the appendix).

Given the relatively low frequency of the resonances, the signal is generated
and readout by a 125 MS/s analog-to-digital and digital-to-analog field-
programmable-gate-array board, allowing to handle the signal processing
digitally. In addition, several environmental parameters, like temperature,
pressure or local magnetic field, are recorded with dedicated sensors,
allowing to disentangle a possible axion signal from environmental
systematics. The setup characterisation relies on the transmission measurement
of the resonances, which gives the resonance frequencies of the arms, and
their linewidth, as presented in Fig. 4a. The north arm has resonance
frequency $\omega_{\mathrm{N}}=5.45\,\mathrm{GHz}$ and linewidth
$k_{\mathrm{N}}=19.4\,\mathrm{kHz}$, while the east arm has
$\omega_{\mathrm{E}}=5.63\,\mathrm{GHz}$ and
$k_{\mathrm{E}}=21.4\,\mathrm{kHz}$, obtained by fitting the transmission
measurement with Lorentzian functions.

The haloscope operation requires to send two tones on resonance with each arm
to achieve the maximum phase modulation, as shown in Fig. 4b and 4c. The
tones’ frequency are $\omega_{\mathrm{N}}$ and $\omega_{\mathrm{E}}$, while
their amplitudes are set to $A_{\mathrm{N}}=A_{\mathrm{E}}=1.0\,\mathrm{V}$.
The two analog signals are summed and down-converted according to Fig. 3,
where the down-conversion frequency
$\omega_{\mathrm{LO}}=(2\pi)f_{\mathrm{LO}}$ needs to be
$\omega_{\mathrm{LO}}=(\omega_{\mathrm{N}}+\omega_{\mathrm{E}})/2$. The
interferometer can suppress common phase or amplitude noise depending on the
phase of the down-conversion tone [54], and given the nature of the axion
signal, we opt for phase noise reduction. Eq. (9) shows that it is possible to
arbitrarily improve the sensitivity by increasing $A_{0}$, i. e. using
stronger tones. However, in a realistic experimental configuration $N_{0}$ is
directly proportional to $A_{0}$, or—in the best case scenario—dominated by
quantum fluctuations. The limit of the present setup is the phase noise of the
signal generators, which is about -100 dBc, and fixes $A_{0}/N_{0}\simeq
10^{-5}\,\mathrm{V/Hz^{1/2}}$. As mentioned in the introduction, the
virialised axion field have $Q_{a}\simeq 10^{6}$, and therefore the optimal
resolution bandwidth to detect it is $\mathrm{RBW}=\omega_{a}/Q_{a}$. For
frequencies below 200 kHz $\mathrm{RBW}$ is above one second, but in the
present setup the resolution bandwidth is set to
$\mathrm{RBW}=1\,\mathrm{kHz}$, the minimum allowed by the electronics. The
duty cycle of the acquisition is also limited to roughly 2%. These last two
limitations are purely technical, and will be optimised in an updated version
of the setup. The experiment is controlled using a dedicated characterisation,
acquisition and analysis code which can be found in Ref. [53].

If one arm is oriented towards Vega [40] its resonance frequency is modulated
by the axion field $b_{a}$, while the one of the remaining arm is unaltered. A
single radiofrequency signal probes the on-resonance transmission of the
ferrite, and indicates whether the resonance frequency shifts, while two
signals, each on one arm of the interferometer, can detect relative shifts of
one spin resonance with respect to the other [54]. The measurement scheme is
interferometric, and therefore resilient to systematic uncertainties. The
output data (see Fig. 3) is a spectrum which contains information on whether
$\omega_{\mathrm{N}}$ shifted with respect to $\omega_{\mathrm{E}}$ or
viceversa, which can be used to put a limit on the presence of a dark matter
wind.

An experimental run consists in the acquisition of an arbitrary number of 1
ms-long spectra which are then averaged to a single spectrum. We collect two
runs of $10^{4}\,\mathrm{s}$, called signal and background. The former is
collected while the signal is at the maximum of its daily modulation and the
latter while no signal is expected, so that by comparing them one can extract
an upper limit on the axion-electron interaction. Signal and background are
found to be compatible on all the measured frequencies, so the result is
consistent with no axion signal. Following Eq. (5), the magnetic field limit
is recast in the $2\sigma$ (90% confidence level) upper limit on $g_{p}$
reported in Fig. 5. The minimum value measured for $|g_{p}|$ is $1.2\times
10^{-7}$ for an axion mass of about $0.36\,\mathrm{neV}$.

Figure 5: Exclusion plot, and axion line [51]. Other limits (not shown) are at
higher frequencies [27, 55, 26], or based on solar axion searches [56].

##  .4 Perspectives and conclusions

A fermionic interferometer is a broadband dark matter axion experiment based
on the modulation of spin resonances. Two light beams interact with a magnetic
material whose properties depend on the local dark matter density to then
interfere, eliminating systematic errors and only preserving the searched-for
signal. The scheme can be realised with radiofrequency, microwave or laser
light. We present the experimental implementation and operation of a
radiofrequency fermionic interferometer with which we perform an axion search
in the sub-neV mass range.

This preliminary setup of the interferometer is realised in a minimal
experimental scheme which is prone to improvements, and which is simple enough
to be openly reproduced [53]. Short term improvements include an apparatus
where the interferometry is analogic, and digitisation happens only after the
down-conversion. This allows for continuous monitoring of the signal, and
therefore an arbitrary resolution bandwidth and a unitary duty cycle, making
it also possible to search for transient signals.

For future axion searches, in particular towards $\mu\mathrm{eV}$-mass axions
(see Table 1), one can envision a laser interferometer probing an infrared
magnetic resonance of e. g. Er:YLF or Er:YSO [57, 58, 59]. This axion
experiment is particularly interesting because it leverages the analogy with
gravitational waves detectors, where the magnetic resonance plays the role of
the Fabry-Pérot cavity. However, a detailed studies of the atomic transition
to be used is required to design and operate such experiment.

Beyond axion searches, an intriguing possibility is the use of this
interferometer to detect gravitational waves effects via the gravitational
Larmor [60] or gravitational precession [61, 62]. No change is necessary on
the experimental setup, as it is already sensitive to any effective magnetic
field modulating the spin resonance, but the analysis procedure would need to
be modified in order to measure transient effects instead of persistent ones.
Eventually, we mention that the magnetic field sensitivity of the setup is
interesting also beyond fundamental physics, as when optimised can compete
with state-of-the-art magnetometers [45, 63, 64, 65, 66].

##  .5 Acknowledgements

The support of Amberlab, in the persons of Davide Fasoli and Manuel Pachera,
is greatly acknowledged for the help and support for building the experimental
apparatus. Federico Chiossi is also acknowledged for the advice on the use of
atomic transitions to realise the fermionic interferometer. Eventually, the
author would like to thank Gianni Carugno and Giuseppe Ruoso for the
discussion on the experimental scheme.

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##  .6 Appendix: open source project

The simple hardware requirements of the experiment make it interesting for
open source projects to be carried out in schools, universities or by
amateurs. The described experimental setup is inexpensive and all of the
necessary code, from drivers to analysis routines, is publicly available [53].
A circuit-board scheme of the experiment is sketched in Fig. 6.

Figure 6: Circuit board scheme of the haloscope.