Abstract
In this paper, we mainly study persistence properties for a generalized Camassa-Holm equation with cubic nonlinearity, and we prove the persistence properties in weighted spaces of the solution to the equation, provided that the initial potential satisfies a certain sign condition. Our results extend the work of Brandolese (Int. Math. Res. Not. 22:5161-5181, 2012) on persistence properties to the Fokas-Olver-Rosenau-Qiao equation. In contrast to the Camassa-Holm equation with quadratic nonlinearity, the effect of cubic nonlinearity of the Fokas-Olver-Rosenau-Qiao equation on the persistence properties is rather delicate.
Highlights
1 Introduction The present paper focuses on the Cauchy problem of the integrable modified CamassaHolm equation with cubic nonlinearity mt + (u – u x)mx + uxm + γ ux =, m = u – uxx, t >, x ∈ R, ( . )
It was shown that Equation ( . ) admits the Lax pair and the Cauchy problem ( . ) may be solved by the inverse scattering transform method
Author details 1College of Mathematics Science, Chongqing Normal University, Chongqing, 41331, China. 2Handan College, College North Road, Handan, Hebei 056005, China. 3College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China
Summary
) was independently proposed by Fokas [ ], Fuchssteiner [ ], and Olver and Rosenau [ ] as a new generalization of an integrable system by applying the general method of tri-Hamiltonian duality to the bi-Hamiltonian representation of the modified Korteweg-de Vries equation. Later, it was obtained by Qiao [ , ] from the two-dimensional Euler equations, where the variables u(t, x) and m(t, x) represent, respectively, the velocity of the fluid and its potential density. It has been shown that this problem is locally well posed for initial data u
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