WKB analysis of the Schrödinger-KdV system

Chi Kun Lin; Jun ichi Segata

doi:10.1016/j.jde.2014.03.001

WKB analysis of the Schrödinger-KdV system

Chi Kun Lin, Jun ichi Segata^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We consider the behavior of solutions to the water wave interaction equations in the limit ε→0⁺. To justify the semiclassical approximation, we reduce the water wave interaction equation into some hyperbolic-dispersive system by using a modified Madelung transform. The reduced system causes loss of derivatives which prevents us to apply the classical energy method to prove the existence of solution. To overcome this difficulty we introduce a modified energy method and construct the solution to the reduced system.

Original language	English
Pages (from-to)	3817-3834
Number of pages	18
Journal	Journal of Differential Equations
Volume	256
Issue number	11
DOIs	https://doi.org/10.1016/j.jde.2014.03.001
Publication status	Published - 1 Jun 2014
Externally published	Yes

Keywords

System of dispersive equations
WKB analysis
Well-posedness
Zero dispersion limit

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10.1016/j.jde.2014.03.001

Cite this

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title = "WKB analysis of the Schr{\"o}dinger-KdV system",

abstract = "We consider the behavior of solutions to the water wave interaction equations in the limit ε→0+. To justify the semiclassical approximation, we reduce the water wave interaction equation into some hyperbolic-dispersive system by using a modified Madelung transform. The reduced system causes loss of derivatives which prevents us to apply the classical energy method to prove the existence of solution. To overcome this difficulty we introduce a modified energy method and construct the solution to the reduced system.",

keywords = "System of dispersive equations, WKB analysis, Well-posedness, Zero dispersion limit",

author = "Lin, {Chi Kun} and Segata, {Jun ichi}",

note = "Funding Information: J.S. is partially supported by MEXT , Grant-in-Aid for Young Scientists (B) 21740122 . Part of this work was done when he visited CMMSC of NCTU in Taiwan, he would like to thank their support and hospitality during his visit. C.K. Lin is partially supported by National Science Council of Taiwan under the grant NSC101-2115-M-009-008-MY2 .",

year = "2014",

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doi = "10.1016/j.jde.2014.03.001",

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T1 - WKB analysis of the Schrödinger-KdV system

AU - Lin, Chi Kun

AU - Segata, Jun ichi

N1 - Funding Information: J.S. is partially supported by MEXT , Grant-in-Aid for Young Scientists (B) 21740122 . Part of this work was done when he visited CMMSC of NCTU in Taiwan, he would like to thank their support and hospitality during his visit. C.K. Lin is partially supported by National Science Council of Taiwan under the grant NSC101-2115-M-009-008-MY2 .

PY - 2014/6/1

Y1 - 2014/6/1

N2 - We consider the behavior of solutions to the water wave interaction equations in the limit ε→0+. To justify the semiclassical approximation, we reduce the water wave interaction equation into some hyperbolic-dispersive system by using a modified Madelung transform. The reduced system causes loss of derivatives which prevents us to apply the classical energy method to prove the existence of solution. To overcome this difficulty we introduce a modified energy method and construct the solution to the reduced system.

AB - We consider the behavior of solutions to the water wave interaction equations in the limit ε→0+. To justify the semiclassical approximation, we reduce the water wave interaction equation into some hyperbolic-dispersive system by using a modified Madelung transform. The reduced system causes loss of derivatives which prevents us to apply the classical energy method to prove the existence of solution. To overcome this difficulty we introduce a modified energy method and construct the solution to the reduced system.

KW - System of dispersive equations

KW - WKB analysis

KW - Well-posedness

KW - Zero dispersion limit

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U2 - 10.1016/j.jde.2014.03.001

DO - 10.1016/j.jde.2014.03.001

M3 - Article

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SN - 0022-0396

VL - 256

SP - 3817

EP - 3834

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 11

ER -

WKB analysis of the Schrödinger-KdV system

Abstract

Keywords

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