Arthur S Peters. # The solution of a certain nonlinear Riemann-Hilbert problem with an application online

. **(page 1 of 2)**

Online Library → Arthur S Peters → The solution of a certain nonlinear Riemann-Hilbert problem with an application → online text (page 1 of 2)

Font size

The Solution of a Certain

Nonlinear Riemann-Hilbert

Problem with an Application

A. S. Peters

IMM 392

November 1971

Courant Institute of

Mathematical Sciences ^"^^ -^cr^i!'^'

f^r'-' '

Prepared under Contract N00014-67-A-0467-0016

with the Office of Naval Research NR 062-160

Distribution of this document is unlimited.

New York University

' o '

m

z-w

dz

and p(w) is such that

+

via = p"(c) = p(c) .

From the above

(w-w ) e

^ o^

2t(w)

h (w)+p(wj

m o

- 1

G(w) =

\

2t(w)

-h (w)-p{w)

e Â° -1

and we find

w in D

+

w in D'

^ h (w)+p(w)

(w-w j e + 1

s(w)+ r(w)+ t(w) Â»

(w-w ) e

^ o '

h (wj+p(wj

w in D

+

m o

- 1

(3-30) F(w) = J

-h^(w)-p(w)

I s(w) -r(w)+t(w). \^ (wj-plw)""^

L e Â° -1

w in D

as the general solution of (3.25) subject to (3.26). The integer m

is the index of N(z) on C and it is given by

1 ,^N'(z)

m =

WIj^ N(z}

C

dz .

17

4. An Application

Let us apply some of the preceding Ideas to the equation

(4.1) a(0 f^^ ^^Mz]i(_zjdz ^ ^(^) ^

L L

We proceed to demonstrate that If

a(C)b(0 = -1

(^.2)

a(0 7^ 1 , -1 ;

then the solution of (4.1) can be presented In closed form because

the analysis of (4.1) can he reduced to the analysis of a linear

harrier equation. We assume that a(C)j t)(^), f(C) a-nd ({)(C) a-^e inte-

grable along L; and that they are Hblder continuous when C denotes

a point in L' = L - a - p.

In order to solve (4.1) introduce

L L

where )(-, (w) and Xp(^) ^^^ required to be analytic in D except for

possible poles at infinity. The Plemelj formulas, with (4.1), show

that if we require Xi (^) ^^'^ Xp(^) ^Â° ^^ ^^Â° f^^mctions, not both

identically zero, which satisfy the simultaneous linear barrier

equations

(^.^) xiiO + ^iOxliO = -[xI(C)+b(C)x2(C)] ;

(^.5) xt(C) -a(C)x2(0 = Xl(0 -a(Ox2(0 ;

18

then x(^) niust satisfy the equation

We also have

L

+[xlio + x~2iOsna .

It is now evident that if Xn (^) ^^^ X2^^) ^^"^ ^^ found, then (4.6)

can be solved for x(w) and hence ^{z) can be deduced from (4.7) by

introducing

(4.8) 0(w) =/ii^

L

which provides

(4.9) â– 27ri MO =0"'(C) -0-(0

and

(4.10) 2/%l|i =o+(C)+o-(C) .

L

In terms of o(w) the equation (4.7) becomes

(4.11) x'"(C) + x"(0 = [2x^(C)+[b(C)-a(0]X2(^)f""'(0

-[b(0 + a(C)]x2(Oo-(0+)x2(^)+xi(Orf(0

and if we use (4.4) and (4.5) we have

19

[2x^(0 +[b(0-a(C)]x2(Olo"'(U

(^.12) +[2x^(C)+ [b(C)-a(C)]x;(C)jo-(C)

= x^(C)+x"(C)-[x2(0+x2(C)U(C) ,

a linear barrier equation which can be solved for o(w) once Xi (w)

and xp('") are known. The equation (4.12) can be solved if we know

only certain particular solutions of (4.4), (4.5) and (4.6).

The discovery of admissible solutions of the system (4.4),

(4.5) can be made to depend on a solution of the nonlinear barrier

equation for

Xi (^)

(4.13) F(W) = -^ r

which is generated by dividing (4.4) by (4.5). We find

(4.14) F+(C)F~(C)+ '^^^^â€¢' l^^^^^ [F'^(C) + F-(C)] +1 = 0.

The pair (4.l4) and either (4.4) or (4.5), is equivalent to (4.4)

and (4.5).

Since (4.2) is in force, we know from Section 3 and (3.16)

that if

(4.15) A(w) ^yiw-aj(w-pj r ^^ U-a(zjJ

"2?r

"1+ a(z)

^ (z-w)/(z-aj(z-Pj

dz ,

where [1+ a(z )]/[l - a(z)] neither vanishes nor becomes infinite for

z on L, then

20

p2A(w)

(4.16) F-l(w) = \^^^^ = coth A(w)

is a particular solution of (4.l4). From this, we find by

substituting

Xi(^) = Fi(w)x2(w)

in (4.4) and (4.5), that Xo(^) ni^st

[p^(c).MiN^(i)ijtx^(c)]2

(4.17)

Equation (4.l4) shows that

(,.,8) Mc)-a(o_t^l(c)p-,(0.ii

^ F+(0 + F-{C)

and therefore equation (4.17) is the same as

(4.19) [[F^(c)]^- ii[x2(c)]^- li^liof- ihxliof =

The function

(4.20) X2^^) " "^ = sinh A(w)

jpfM -1

satisfies (4.19); it is analytic in D and behaves like a constant at

infinity. Corresponding to this Xo^^^' ^^^ function Xi (^) i^

(4.21) Xi(^) = Fj_(w)x2(w) = cosh A(w)

21

and it also satisfies the original conditions imposed on )(-, and y^

by being analytic in D and behaving like a constant at infinity.

Notice that if we defnne

(4.22)

and

(4.23)

then

and

A(0 =

v(0 = ^ ^n

27ri

1+ a(0

l-a(a

Hn

1 + a(z)

LI- a(z)

(z-C)/(z-aj(z-fij

dz ;

A^O = v(C) + A(0

A"(C) = v(0- A(0 .

Notice also that (4.6) and (4.20) yield

+

X (C)-x"(0

â– v(C) ^ -v(C)

e ^ ^ ^ + e

f(C) sinh A(C)

_ 2f(C) sinh A(C)

Jl-a2(C)

Hence

(4.24)

X(w) =-4- r f(^) 3inh A(z2

L (z-w),Jl - a'^(z)

is the only solution of (4.6) with properties that match those of the

representation (4.3).

We are now in a position to solve (4.12) forO(w). If we use

(4.13), (4.18) and (4.6), then (4.12) becomes

22

([F^(C)]^ -l]X2(0o"'(0 + [[F-(C)]^- l}xi(Oo-(0

,+

+

+

= FlU)ix U)-X2(^)^i^)) +Fl(0[x"(0-x;(Of(0}

This leads to

1

0(w)

sinh a(w j

- x('^) coth a(w) j'

> +

2f(C) cosh a(C)

+ [ si,Â£'"M -y(") =othA(w)f'

Nonlinear Riemann-Hilbert

Problem with an Application

A. S. Peters

IMM 392

November 1971

Courant Institute of

Mathematical Sciences ^"^^ -^cr^i!'^'

f^r'-' '

Prepared under Contract N00014-67-A-0467-0016

with the Office of Naval Research NR 062-160

Distribution of this document is unlimited.

New York University

' o '

m

z-w

dz

and p(w) is such that

+

via = p"(c) = p(c) .

From the above

(w-w ) e

^ o^

2t(w)

h (w)+p(wj

m o

- 1

G(w) =

\

2t(w)

-h (w)-p{w)

e Â° -1

and we find

w in D

+

w in D'

^ h (w)+p(w)

(w-w j e + 1

s(w)+ r(w)+ t(w) Â»

(w-w ) e

^ o '

h (wj+p(wj

w in D

+

m o

- 1

(3-30) F(w) = J

-h^(w)-p(w)

I s(w) -r(w)+t(w). \^ (wj-plw)""^

L e Â° -1

w in D

as the general solution of (3.25) subject to (3.26). The integer m

is the index of N(z) on C and it is given by

1 ,^N'(z)

m =

WIj^ N(z}

C

dz .

17

4. An Application

Let us apply some of the preceding Ideas to the equation

(4.1) a(0 f^^ ^^Mz]i(_zjdz ^ ^(^) ^

L L

We proceed to demonstrate that If

a(C)b(0 = -1

(^.2)

a(0 7^ 1 , -1 ;

then the solution of (4.1) can be presented In closed form because

the analysis of (4.1) can he reduced to the analysis of a linear

harrier equation. We assume that a(C)j t)(^), f(C) a-nd ({)(C) a-^e inte-

grable along L; and that they are Hblder continuous when C denotes

a point in L' = L - a - p.

In order to solve (4.1) introduce

L L

where )(-, (w) and Xp(^) ^^^ required to be analytic in D except for

possible poles at infinity. The Plemelj formulas, with (4.1), show

that if we require Xi (^) ^^'^ Xp(^) ^Â° ^^ ^^Â° f^^mctions, not both

identically zero, which satisfy the simultaneous linear barrier

equations

(^.^) xiiO + ^iOxliO = -[xI(C)+b(C)x2(C)] ;

(^.5) xt(C) -a(C)x2(0 = Xl(0 -a(Ox2(0 ;

18

then x(^) niust satisfy the equation

We also have

L

+[xlio + x~2iOsna .

It is now evident that if Xn (^) ^^^ X2^^) ^^"^ ^^ found, then (4.6)

can be solved for x(w) and hence ^{z) can be deduced from (4.7) by

introducing

(4.8) 0(w) =/ii^

L

which provides

(4.9) â– 27ri MO =0"'(C) -0-(0

and

(4.10) 2/%l|i =o+(C)+o-(C) .

L

In terms of o(w) the equation (4.7) becomes

(4.11) x'"(C) + x"(0 = [2x^(C)+[b(C)-a(0]X2(^)f""'(0

-[b(0 + a(C)]x2(Oo-(0+)x2(^)+xi(Orf(0

and if we use (4.4) and (4.5) we have

19

[2x^(0 +[b(0-a(C)]x2(Olo"'(U

(^.12) +[2x^(C)+ [b(C)-a(C)]x;(C)jo-(C)

= x^(C)+x"(C)-[x2(0+x2(C)U(C) ,

a linear barrier equation which can be solved for o(w) once Xi (w)

and xp('") are known. The equation (4.12) can be solved if we know

only certain particular solutions of (4.4), (4.5) and (4.6).

The discovery of admissible solutions of the system (4.4),

(4.5) can be made to depend on a solution of the nonlinear barrier

equation for

Xi (^)

(4.13) F(W) = -^ r

which is generated by dividing (4.4) by (4.5). We find

(4.14) F+(C)F~(C)+ '^^^^â€¢' l^^^^^ [F'^(C) + F-(C)] +1 = 0.

The pair (4.l4) and either (4.4) or (4.5), is equivalent to (4.4)

and (4.5).

Since (4.2) is in force, we know from Section 3 and (3.16)

that if

(4.15) A(w) ^yiw-aj(w-pj r ^^ U-a(zjJ

"2?r

"1+ a(z)

^ (z-w)/(z-aj(z-Pj

dz ,

where [1+ a(z )]/[l - a(z)] neither vanishes nor becomes infinite for

z on L, then

20

p2A(w)

(4.16) F-l(w) = \^^^^ = coth A(w)

is a particular solution of (4.l4). From this, we find by

substituting

Xi(^) = Fi(w)x2(w)

in (4.4) and (4.5), that Xo(^) ni^st

[p^(c).MiN^(i)ijtx^(c)]2

(4.17)

Equation (4.l4) shows that

(,.,8) Mc)-a(o_t^l(c)p-,(0.ii

^ F+(0 + F-{C)

and therefore equation (4.17) is the same as

(4.19) [[F^(c)]^- ii[x2(c)]^- li^liof- ihxliof =

The function

(4.20) X2^^) " "^ = sinh A(w)

jpfM -1

satisfies (4.19); it is analytic in D and behaves like a constant at

infinity. Corresponding to this Xo^^^' ^^^ function Xi (^) i^

(4.21) Xi(^) = Fj_(w)x2(w) = cosh A(w)

21

and it also satisfies the original conditions imposed on )(-, and y^

by being analytic in D and behaving like a constant at infinity.

Notice that if we defnne

(4.22)

and

(4.23)

then

and

A(0 =

v(0 = ^ ^n

27ri

1+ a(0

l-a(a

Hn

1 + a(z)

LI- a(z)

(z-C)/(z-aj(z-fij

dz ;

A^O = v(C) + A(0

A"(C) = v(0- A(0 .

Notice also that (4.6) and (4.20) yield

+

X (C)-x"(0

â– v(C) ^ -v(C)

e ^ ^ ^ + e

f(C) sinh A(C)

_ 2f(C) sinh A(C)

Jl-a2(C)

Hence

(4.24)

X(w) =-4- r f(^) 3inh A(z2

L (z-w),Jl - a'^(z)

is the only solution of (4.6) with properties that match those of the

representation (4.3).

We are now in a position to solve (4.12) forO(w). If we use

(4.13), (4.18) and (4.6), then (4.12) becomes

22

([F^(C)]^ -l]X2(0o"'(0 + [[F-(C)]^- l}xi(Oo-(0

,+

+

+

= FlU)ix U)-X2(^)^i^)) +Fl(0[x"(0-x;(Of(0}

This leads to

1

0(w)

sinh a(w j

- x('^) coth a(w) j'

> +

2f(C) cosh a(C)

+ [ si,Â£'"M -y(") =othA(w)f'

1 2

Online Library → Arthur S Peters → The solution of a certain nonlinear Riemann-Hilbert problem with an application → online text (page 1 of 2)