数理解析研究所講究録
1111 巻 1999 年 78-96

78

A SURVEY ON SINGULARITIES OF SOLUTIONS FORFIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS

周 $-$ )SHYUICHI IZUMIYA (泉屋HOKKIADO UNIVERSITY (北海道大学大学院理学研究科)

1. INTRODUCTION
This is a survey article on recent results about the singularities of solutions for first orderpartial differential equations. Firstly we consider the following two kinds of first order partial
differential equations:(Q)$\sum_{i=1}^{n}a_{i}(x, y)\frac{\partial y}{\partial x_{i}}-b(x, y)=0$(H)$H(x_{1}, \ldots, X_{n}, \frac{\partial y}{\partial x_{1}}, \ldots, \frac{\partial y}{\partial x_{n}})=0$,-functions. Here the equation (Q) is called
where $a_{i}(x, y),$ $b(x, y)$ and $H(x,p)$ arequasilinear first order partial differential equation (briefly, a quasilinear equation) and (H) isare well studied in several articles ([2-9,called a Hamilton-Jacobi equation. These
11-14, 19-22, 24-28], etc.). For the study of quasilinear equations, the theory of entropy
solutions has provided the right weak setting (see, for example [22]). For Hamilton-Jacobi
equations, the theory of viscosity solutions is appropriate one ([5-7]). However, these notionsof weak solutions have quite different features. Under the some assumptions, the entropy
solutions are discontinuous and the viscosity solutions are continuous.
We refer the following two typical examples of these equations.
$a$$C^{\infty}$$\mathrm{e}\mathrm{q}\mathfrak{U}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{s}}$Example 1.1. We consider the following equations.
$\frac{\partial y}{\partial x_{1}}+2y\frac{\partial y}{\partial x_{2}}=0$$(\mathrm{Q}^{})$(H)$\frac{\partial y}{\partial x_{1}}+(\frac{\partial y}{\partial x_{2}})^{2}=0$.

We can explicitly solve these equations by the classical method of characteristics, when the. The pictures of the graph of geometric (multi-valued)
initial condition issolutions of these equations are given in Figure 1. These pictures are useful to understand$y(\mathrm{O}, x_{2})=\sin x_{2}$Typeset by$A_{\lambda 4}S-\mathrm{T}ffi$

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the difference between these two equations. We can observe that the geometric solution for(Q) is a smooth submanifold but for (H) is not smooth in the-space.
$(x_{1}, x_{2}, y)$$\mathrm{Q}’$$\mathrm{H}’$Figure 1
We can easily choose the continuous branch of the multi valued solution for $(H’)$ . However,
we cannot choose the continuous branch of the multivaleud solutions for $(Q’)$ .
2. GEOMETRIC FRAMWORK FOR TIME-DEPENDENT$\mathrm{H}\mathrm{A}\mathrm{M}\mathrm{I}\mathrm{L}\mathrm{T}\mathrm{O}\mathrm{N}-\mathrm{J}\mathrm{A}\mathrm{C}\mathrm{o}\mathrm{B}\mathrm{I}$

EQUATIONSIn which we give a brief review of the geometric hamework for the study of singularities
geometric solutions of the time-evolutional Hamilton-Jacobi equations ([14-17]):$\frac{\partial y}{\partial t}+H(t, x_{1}, \ldots, xn’\frac{\partial y}{\partial x_{1}}, \cdots, \frac{\partial y}{\partial x_{n}})=0$$(\mathrm{P}^{})$$\{$$y(0, x_{1}, \cdots, x_{n})=\phi(X_{1}, \cdots, x_{n})$,We describe the theory for the general case here.
Letbe the 1-jet bundle of functions of -variables which may be consideredaswith a natural coordinate system $(x_{1}, \ldots, x_{n}, y,p1, \ldots,pn)$ , whereis acoordinate system of. We also have a natural projection :given
by $\pi(_{X}, y,p)=(x, y)$ .
An immersion germis said to be a Legendrian immersion germ
(i.e., Legendrian submanifold germ) if $\dim L=n$ and, where $\theta=dy-\sum_{i=}^{n}1pi$ .
.
The image ofis called the wave front set of and it is denoted by $W(i)$ . We also
consider the 1-jet bundle
and the canonical 1-form $$ on that space. Let
be a canonical coordinate system on
and $(t, x_{1}, ..,., x_{n}, y, s,p1, \ldots,p_{n})$the corresponding coordinate system on. Then, the canonical 1-form is given
by $=dy- \sum_{i=}^{n}1pi$ . $d_{X_{i^{-S}}}\cdot dt=\theta-s\cdot dt$ .
$J^{1}(\mathbb{R}^{n}, \mathbb{R})$$n$$\mathbb{R}^{2n+1}$$(x_{1}, \ldots, x_{n})$$\mathbb{R}^{n}$$\pi$$J^{1}(\mathbb{R}^{n}, \mathbb{R})arrow \mathbb{R}^{n}\cross \mathbb{R}$$i:(L_{0}, u\mathrm{o})arrow J^{1}(\mathbb{R}^{n}, \mathbb{R})$$i^{*}\theta=0$$\pi \mathrm{o}i$$i$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$(t, x_{1}, \ldots, x_{n})$$\mathbb{R}\mathrm{x}\mathbb{R}^{n}$$J^{1}(\mathbb{R}\chi \mathbb{R}^{n}, \mathbb{R})$$d_{X}i$

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We define the natural projection :$(t, x, y)$ . We call the above 1-jet bundle an unfolded 1-jet bundle.
A Hamilton-Jacobi equation is defined to be a hypersurface$\Pi$(G-H-J)$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})arrow(\mathbb{R}\mathrm{X}\mathbb{R}^{n})\mathrm{x}\mathbb{R}$by $\Pi(t, x, y, s,p)=$$E(H)=\{(t, x, y, s,p)\in J^{1}(\mathbb{R}\mathrm{X}\mathbb{R}^{n}, \mathbb{R})|_{S+}H(t, x,p)=0\}$

A geometric (multi-valued) solution of $E(H)$ is a Legendrian submanifold
lying in $E(H)$ . In this case the wave front set $W(i)$ is “the graph” of theingeometric solution which is generally a hypersurface with singularities.
, we define
In order to study (P) we need the following hamework: For any

in$L$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R}).$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$c\in(\mathbb{R}, 0)$$E(H)_{c}=\{(_{C,x,y,-H}(C, x,p),p)|(x, y,p)\in J1(\mathbb{R}n, \mathbb{R})\}$.

and $\mathrm{O}-_{C}=\Theta|E(H)c=$
Then, $E(H)_{c}$ is a $(2n+1)$ -dimensional submanifold of:$dz- \sum_{i=}^{n}1p_{i}dxi$ gives a contact structure on $E(H)_{c}$ . We define a mapping
is a contact diffeomorphism$E(H)_{c}$ by $\iota_{c}(x, y,p)=(C, x, y, -H(C, x,p),p)$ . The mapping
and the following diagram is commutative:$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$\iota_{c}$$J^{1}(\mathbb{R}^{n}, \mathbb{R})arrow$$i_{c}$$J^{1}(\mathbb{R}^{n}, \mathbb{R})arrow\iota_{c}E(H)_{c}$$\downarrow\pi_{\mathrm{C}}$$\pi\downarrow$$\mathbb{R}^{n}\cross \mathbb{R}$$–\mathbb{R}^{n}\mathrm{x}\mathbb{R}$.

We say that a geometric Cauchy problem (with initial condition $L’$ ) associated with thetime parameter $(GcP\tau)$ is given for an equation $E(H)$ if there is given an n-dimensional. Sincesubmanifold : $L’\subset E(H)$ with $i^{*}\Theta=0$ and $i(L’)\subset E(H)_{c}$ for someis the characteristic vector field given by$X_{H}\not\in TE(H)_{c}$ , we have $X_{H}\not\in TL’$ , where$c\in(\mathbb{R}, 0)$$i$$X_{H}$$X_{H}= \frac{\partial}{\partial t}+\sum_{i=1}n\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial x_{i}}+(\sum_{=i1}p_{i}\frac{\partial H}{\partial p_{i}}-Hn)\frac{\partial}{\partial y}-\frac{\partial H}{\partial \mathrm{t}}\frac{\partial}{\partial s}-\sum^{n}\frac{\partial H}{\partial x_{i}}\frac{\partial}{\partial p_{i}}i=1^{\cdot}$

By using the classical characteristic method, we can show that there exists a unique geometricsolutions around $L’$ .
We remark that Cauchy problem (P) is a GCPT. The initial submanifold is given by.

$L_{\phi,0}= \{(0, x, \phi(x), -H(0, x, \frac{\partial\phi}{\partial x}), \frac{\partial\phi}{\partial x})|x\in \mathbb{R}^{n}\}\subset E(H)_{0}$The problem of studying the singularities of the graph of the geometric solution is formulated as follows:
Geometric Problem. Classify th generic bifu rcations of wave fronts of$\mathrm{e}$$\pi_{t}|$:$L\cap E(H)_{t}arrow \mathbb{R}^{n}\cross \mathbb{R}$81with resp $ect$ to th parameter (i.e., th generic bifurca tions of wave fronts of geome tric
solutions along th time parameter).
$t$$\mathrm{e}$$\mathrm{e}$$\mathrm{e}$Following [16], in order to study the singularities of the geometric solution we identify geometric solutions with one-parameter Legendrian unfoldings. Let be an $(n+1)$ -dimensionalsmooth manifold, :be a submersion germ and :bea smooth map germ. We say that the pairis a Legendrian family if
is aLegendrian immersion germ for any. Then there exist a unique element
such that, whereis the ring of smooth function germs at . Define a
map germ :by$R$$(R, u\mathrm{o})arrow(\mathbb{R}, t_{0})$$\mu$$\ell$$(R, u_{0})arrow J^{1}(\mathbb{R}^{n}, \mathbb{R})$$(\mu, \ell)$$\ell_{t}=\ell|\mu^{-1}(t)$$t\in(\mathbb{R}, t_{0})$$\ell^{*}\theta=h\cdot d\mu$$h\in C_{u_{\mathrm{O}}}^{\infty}(R)$$C_{u_{0}}^{\infty}(R)$$u_{0}$$(R, u_{0})arrow J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$\mathcal{L}$$\mathcal{L}(u)=(\mu(u), x\circ l(u),$ $y\circ\ell(u),$ $h(u),p\mathrm{o}\ell(u))$.

We can easily show that is a Legendrian immersion germ. If we fix 1-forms and , theLegendrian immersion germ is uniquely determined by the Legendrian family. We
call a Legendrian unfolding associated with the Legendrian family.
We have to study how various branches of the multi-valued graph$W(i)$ intersecting at a point bifurcate in time for an arbitrary Hamiltonian $H(t, x,p)$in [17].
We classify the bifurcations of the branches of the graph by classifying the bifurcations of
singularities of multi-Legendrian unfoldings which are expressed in terms of multi-germs.
Let:$z_{i})(i=1, \ldots, r)$ be Legendrian unfoldings with$\Pi(z_{i})=0$ whereare distinct. We calla multi-Legendrian unfolding. Letandbe multi-Legendrian unfoldings. We say that theseare-Legendrian equivalent if there exist contact diffeomorphism germs$\mathcal{L}$$\Theta$$\mathcal{L}$$\theta$$(\mu, l)$$\mathcal{L}$$(\mu, \ell)$$W_{t}=(\{t\}\cross \mathbb{R}^{n}\cross \mathbb{R})\cap$$\mathcal{L}_{i}$$(R, u\mathrm{o})arrow(J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R}),$$z_{1},$$\ldots,$$(\mathcal{L}_{1}, \ldots, \mathcal{L}_{r})$$z_{r}$$(\mathcal{L}_{1}, \ldots , L_{r})$$(\mathcal{L}_{1}’, \ldots, \mathcal{L}_{r}’)$$P_{(r)}$$K_{i}$:$(J^{1}(\mathbb{R}\cross \mathbb{R}n, \mathbb{R}),$$zi)arrow(J^{1}(\mathbb{R}\cross \mathbb{R}n, \mathbb{R}),$$Z’)i$$(i=1, \ldots, r)$of the form $K_{i}(t, x, y, s,p)=(\phi_{1}(t), \phi 2(t, x, y), \phi_{\mathrm{s}}(t, x, y), \phi_{4}^{i}(t, x, y, s,p), \phi_{5}i(t, X, y, s,p))$ anda diffeomorphism germ
: $(R, u_{0})arrow(R, u_{0}’)$ such thatfor any $i=$
, . It is clear that if two multi-Legendrian unfoldings are-Legendrian equivalent,
then there exists a diffeomorphism germ :of the form$\Phi(t, x, y)=(\phi_{1}(t), \phi 2(t, X, y), \phi 3(t, x, y))$ such that
. Thus theabove equivalence describes how bifurcations of wavefronts (i.e. graphs of solutions) interact.
We can define the notion of stability with respect to the-Legendrian equivalence inthe same way as for the ordinary Legendrian stability (see [1,29]). Motivated by Arnol’dZakalyukin’s theory ([1, 29]), we can construct multi-generating families of multi-Legendrian
unfoldings and give a classification of-Legendrian stable Legendrian unfoldings by using
the classification of multi-families of function germs in Zakalyukin [29]. We get a list ofclassifications for $n=1,2,3$ in [17]. However, we only present the list of classifications for$n=1$ . For the case $n=2,3$ , see [17].
$\Psi$$1,$$\ldots$$K_{i}\circ \mathcal{L}_{i}=\mathcal{L}_{i}’\circ\Psi$$r$$P_{(r)}$$\Phi$$(\mathbb{R}\cross(\mathbb{R}^{n}\cross \mathbb{R}), 0)arrow(\mathbb{R}\cross(\mathbb{R}^{n}\cross \mathbb{R}), 0)$$\Phi(\bigcup_{i1}^{r}=W(\mathcal{L}_{i}))=\bigcup_{i=1}^{r}W(\mathcal{L}_{i})$$P_{(r)}$$P_{(r)}$

Theorem 2.1 [1]. Suppos that $n=1$ . Then a generic multi-Legendrian unfolding is
egendrian uivalent to one of th multi-Legendrian unfoldings in th following list:$\mathrm{e}$$L$$\mathrm{e}q$$\mathrm{e}$$r=1$ ;
$0A_{1}$$0A_{2}$::$(t, u, 0,0, \mathrm{o})$;$(t, 3u^{2},2u3, \mathrm{o}, u)$;$e$$P_{(r)^{-}}$

82$1A_{3}$: $(t, 4u^{3}+2ut, 3u^{4}+u^{2}t, -u^{2}, u)$ .

$r=2$ ;
$0(^{0}A1A01)$

1 $(^{0}A_{1}0A_{1})$$1A_{2}0A_{1}$: $((t, u, -u, \mathrm{O}, -1), (t, u, u, \mathrm{O}, 1))$ ;: $((t, u, t\pm u^{2},1, \pm 2u), (t, u, 0,0,0))$ ;: $((t, 3u-2t, 2u3, u, u), (t, u, -u, \mathrm{o}, -1))$ .

$r=3$ ;.
1from theWhen we consider the geometric solution, we can get rid of the germabove list because the geometric solution is a one-to-one immersions into the unfolded l-jet
space. For the purpose, we need a kind of non-degeneracy condition on the Hamiltonian
function. We say that a Hamiltonian function $H(t, x,p)$ is non-degenerate at $(t_{0,x_{0},p0})$ if
for some $1\leq i,j\leq n$ . This condition is weaker than the condition
itfor-variables atthat $H(t, x,p)$ is convex (or concave) with respect to
given
for
a
singularities
generic$n\geq 2$ . The following theorem is a realization theorem for
Hamilton-Jacobi equation.
Theorem 2.2 ([17,18]). Let $H(t, x, p)$ be a non-degenerate Hamiltonian function germ ategendrian stablebe a$(t_{0}, x0,p_{0})$ and:. Then there exis a Legendrian unfolding
Legendrian unfolding associ ated withuation $s+H(t, x,p)=0$ such thattion of the Hamilton-Jacowhich is a geometric-Legendrian $eq$ uivalent.
areandsingularity (even for general ) describes how the singularity appears
We remark that
-Legendrian stable Legendrian unfoldings, so that thesefrom a smooth solution. These arecan be realized as geometric solutions at the non-degenerated point for a given HamiltonJacobi equation. We can asserts the detailed statement for the case that the Hamiltonian-variables. In this case the Cauchy problem is given byfunction depends only on$0A_{1}0A_{1}0A_{1}$:$((t, u, t-u, 1, -1), (t, u, \mathrm{O}, \mathrm{O}, \mathrm{o}), (t, u, u, \mathrm{o}, 1))$$(^{0}A_{1}0A_{1})$$\frac{\partial^{2}H}{\partial p_{i}\partial p_{j}}(t_{0,0}x,p0)\neq 0$$(t_{0}, x_{0,p_{0}})$$(p_{1}, \ldots,p_{n})$$\mathcal{L}$$(R, u_{0})arrow(J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R}),$ $(t_{0}, x_{0}, y_{0}, s_{0},p\mathrm{o}))$$P_{(1)^{-}}L$$\mathcal{L}’$$(\mu, \ell)$$\mathrm{t}s$$\mathcal{L}$$sol\mathrm{u}$$\mathcal{L}’$$\mathrm{b}i\mathrm{e}q$$P_{(1)}$$1A_{3}$$n$$P_{(1)}$$(p_{1}, \ldots,p_{n})$$\frac{\partial y}{\partial t}+H(\frac{\partial y}{\partial x_{1}}, \cdots, \frac{\partial y}{\partial x_{n}})=0$

(P)

$\{$

,
-functiions. Then we have the following proposition.
where $H$ and are
Proposition 2.3. Let $s+H(p)=0$ be a Hamilton-Jacobi equation. If a singularity ofears at a point $(t_{0}, x_{0},p0)$ , then $H$ isgeome tric solution for the Cauchy problem.
non-degenrated at
In this case the characteristic equation is given by$y(0, x_{1}, \cdots, x_{n})=\phi(X_{1}, \cdots, x_{n})$$\phi$$c\infty$$(P)\mathrm{a}pp$$(t_{0}, x\mathit{0},p_{0})$$(\mathrm{C}^{})$

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We can explicity solve the charcteristic equation as follows:$x_{i}(t, u)=u_{i}+t \frac{\partial H}{\partial p_{i}}(\frac{\partial\phi}{\partial u}(u))(i=1, \ldots, n)$(S)$p_{i}(t, u)= \frac{\partial\phi}{\partial u_{i}}(u)(i=1, \ldots, n)$
$\{$

,

,

$y(t, u)=t \{-H(\frac{\partial\phi}{\partial u}(u))+\sum_{i=1}n\frac{\partial\phi}{\partial u_{i}}(u)\cdot\frac{\partial H}{\partial p_{i}}(\frac{\partial\phi}{\partial u}(u))\}+\phi(u)$.

3. VISCOSITY SOLUTIONS
The viscosity solutions for nonlinear equations of first order have been introduced byCrandall and Lions [7]. Such solutions need not be differentiable everywhere, as the only
regularity required in the definition is that of continuity. The function
is aviscosity solution of$y_{\mathfrak{h}}\in C(\mathcal{O})$(H-J)$\frac{\partial y}{\partial t}+H(t, X, \frac{\partial y}{\partial x_{1}}, \ldots, \frac{\partial y}{\partial x_{n}})=0$

in the open domain$\mathcal{O}\subset \mathbb{R}^{+}\cross \mathbb{R}^{n}$provided$\frac{\partial\psi}{\partial t}(t, x)+H(t, x, \frac{\partial\psi}{\partial x_{1}}(t, x), \ldots, \frac{\partial\psi}{\partial x_{n}}(t, X))\leq 0$

for any, (resp.

$\geq 0$)for whichattains a local maximum (resp. local minimum) at thepoint. The function
is a viscosity solution of the Cauchy
problem $(P)$ if and only if it is a viscosity solution of (H-J) in the domain
andsatisfies the initial condition. The above inequality will be referred as
the viscosity criterion at the point $(t, x)$ . We next state the viscosity criterion in a form which
is more useful for the construction of the solution. To this end, assume thatis open and that there is a smooth hypersurface of, which divides into two open
sets
and
. Then we have the following theorem.
$\psi\in C^{1}(\mathcal{O})$$y_{\mathfrak{v}}-\psi$$(t, x)\in \mathcal{O}$$y_{\mathfrak{v}}\in C([0, \infty)\cross \mathbb{R}^{n})$$(0, \infty)\cross \mathbb{R}^{n}$$\lim_{tarrow 0+}y\mathfrak{v}(t, x)=\phi(x)$$\mathcal{O}\subset(0, \infty)\cross \mathbb{R}^{n}$$\Gamma$$\mathcal{O}^{+}$$\mathcal{O}^{-},$$\mathbb{R}^{+}\cross \mathbb{R}^{n}$$\mathcal{O}$$\mathcal{O}=\Gamma\cup \mathcal{O}^{+}\cup \mathcal{O}^{-}$

Theorem 3.1. Let
andin. Thenis a viscosity solution of (H-J) in
conditions hold:a)andare classical solu tions of (H-J) inand
If the vector, then$y_{\mathfrak{v}}\in C(\mathcal{O})$$y_{\mathfrak{v}}=y_{\mathfrak{v}}^{+}$$O^{+}\cup\Gamma,$$C^{1}(\mathcal{O}^{\pm}\cup\Gamma)$

in
where
if and only if the following$y_{\mathfrak{h}}=y_{\overline{\mathfrak{v}}}$$\mathcal{O}$$y_{\mathrm{b}}$$y_{\mathfrak{v}}^{+}$$\mathcal{O}^{+}$$y_{\overline{\mathfrak{v}}}$$b)$$\mathcal{O}^{-}\cup\Gamma$$y_{\mathfrak{h}}^{\pm}\in$,$\mathcal{O}^{-}r\mathrm{e}spectiv\mathrm{e}\iota_{y}$$\tilde{\eta}=(H(t, x, \frac{\partial y^{+}}{\partial x})-H(t, x, \frac{\partial y^{+}}{\partial x}), -(\frac{\partial y^{+}}{\partial x_{1}}-\frac{\partial\overline{y}}{\partial x_{1}} , .

.

.

points in to, \frac{\partial y^{+}}{\partial x_{n}}-\frac{\partial y^{-}}{\partial x_{n}}))$$\mathcal{O}^{+}$$H(t,$ $x,$

(resp.

$\geq 0$$(1- \lambda)\frac{\partial y_{\mathrm{b}}^{+}}{\partial x}+\lambda\frac{\partial y_{\mathrm{U}}}{\partial x})-(1-\lambda)H(t, X, \frac{\partial y_{\mathfrak{v}}^{+}}{\partial x})-\lambda H(t, X, \frac{\partial y_{\overline{\mathfrak{v}}}}{\partial x})\leq 0$), where $\lambda\in[0,1]$ .

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In particular, th graph of $H$ lies resp tively below or above the line egment joining the.
andpointsThe proof of Theorem 3.1 is given in ([20, 21]) as a direct application of Theorem 1.3 in [5].
The condition b) will be referred in the sequel as the viscosity criterion. The hypersurface
has the properties specified in the above theorem is thein the neighbourhood of whichshock surface. If the Hamiltonian is uniformly convex (or concave), we can automaticallyconstruct viscosity solutions from our normal forms, so that we can easily draw the picturesof shock surfaces for lower dimensional cases. In [4] Bogaevskii has shown that the potential
solution of the Burgers system with vanishing viscosity is given by the minimum function of acertain family of smooth functions. It corresponds to the viscosity solution of the Hamilton. He has
Jacobi equation when the Hamiltonian is given bydrawn the pictures of shocks for this case. Our pictures are same as his pictures, so we do
not present these in here (see [4]).
On the other hand, Bogaevskii used Florin-Hopf-Cole method ([10, 12]) to detect thesolution for the Hamilton-Jacobi equation correspoding to the Burgers system. However,his method works for geral Hamilton-Jacobi eqauations which are convex with respect toto our situations in stead-variables. In this case we apply Bardi-Evans’is given bymethod. The geometric solution forof$s$$\mathrm{e}c$$\mathrm{e}$$( \frac{\partial y}{\partial}x^{\mathrm{L}}H+,(t, x, \frac{\partial y_{\mathrm{r}}^{+}}{\partial x}))$$( \frac{\partial y_{0}^{-}}{\partial x},$$\Gamma$$H(t, x, \frac{\partial}{\partial}\overline{y_{-}\Delta})x)$$y_{0}$$H(p_{1}, \ldots p_{n})=\frac{1}{2}p_{1}^{2}+\cdots+\frac{1}{2}p_{n}^{2}$$\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}\mathrm{t}[2]$$(p_{1}, \ldots,p_{n})$$(\mathrm{P}’)$$\mathrm{F}\mathrm{l}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}- \mathrm{H}_{0}\mathrm{p}\mathrm{f}_{-}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{e}$(S)$L_{\phi,t}=\{(t, x(t, u), y(t, u), -H(p(t, u)),p(t, u))|u\in \mathbb{R}^{n}\}$

,

where

We consider a family of functions $F(t, x,p, q)=\phi(q)+<p,$ $(x-q)>-H(p)t$ , where. We havethe canonical inner product on
and $<,$$\mathbb{R}^{n}$$>\mathrm{i}\mathrm{s}$$(t, x, p, q)\in \mathbb{R}\cross \mathbb{R}^{n}\cross(\mathbb{R}^{n}\cross \mathbb{R}^{n})$$\Sigma(F)=\mathrm{f}(t, q+\frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial q}(q))t,$

where$\Sigma(F)$is the set defined to beby$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$\frac{\partial F}{\partial p_{i}}=0$

and$\frac{\partial\phi}{\partial q}(q),$$\frac{\partial F}{\partial q_{i}}=0$,. We now define a map. It follows that$\Phi_{F}$:$\Sigma(F)arrow$$\Phi_{F}(t, X,p, q)=(t, x, F(t, x,p, q), \frac{\partial F}{\partial t} , \frac{\partial F}{\partial x})$$\Phi_{F}(t, q+\frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial q}(q))t,$$\frac{\partial\phi}{\partial q}(q),$$q)=$$(t, q+ \frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial q}(q))t,$$-H( \frac{\partial\phi}{\partial q}(q))t+<\frac{\partial\phi}{\partial q}(q),$ $\frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial q}(q))>+\phi(q)$$-H( \frac{\partial\phi}{\partial q}(q)),$$\frac{\partial\phi}{\partial q}(q))$

is equal toThis shows that the image of the map.
family of
We refer the following result of Bardi-Evans [2].
$\Phi_{F}$$L_{\phi,t}$$q)|(t, q)\in \mathbb{R}\mathrm{x}\mathbb{R}^{n}\}$,.

$L_{\phi,t}$, namely, $F$ is a global generating

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Theorem 3.2.

$Ass\mathrm{u}m\mathrm{e}$that th Hamiltonian $H(p_{1}, \ldots,p_{n})$ is convex, then$\mathrm{e}$$y(t, x) \equiv\inf_{q}\sup_{p}\{\phi(q)+<p, (x-q)>-H(p)t\}$is the unique$vi\mathrm{s}Co\epsilon i\mathrm{t}_{\mathrm{J}^{\Gamma}}sol$

ution of (P).

Then we have the following theorem as a corollary of the above theorem.

Theorem 3.3. Assume th at $H$ is uniformly convex andas th minim um. Let
th geome triction (S) of th Cauchy problem $(rmP’)$ . Then$\phi h$$\mathrm{e}$$L_{\phi,t}$be$sol\mathrm{u}$$\mathrm{e}$$\mathrm{e}$$y(t, x) \equiv\min_{y}\{y|(t, x, y)\in\Pi(L_{\phi,t})\}$

is th uniq ue viscosity solu tion of (P).
$\mathrm{e}$However, for general (non-convex) Hamiltonian, situations are quite different.
4. NON CONVEX HAMILTONIANS IN ONE SPACE VARIABLE

In this section we stick to the Cauchy problem of Hamilton-Jacobi equation in one spacevariable as follows:$\frac{\partial y}{\partial t}+H(\frac{\partial y}{\partial x})=0$

(P)

$\{$

$y(0, x)=\phi(_{X)}$,where $H$ and are-functions. Since $H(p)$ is not assumed to be uniformly convex (orwe cannot use Theorem 3.3, so that the situations should be quite complicated evenfor the one space variables case.
In this case the geometric solution is given by$\phi$$C^{\infty}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{C}\mathrm{a}\mathrm{v}\mathrm{e}))$$L_{\phi,t}=\{(t, X(t, u), y(t, u), -H(p(t, u)),p(t, u))|u\in \mathbb{R}\}$

where

,

$x(t, u)=u+tH’(\phi^{J}(u))$ ,$\{$$p(t, u)=\phi/(u)$$y(t, u)=t\{-H(\phi/(u))+\phi’(u)H’(\phi/(u))\}+\phi(u)$ .

Before the first critical time that characteristics cross in the $(t, x)$ -plane,is the graphof the viscosity solution . After the characteristics cross,becomes singular. Theorem
2.1 describes the generic singularities of . The first singularity appears in the form of.
See Figure , where we show the shape of the appearing singularity. By Proposition 2.3,
these appear at the convex or the concave points of the Hamiltonian function. Away fromthe singularity, the viscosity solution is given by. In ([17], [18]) we have constructed theunique viscosity solution past the ffist critical time by selecting a single-valued branch of.
$(t_{0},x_{0},p0)$
Assume that the singularity of typeappears at the point. After the criticaltime , the wave front
is three-valued on an interval $(x_{1}(t), X_{2}(t))$ ; see Figure . Let ,$i=1,2,3$ be the three branches of, whereis defined on a neighborhood ofandon a neighborhood of$\chi(t)\in(x_{1}(t),x2(t))$. Thenintersect at one point, for$t>t_{0}$ . We define the viscosity solution pastby selecting a continuous single-valued branchof
as follows:$W_{t}$$W_{t}$$y_{\mathfrak{d}}$$1A_{3}$$W_{t}$$2\mathrm{a}$$W_{t}$$W_{t}$$1A_{3}$$t_{0}$$W_{t}$$2\mathrm{b}$$y_{i}$$W_{t}$$x_{2}(t)$$y_{2}$$y_{1}$$y_{1},$$y_{2}$$t_{0}$$W_{t}$$x_{1}(t)$86Theorem 4.1. There exis$(x_{1}(t), x2(t)),$ deBned by$\mathrm{t}s$an$\epsilon>0$such that th function$\mathrm{e}$$y_{1}(t, x),$$x\leq x(t)$$y_{3}(t, x),$$x\geq x(t)$$y_{\mathfrak{g}}(t, x)=\{$$(^{*})$

is the viscosity$sol\mathrm{u}$tion of $(P)$ in a neighborhood of$x_{0}$$y_{\mathfrak{y}}(t, x),$$(t, x)\in(to, t_{0}+\epsilon)$$\cross$,

past th time$\mathrm{e}$$t_{0}$.

In view of Proposition 2.3 the viscosity criterion (see Section 3) is satisfied across. Hence, by the uniqueness of
is a classical solution away fromdefined bywhile
gives the viscosity solution of (P) past .
the viscosity solution,
By this construction, we have extended the viscosity solution beyond the first critical time$\chi(t)$$y_{0}$$\chi(t)$$(^{*})$$t_{0}$$(^{*})$$t_{0}$.

$\mathrm{x}\mathrm{r}_{\mathrm{f}11})-$FigureFigureFigureconcave domains of $H(p)$ ,or
According to Theorem 2.5 the shock is generated in a convex
so the viscosity criterion is automatically satisfied. The graph of the viscosity solution past
$H$ isthe first critical time is depicted by a full line in Figure , where we assume thatconvex in the neighborhood of the appearing singularity 1A3. The shock corresponds to theintersection of the two branches and it is called a genuine shock. The genuine shock is definedas the intersection of two incoming characteristics (or waves) and its speed is given by the
Rankine-Hugoniot condition$2\mathrm{a}$$2\mathrm{c}$$2\mathrm{b}$$t\mathrm{c}$$\chi’(t)=’\frac{H(y_{\mathfrak{v}}^{+}x(t,x(t)))-H(y\overline{\mathfrak{v},}x(t,\chi(t)))}{y_{\mathfrak{v}x}^{+})(t,\chi(t))-y^{-}\mathfrak{o},x(t,\chi(t))}$,. Therefore in order to follow the evolution of the shock
and
where
we have to study the following questions:intersecting at one point bifura) How different branches of the multi-valued graph ofcate in time.
b) If the two branches initially defining the shock continue to cross, whether the viscosity
criterion is satisfied across the intersection.
, we can choose the correct
If the viscosity criterion is satisfied at the time
branch of the graphs of the geometric solutions as viscosity solutions.
$y_{\mathrm{U}_{)}x}^{\pm}= \frac{\partial y}{\partial}x\mathrm{g}_{-}\pm$$\chi’(t)=\underline{d}_{X}dt(t)$$W_{t}$$t_{\alpha}=t_{0}+\epsilon$

87
We will now investigate how the viscosity criterion can be violated across the intersection of
two branches. Assume that a generated shock is defined by two intersecting branches
and. We denote by(resp.
) the branch representing the viscosity solution for $x<\chi(t)$(resp. $x>\chi(t)$ ). If the two branches remain intersected they evolve according to.
We denote by
the intersection of the two branches. In the case when $H(p)$ has only one
inflection point Kossioris [20] studied this problem and constructed the viscosity solutions.
We consider the general situation here. It is clear that for generic Hamiltonian function$H(p),$ $H$ has only Morse type critical points and no tritangent lines. So we assume
that theHamiltonian has the above properties. By Theorem 2.1, we have the following theorem.
Theorem 4.2. For a generic in itial function , if th viscosity criterion is violated at ,then the only following 8 cas es may occur:(1) The normal form isandangent to the graph of $H(p)$ at only on of
the points
and the line is not tangent to the graph at other points between these$y^{-}$$y^{+}$$y^{+}$$\overline{y}$$\mathit{0}_{(^{0}A_{1}A_{1})}0$$\chi(t)$$\phi$$\mathrm{e}$$t_{\alpha}$$\overline{P^{+}P^{-}}$$\mathit{0}(0A_{1}0A_{1})$$\mathrm{e}$$P^{+},$$P^{-}$

points.

(2) The normal form is $0(0A_{1}0A_{1})$ andnot tangent to th graph of $H(p)$ at eachpoint
and there exists only one another point be tween these points at where theabove line is tangent to the graph.
(3) The normal form isandtangent to th graph of $H(p)$ at only one of th
pointsand the line is not tangent to the graph at other points between these points.
We enote
line throughin the $(p, H(p))$ -plane, where$\overline{P^{+_{P}}-}i_{S}$$\mathrm{e}$$P^{+},$$P^{-}$$1A_{2}0\mathrm{A}_{1}$$\overline{P^{+}P-}i_{\mathrm{S}}$$\mathrm{e}$$P^{+},$$d$$\mathrm{e}$$P^{-}$$\overline{P^{+}P-}\mathrm{t}he$$P^{+},$ $P^{-}$$P^{+}=(y_{x}^{+}(t_{\alpha}, \chi(t\alpha)),$ $H(y^{+}x(t\alpha’\chi(t_{\alpha})))$$P^{-}=(y_{x}^{-}(t\alpha’\chi(t\alpha)), H(y_{x}^{-(}t\alpha’ x(t_{\alpha})))$,.

We can show that the case 3) cannot ocur if the viscosity criterion is satisfied before theperestroika time . We can solve local Riemann problems and construct viscosity solutions
for each case in the above theorem. However, we only consider the cases (1) in this note. For$t_{\alpha}$

the deatiled consideration , please refere [16]Case (1). We assume that the graph of the viscosity solution at the time
as in Figure .

$t\leq t_{\alpha}$

is depicted$3\mathrm{a}$$\mathrm{H}(\mathrm{P})$FigureWithout the loss of generality, we may assume thatat the point (andb’ligure 3$3\mathrm{a}$$y_{x}^{-}(t_{\alpha’ x}(t_{\alpha})),$$\overline{P^{+}P^{-}}\mathrm{i}\mathrm{S}$$H(yx(-t_{\alpha}, \chi(t_{\alpha})))$$H”((y_{x}^{-}(t_{\alpha}, \chi(t_{\alpha})))<0$$\mathrm{b}$tangent to the graph of $H(p)$88). As we already mentioned that the genuine shocks satisfies the Rankine(see Figure
Hugoniot condition. So we should construct new characteristics which satisfies both of the
Rankine-Hugoniot condition and the viscosity criterion. In this case we have
$3\mathrm{b}$$H’(y_{x}^{-}(t \alpha’\chi(t\alpha)))=\frac{H(y_{x}(+t\alpha’\chi(t\alpha)))-H(y_{x}-(t\alpha\chi(t_{\alpha})))}{y_{x}^{+}(t_{\alpha},\chi(t\alpha))-y_{x}(-t\alpha’ x(t\alpha))},=x(t_{\alpha})/$.

We now distinguish two cases as follows:a) If$H’(y_{x}^{-}(t, x(t))) \geq\frac{H(y_{x}^{+}(t,\chi(t)))-H(y_{x}^{-((t)))}t,x}{y_{x}^{+}(t,\chi(t))-y^{-(t,\chi(}xt))}$, then we can easily show that the viscosity
for sufficiently small
for. So we can choose single valued continuous branches of thecriterion is satisfied forgeometric solution as the viscosity solution.
b) If$\epsilon>0$$t_{\alpha}\leq t<t_{\alpha}+\Xi$$t<t_{\alpha}+\epsilon$$H’(y_{x}^{-((}t, xt)))< \frac{H(y_{x}^{+}(t,x(\iota)))-H(y_{x}-(t,\chi(t)))}{y_{x}^{+}(t,\chi(t))-y^{-(t,\chi(}x\mathrm{t}))}$, then we can easily show that the viscosity
for sufficiently small
for, so that a new way to build the solution is required
criterion is violated for(cf., Figure 4).
$\epsilon>0$$t_{\alpha}\leq t<\mathrm{t}_{\alpha}+\epsilon$$t_{\alpha}<t<t_{\alpha}+\epsilon$$\mathrm{H}(\mathrm{p})$$\mathrm{H}(\mathrm{P})$FIGURE 4
In this case we can use the techniques in [20] to construct the contact discontinuity shock
curve and then obtain new characteristics. Lets consider the relation $H’(q)= \frac{H(p)-H(q)}{p-q}$
and $H^{;/}(q_{0})\neq 0$ . By the implicit function
witharoundtheorem, there exists a smooth function around such that the above relation is equivalent
to $q=\psi(p)$ . We will first construct the contact discontinuity as the solution of the following
initial value problem.
$\chi_{C}’(t)=H’(\psi(yx+(t, \chi c(t))))$ ,$(q0,p\mathrm{o})$$q_{0}\neq p_{0},$$H’(q_{\mathit{0}})= \frac{H(p\mathrm{o})-H(q\mathrm{o})}{p_{0}-q_{\mathrm{O}}}$$\psi$$\{$$\chi_{C}(\mathrm{t}_{\alpha})=\chi(t\alpha)$$p_{0}$.

89

The characteristic which is started at a point$(\tau, \chi_{c}(\tau))$should be satisfied the following:$X’(t)=H;(p(t))$ ,$\{$$p’(t)=0$$y’(t)=-H(p(t))+p(t)H’(p(t))$ ,with the initial condition$x(\tau)=xc(\mathcal{T}),$$y(\tau)=y^{+}(\tau, \chi_{c}(\mathcal{T}))$and $p(\tau)=\psi(y_{x}^{+}(\tau, xC(\tau)))$ .

So the solution is exactly given as follows:$\tilde{x}(t)=x_{c}(\tau)+(t-\mathcal{T})H’(\psi(y^{+}x(\mathcal{T}, \chi c(\mathcal{T}))))$$\tilde{p}(t)=^{\psi(y(\mathcal{T}}x+,$,$\chi_{\mathrm{C}}(_{\mathcal{T})}))$$\{$$\tilde{y}(t)=y(+\tau, x_{c}(\tau))$.

$+(t-\tau)\{-H(\psi(y_{x}^{+}(_{\mathcal{T}}, \chi c(_{\mathcal{T})})))+\psi(y_{x}+(\tau, \chi_{C}(\mathcal{T})))H’(\psi(y_{x}^{+}(\tau, x_{C}(\tau))))\}$

By definition of the contact discontinuity, we have$\chi_{c}’’(t)=H’’(\psi(\phi(u+(t))\frac{\partial\psi}{\partial p}(\phi’(u_{+}(t))\phi’’(u_{+())(t)}tu’+$where $\chi_{c}(t)=u_{+}(t)+tH’(\phi(u_{+}(t)).$ Since$\frac{\partial\psi}{\partial p}=,,\frac{H’(p)-H’(q)}{H(q)(p-q)}$’, we have$\chi_{\mathrm{C}}’’(t)=,\frac{H’(\phi’(u_{+(t))H’}-(\psi(\phi/(u_{+(}t))))}{\phi(u_{+}(t))-\psi(\phi(u+(t)))},\phi JJ(u+(t))u’(+t)$.

We also have$\chi’(t)=u_{+}’(t)\{1+tH’’(\phi’(u+(t))\phi’/(u_{+}(t))\}+H’(\phi’(u_{+(}t)))$.

It follows that.

$\chi_{C}’’(t)=-,\frac{(H’(\phi’(u+(t)))-H/(\psi(\phi/(u_{+}(t)))))2}{\phi(u_{+}(t))-\psi(\phi(u+(t)))},\frac{\phi^{\prime/}(u+(t))}{1+tH^{\prime l}(\phi(u_{+(}t))\phi^{\prime/}(u+(t))},$

Since$\frac{\partial x}{\partial u}(t, u_{+}(t))=1+tH’’(\phi/(u+(t)))\phi/’(u_{+}(t))$,

we may assume that $1+tH^{\prime/}(\phi’(u_{+}(t)))\phi’/(u_{+}(t))>0$ . So
is convex if and only if$\phi’’(u_{+}(t))>0$ . We suppose thatand denote $\chi_{c}(t)=u_{+}(t)+tH’(\phi(u_{+(}t))=$$u_{-}(t)+tH/(\phi(u_{-(}t)))$ , where(resp.
) is the point corresponding to the characteristic from the right (resp. left) side of. We distinguish two cases as follows:$\chi_{c}(t)$$\phi^{\prime/}(u_{+(}t))\leq 0$$u_{-}(t)$$u_{+}(t)$$(t, \chi_{c}(t))$

90
is monotone. Since $u_{-}’(t)<0,$ $\phi_{-}’(u(t))$ moves to the leftb-l) If $\phi’’(u_{-(}t))>0$ , thendirection, so that the viscosity criterion is satisfied across .
, then 1+
and the viscosity criterion is violated across for$\chi_{c}(t)=u-(t)+tH’(\phi(u_{-}(t))$$tH”(\phi’(u-(t)))\phi^{\prime J}(u_{-}(t))>0$ near
. Differentiate the equality
we
have
,
with respect to then$\phi’$$\chi$$\mathrm{b}-2)\mathrm{I}\mathrm{f}\phi/’(u-(t))<0$$t>t_{\alpha}$$\chi$$t_{\alpha}$$t$$x’(t)-H’(\phi’(u-(t)))=\mathrm{t}1+tH/f(\phi’(u_{-(t))})\phi^{\prime/}(u-(t))\}u^{J}-(t)$

.

Since
$\chi’(t_{J}\backslash =,\frac{H(\phi’(u_{+(t))})-H(\phi^{;}(u_{-}(t)))}{\phi(u_{+}(t))-\phi’(u-(t))}>H’(\phi’(u-(t)))$,is increase, which is a contradiction.
is violated for $t>t_{\alpha}$ , the contact discontinuuity curve
viscosity
criterion
Hence, if theis convex and the viscosity solution can be constructed. We draw the picture which isillustrating the situations as follows :we have $u_{-}’(t)>0$ , so that$u_{-}(t)$$\chi$FIGURE 5
Then we can draw the picture of the graph of the viscosity solution forshock curve around .
$t_{\alpha}$$’;’/\prime\prime$$\backslash _{\mathrm{s}}\iota \mathrm{c}\backslash \iota$’$.\backslash \prime_{\backslash }$$\cap’//’\backslash$

FIGURE 6$t>\mathrm{t}_{\alpha}$

and the91

5. BIG RAY$\mathrm{T}\mathrm{R}\mathrm{A}\mathrm{C}\mathrm{I}\mathrm{N}\mathrm{G}:\mathrm{T}\mathrm{H}\mathrm{E}\mathrm{B}\mathrm{E}\mathrm{N}\mathrm{A}\mathrm{M}\mathrm{o}\mathrm{U}’ \mathrm{S}$PROJECT

Consider the following Helmholtz equation$\Delta u(x, z)+k22(\eta Z)u(x, Z)=0$ ,whereis a piecewise smooth continuous function. This equation appears in the theoreyof underwateracoustics and seismology. The orresponding eikonal euation is$\eta(z)$$( \frac{\partial u}{\partial x}(_{X,Z}))2+(\frac{\partial u}{\partial z}(x, z))2-\eta 2(z)=0$.

Here, we consider the point source case. The source point is. The classicalray tracing is the integaration of the ray eaquation (i.e., characteristic equation) for the$(z_{0},0)\in \mathbb{R}^{2}$

Hamiltonian function$H(_{X,z,p}, q)= \frac{1}{2}\{p^{2}+q^{2}-\eta^{2}(z)\}$which is an ordinary differential equation:$\frac{dx}{d\tau}=p,$$\frac{dz}{d\tau}=q,$$\frac{dp}{d\tau}=0,$$\frac{dq}{d\tau}=\eta(_{Z})\eta’(Z)$with the initial data$x(0)=0,$ $z(0)=z0,$ $p(0)=\eta(z\mathrm{o})\cos\theta,$ $q(0)=\eta(Z0)\sin\theta$.

Therefore, we have the solution of the ray equation of the form
$x(\tau, \theta)=\eta(z\mathrm{o})\cos\theta \mathcal{T},$$z(_{\mathcal{T}}, \theta)=z(\tau, \theta),$$p(\tau, \theta)=\eta(z_{0)\mathrm{c}}\mathrm{o}\mathrm{s}\theta,$$q(_{\mathcal{T}}, \theta)=q(\mathcal{T}, \theta)$.

By allowing to vary and computing a (necessarily finite) number of corresponding ray,we want to cover the region as besta as possible (in order to compute the taravel time etc.)
In the classical results, an interpolation process has to be used. However, for hetrogeneous
media (i.e.,), this process may be difficult by the following reason:(a) zones where few rays enter appear (low density zone) (cf., Fig??)(b) zones with complex multivalued travel tie fields appear (different rays cross) (cf., Fif??).
An alternative method for the ray tracing propsing by
is to solve the eikonal
equation directly by finite difference or finite element schees (i.e., the eikonal solver). These
scheme, however, only compute a single valued viscosity soloutions.
The algorithm given by Benamou is as follows:(1) Shoot a given number of rays, say $M$ , in regularly spaced directions. We denote thesebyand call this step the ray sicretization.
(2) Define around eah ray
a local domain , also called a big ray.
(3) Compute the viscosity solution of the eikonal equation on each .
The difficulty lies in step (2).
hae to satisfy two conflicting properties:a) They have to be big enough to cover the domain.
b) They have to be small enough so that they do not contains several rays whih intersect.
In [3] Benamou preseted an example as follows: He considered the case when the graph ofteh velocity index
is depicted in Fig. 7. He used a third-order Runge-Kutta algorithm tointegarate the ray equations. We first shoot 200 rays (Fi. 8), and the 100 rays (Fig. 9). Here,
we only put the pictures of Big rays andtimes given byin the remaining
pages.
$\theta$$\eta(z)\neq \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{o}\mathrm{u}[3]$$(R_{i})_{i=1,\ldots,M}$$R_{i}$$\Omega_{i}$$\Omega_{i}$$\Omega_{i}$$\eta(z)$$r_{\mathrm{b}\mathrm{a}\mathrm{v}\mathrm{e}1}$$\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{o}\mathrm{u}[3]$

92

FIG.

$7\mathrm{V}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}$FIG. $g$ Twenry rays shot with regularly spaced initial directions.

profile; the honzontal axis is .
$\sim’$

FIG.

$\mathrm{q}$One hundred ravs shot wirh regularly spaced initial direccions$\cdot$93$\neg.-..--.’:.

1^{\backslash }||!$.

$\mathrm{r}..-.\cdot,./\cdot.\backslash .\underline’$$3\zeta_{1\overline{-}}^{-}\mathfrak{t}\iota 7$.

$-^{1}.:...:’-.\vee\cdot\sim|$$\neg.,;_{\overline{c}_{1}\mathrm{d}}$$\vee\wedge \mathrm{k}_{1\overline{\mathrm{d}}l\dot{\mathrm{b}}}"\gamma\int$$.\cdot|_{-}.’--’\grave{\mathrm{L}}|\cdot \mathrm{Q}$.

.

$\wedge.‘ \mathrm{r}.-\vee\wedge’.\sqrt \mathrm{I}\eta$$:_{\mathrm{I}^{\wedge}/_{\mathrm{t}}}^{\mathrm{i}_{\Gamma^{-}’}.\mathrm{t}}.\cdot.$$j_{\mathrm{I}}.\cdot|^{\prime!^{\cap}}".’$.\cdot\cdot..-\llcorner’/^{\mathrm{t}\{}’$;$:^{1\mathrm{q}\prime}..iarrow,l^{}\hat{0}$.
$.\backslash \sigma^{\sim_{1^{1}}},-l$FIG. [
enve$’)\sim 3\mathrm{i}_{arrow},$$.\wedge\wedge\cdot.J\mathrm{s}:$$1_{\mathrm{o}\mathrm{c}\mathrm{e}}‘ 3-\backslash ’\veearrow\gamma_{\vee^{\wedge}}r\negarrow$

Led$;_{\vee\wedge}^{\prime\cdot-}."’ 3\wedge$

J.

$N\mathrm{h}\mathrm{i}\iota \mathrm{e}\iota \mathrm{h}\mathrm{e}\mathfrak{c}\mathrm{w}\mathrm{o}$$‘\urcorner$

olack).

ravs$\mathrm{a}\mathrm{r}\mathrm{o}\mathrm{u}.\urcorner....\mathrm{v}\dot{\mathrm{n}}$ic.r ‘he$\dot{\mathit{0}}i\mathrm{g}$rav^{l}.d$94. .,$81\lambda 003\{07$elkonal Ol$0.\mathrm{N}00\phi 02$$.....\prime\prime$$\backslash \backslash \backslash \backslash \backslash \backslash \backslash \backslash ^{\backslash }$,$\prime\prime$

elkonaf08$.’,,\backslash ’\backslash J\backslash ’\prime_{\backslash },\backslash ^{\backslash ^{\backslash }}\backslash \backslash \backslash ^{\backslash ^{\backslash ^{\backslash }}}\backslash ^{\backslash ^{\backslash ^{\backslash }}}\backslash$$\backslash ^{\backslash }\backslash$,$’$$\backslash ^{\backslash }\backslash$$\backslash \backslash$$.::\overline{\sim_{-}\sim\overline{\sim}}\sim_{--}$$’,,..\backslash ’\backslash \backslash$’,$\prime\prime\prime\vee$elkona109$\#\mathrm{t}\vee’\phi 0\mathrm{d}0\not\in$$\underline{\beta}\mathrm{I}1\ovalbox{\tt\small REJECT} 03$.:,$\mathfrak{g}\iota\ovalbox{\tt\small REJECT} 1\mathfrak{g}$$....’..’,,,\backslash ^{\backslash }’\backslash ’\backslash ’\backslash ’\backslash J,\backslash l\backslash \text{ノ}\backslash J_{J_{J_{\text{ノ},\backslash }\backslash }\backslash }\backslash \backslash \backslash \backslash \backslash \backslash \backslash ^{\backslash \backslash }\backslash \backslash$$\prime\prime$””$\prime \text{ノ_{ノ}/\text{ノ_{ノノノ_{ノ}}}}/$

6MOd12$\mathrm{c}.|\mathrm{x}_{\mathrm{C}\Uparrow}8\mathrm{I}\uparrow.6$$\hat{\mathrm{c}}\ovalbox{\tt\small REJECT} 0\dot{\mathrm{j}}$$.\mathrm{d}.|\mathrm{k}\mathrm{n}\cap \mathrm{a}\mathrm{I}\mathrm{l}\mathrm{l}$....l,$\backslash \backslash ^{\backslash }\backslash ^{\backslash }$.,$....==$

:.

,

$”$

FIG.
0.025 .

$|\mathit{3}/$Travel times computed in each big.

$\backslash \cdot\backslash \backslash \backslash \backslash \backslash$$\mathrm{r}\mathrm{a}\mathrm{v}$$\prime_{\text{ノ},},\text{ノ_{}\prime \text{ノ},\backslash \backslash ^{\backslash }},’,\backslash \text{ノノ},\backslash \backslash \backslash \text{ノ_{ノ}\backslash \backslash }\text{ノ}/\text{ノノ_{ノノ_{}/}\cdot 1\backslash }\backslash \backslash \backslash \backslash$Contour lines every

FIG.

. 0.025 .

$\mathrm{s}$$/\#$Travel tlmes computed in each$\mathrm{s}$$:_{\vee}$$’\wedge.|\mathrm{Y}|’ 00\backslash d14$$\sim \mathrm{k}\wedge \mathrm{o}\mathrm{M}\vee l3$$\prime\prime\prime\prime_{J_{}},\mathit{1}_{\text{ノ},\text{ノ_{ノ}}ノ_{ノ}ノ_{ノ}ノ}\prime \text{ノ_{ノ}},\backslash \text{ノノ_{ノ}\backslash }\backslash /\text{ノ}\backslash \backslash \backslash$$\cdot.\prime\prime,’,_{j}\prime \text{ノ_{ノ}ノ_{ノ_{ノ_{ノ}}ノノノ}’}\text{ノ}//\text{ノ}/\text{ノ_{}//1_{1}^{1}}/////\cdot,\})))$.

$\frac{\wedge}{\vee}\mathrm{i}_{\dot{\mathrm{K}}\mathrm{t}0\mathrm{d}}’\{|6$$-\vee\cdot \mathrm{i}_{\mathrm{K}\mathfrak{g}\Uparrow}\mathrm{t}’ \mathrm{d}l\mathrm{I}^{\zeta}$$\cdot..’,\prime\prime\prime’\prime_{J\prime_{J\prime_{JJ/\mathit{1}}}}J//////////////////\mathit{1}$$’,,,.,\prime\prime,_{\text{ノ},_{J_{J\text{ノ_{ノ}}}}ノ}’_{\mathit{1}/}\text{ノノ}///\text{ノ}///////////$,$\hat{d\vee}..(\mathrm{c}\mathrm{r}\mathrm{I}\hat{c}||\int$$\frac{\wedge}{\vee}|,\cdot \mathrm{k}\mathfrak{g}\lceil_{1\hat{\mathrm{Q}}}|^{l}|7$$\cdot,,\prime\prime\prime\prime l’;\prime\prime l\prime \mathit{1}’/////////,|//,$,$.,,\prime\prime\prime\prime\prime\prime\prime\prime l\prime\prime\prime//\mathit{1}\prime\prime\prime’///////////$

FIG. [$.\mathrm{J}^{\sim}$

0.025 .
$\mathrm{s}$’Travel (lmes computed in each big ray. Contour lines every$\mathrm{b}\mathrm{i}_{3}\sigma$

ray. Contbur lines everv

95

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$Characte\dot{\mathrm{K}}StiCS$$\mathit{8}pace$