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Instability of a class of dispersive solitary waves. (English) Zbl 0713.35108

The authors consider the equation in \({\mathbb{R}}^ 2\), \(Mu_ t+f(u)_ x=0\), where M is a pseudo-differential operator with elliptic symbol m(\(\xi\)) of order \(\geq 1\). The stability and instability properties of solitary wave solutions g(x-ct) are studied in detail, and the obtained results are tested on some interesting examples. The equation \(u_ t- Mu_ x+f(u)_ x=0\) is then also considered.
Reviewer: L.Rodino

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35Q51 Soliton equations
35B35 Stability in context of PDEs
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References:

[1] DOI: 10.1098/rspa.1987.0073 · Zbl 0648.76005 · doi:10.1098/rspa.1987.0073
[2] DOI: 10.1098/rsta.1972.0032 · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032
[3] DOI: 10.1016/0022-0396(86)90057-4 · Zbl 0596.35109 · doi:10.1016/0022-0396(86)90057-4
[4] DOI: 10.1016/0022-1236(87)90044-9 · Zbl 0656.35122 · doi:10.1016/0022-1236(87)90044-9
[5] DOI: 10.1080/03605308708820522 · Zbl 0657.73040 · doi:10.1080/03605308708820522
[6] DOI: 10.1007/BF01212446 · Zbl 0603.35007 · doi:10.1007/BF01212446
[7] Reed, Methods of Modern Mathematical Physics IV (1978)
[8] Yosida, Functional Analysis (1965)
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