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# 1. Introduction. In this paper we consider the nonlinear partial differ- ential equations of the form

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(1)

m=1

m

m

1

1

n

1

[99]

(2)

1

∞ m=2

m

m

1

n

n

1

1

n

1

n

1

n

1

n

\

−∞

## e

−z1(k)x1−...−zn(k)xn

1

n

c1

1

m

m

1

m

m

1

1

m

x

n

1

m

x1

xm

∞ m=2

m

∞ j=2

j

j

∞ j=2

j

2

∞ m=3

m j=2

|p|=m, pk≥1

sj+1

j

p1

pj

1

1

j

j

k

(3)

2

∞ m=3

m j=2

|p|=m, pk≥1

sj+1

j

p1

pj

∞ j=2

j

j

j

2

∞ m=3

m j=2

|p|=m

sj+1

j

p1

pj

∞ j=2

j

2

2

j

2

2

∞ m=2

m j=2

|p|=m

sj+1

j

p1

pj

j

p1

pj

j

p1

pj

m

m

m j=2

|p|=m

sj+1

j

p1

pj

j

p1

pj

j

p1

pj

m

q

−qx

0q

−qτ

0q

q

m

1

m

m

0q1

0qm

m

1

m

1

m

m

0q1

0qm

j

n

(4)

0q1

0qm

2

1

2

2

1

2

1

2

12

1

2

l

1

l

m

1

m

m j=2

|p|=m

sj+1

j

1

p1

p

j−1+1

m

p1

1

p1

pj

pj−1+1

m

1

m

1

p1

p

j−1+1

m

j

1

j

1

j

1j

1

j

m

1

m

2

2

2

1

m

m

m

1

m−1

j=2

|p|=m

sj+1

j

1

p1

p

j−1+1

m

p1

1

p1

pj

p

j−1+1

m

j

1

p1

p

j−1+1

m

m

m

\

−∞

−z(k1)(x+τ1)

1

1

\

−∞

−z(km)(x+τm)

m

m

\

Rm

## e

−[z(k1)+...+z(km)]x

1

m

m

z(k)

1

m

\

Rm

m

## (z(k))e

−[z(k1)+...+z(km)]x

1

m

j

1

j

j

1

j

1

j

k

k

(5)

1

j

j

∞ m=1

\

Rm

m

1

m

## )e

−[z(k1)+...+z(km)]x

m

2

21

2

22

∞ j=1

j

j

∞ j=2

j

j

1

12

22

2

2

2

2

n=2

2

2

1

l

1

l

2

21

l2

2

1≤m<j≤l

2m

2j

m

j

2

l

l

1

r

r

1

r

r

1

p1

p

r−1+1

m

1

m

1

p1

p

r−1+1

m

2

2

n

n

n+

n+

−1

n+

(ω)

n+

(6)

n

n+

(ω)

n+

(ω)

n+

1

(ω)

n+

1

\

−∞

1

−zx

1

1

1

n

1

−(z−w)2

1

n

n+

−1

n

n+

n

n+

+

n

1

n

\

−∞

−(z−z(k))2

1

(ω)

n+

1

σ∈{−1,1}n

ε→0+

\

Rn+

1

1

−zx

σ∈{−1,1}n

ε→0+

n

\

Rn+

\

−∞

## e

−zx−(z+iσε−z(k))2

1

−zx

m

m

\

Rm

1

m

1

m

m

m

m

m

n

m

−(z−w)2

1

n

\

Rm

## e

−(z−(z(k1)+...+z(km)))2

1

m

1

1

m

m

(7)

m

−zx

σ∈{−1,1}n

ε→0+

\

Rn+

−zx

m

m

m

m

−zx

∞ m=1

m

−zx

−zx

(ω)

n+

n+

m

N−1

m=1

m

## [φ].

n+

### R´evis´e le 5.11.1997

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