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A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line
In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation i ∂ t u + ∂ | u | 2 ∂ x u = 0 on the half line (−∞, 0] with decaying initial value. Assuming that the solution u ( x , t ) exists, we show that it can be represented in terms of the s...
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| Published in: | Acta Mathematicae Applicatae Sinica 2018-07, Vol.34 (3), p.493-515 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation
i
∂
t
u
+
∂
|
u
|
2
∂
x
u
=
0
on the half line (−∞, 0] with decaying initial value. Assuming that the solution
u
(
x
,
t
) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter
λ
. The jump matrix has explicit (
x
,
t
) dependence and is given in terms of the spectral functions {
a
(
λ
),
b
(
λ
)} and {
A
(
λ
),
B
(
λ
)}, which are obtained from the initial data
u
0
(
x
) =
u
(
x
, 0) and the boundary data
g
0
(
t
) =
u
(0,
t
),
g
1
(
t
) =
u
x
(0,
t
), respectively. The spectral functions are not independent, but satisfy a so-called global relation. |
|---|---|
| ISSN: | 0168-9673 1618-3932 |
| DOI: | 10.1007/s10255-018-0765-7 |