Advances in Differential Equations

Instability of solitary-wave solutions of the 3-dimensional Kadomtsev-Petviashvili equation

Jerry L. Bona and Yue Liu

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The generalized Kadomtsev-Petviashvili system of equations in three space dimensions, $$ \begin{cases} u_t + u^p u_x + u_{xxx} - v_y - w_z = 0, \\ v_x = u_y, \\ w_x = u_z, \end{cases} \tag*{(*)} $$ has been shown by de Bouard and Saut to possess solitary-wave solutions if and only if $ 1 \le < 4/3. $ It is demonstrated here that these localized traveling-waves, when considered as solutions of the initial-value problem for $ (*) $, are dynamically unstable to perturbations.

Article information

Adv. Differential Equations Volume 7, Number 1 (2002), 1-23.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35R25: Improperly posed problems 37K40: Soliton theory, asymptotic behavior of solutions 37N10: Dynamical systems in fluid mechanics, oceanography and meteorology [See mainly 76-XX, especially 76D05, 76F20, 86A05, 86A10] 76B25: Solitary waves [See also 35C11] 76E30: Nonlinear effects


Bona, Jerry L.; Liu, Yue. Instability of solitary-wave solutions of the 3-dimensional Kadomtsev-Petviashvili equation. Adv. Differential Equations 7 (2002), no. 1, 1--23.

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