Differential and Integral Equations

The Cauchy problem and stability of solitary waves for a 2D Boussinesq-KdV type system

José Raúl Quintero

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Abstract

We address the well posedness of the Cauchy problem and the stability of solitary waves for a Boussinesq system in $\mathbb{R}^{1+2}$. We exploit the fact that this 2D system has a ``KdV'' structure in the sense that it takes the form $U_t =\mathcal A_0U+ \mathcal{\mathcal A}(U) U$, where $\mathcal A_0$ is a third-order linear operator and the entries of the operator $\mathcal{A}(U)(U)$ are linear combinations of products of powers of components of $U$ with its order one spatial derivatives, as in the well-known 1D-KdV model. Using this ``2D-KdV'' structure, we establish existence and uniqueness for the Cauchy problem associated with the Boussinesq type system by following Kato's approach for the generalized KdV equation. By a variational argument, we obtain global well posedness in time for small initial data. We prove orbital stability of solitary waves directly, by using a variational approach involving the characterization of the ground state solutions, as is done for some 2-D models.

Article information

Source
Differential Integral Equations, Volume 24, Number 3/4 (2011), 325-360.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019035

Mathematical Reviews number (MathSciNet)
MR2757463

Zentralblatt MATH identifier
1240.35415

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics 35Q51: Soliton-like equations [See also 37K40] 76B03: Existence, uniqueness, and regularity theory [See also 35Q35]

Citation

Quintero, José Raúl. The Cauchy problem and stability of solitary waves for a 2D Boussinesq-KdV type system. Differential Integral Equations 24 (2011), no. 3/4, 325--360. https://projecteuclid.org/euclid.die/1356019035


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