Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Natalia Kolkovska, Milena Dimova and Nikolai Kutev
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Natalia Kolkovska: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl.8, 1113 Sofia, Bulgaria
Milena Dimova: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl.8, 1113 Sofia, Bulgaria
Nikolai Kutev: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl.8, 1113 Sofia, Bulgaria

Mathematics, 2021, vol. 9, issue 12, 1-19

Abstract: We consider the orbital stability of solitary waves to the double dispersion equation u t t ? u x x + h 1 u x x x x ? h 2 u t t x x + f ( u ) x x = 0 , h 1 > 0 , h 2 > 0 with combined power-type nonlinearity f ( u ) = a | u | p u + b | u | 2 p u , p > 0 , a ? R , b ? R , b ? 0 . The stability of solitary waves with velocity c , c 2 < 1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d ( c ) , related to some conservation laws. We derive explicit analytical formulas for the function d ( c ) and its second derivative for quadratic-cubic nonlinearity f ( u ) = a u 2 + b u 3 and parameters b > 0 , c 2 ? 0 , min 1 , h 1 h 2 . As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f ( u ) = b u 3 .

Keywords: double dispersion equation; combined power-type nonlinearity; solitary waves; orbital stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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