NONLINEAR STABILITY OF A ONE-DIMENSIONAL BOUSSINESQ EQUATION ´ RAUL ´ QUINTERO1 JOSE
1. Introduction In this paper we shall study nonlinear stability and instability of solitary waves of the nonlinear one-dimensional Boussinesq type equation or Benney-Luke equation Φtt − Φxx + aΦxxxx − bΦxxtt + Φt Φxx + 2Φx Φxt = 0,
(1.1)
where a and b are positive numbers such that a − b = σ − 1/3 (σ is named the Bond number). We will consider the study of traveling waves of lowest energy in the energy norm or simply solitary waves for the equation (1.1) of the form Φ(x, t) = u(x − ct), where c > 0 satisfies c2 < min(1, a/b). In this case, the traveling wave profile u should satisfy the equation (c2 − 1)uxx + (a − bc2 )uxxxx − 3cuxx ux = 0.
(1.2) p We will show that traveling wave solutions are orbitally stable when 0 < c < 1 < a/b, pwhich corresponds to the case Bond number σ > 1/3 and are orbitally unstable when 0 < c < a/b < 1, which corresponds to the case Bond number σ < 1/3. The Boussinesq type equation (1.1) fits into the class of abstract Hamiltonian system studied by Grillakis, Shatah and Strauss in Grillakis et al., (1987). The condition of orbital stability and instability is characterized by the convexity and the concavity respectively of the function defined by Z 1 c 2 d(c) = (1 − c2 )(vxc )2 + (a − bc2 )(vxx ) + c(vxc )3 dx, 2 R where v c is a least energy solitary wave solution of the equation (1.2). Under the assumption of decay of the solutions of (1.2) at ±∞, we find all the ground state solutions, up to translations. The advantage of (1.1) is that it is a formally valid approximation for describing two-way water wave propagation, whereas the KdV and other long-wave approximations such as the regularized long wave equation and regularized Boussinesq equation (see for example equation (1.1.B1) in Pego and Weinstein, (1997)) are formally valid only for one-way propagating waves. In fact, note that quation (1.1) is the one-dimensional version (after rescaling) of the Benney-Luke equation ¡ ¢ Φtt − ∆Φ + µ a∆2 Φ − b∆Φtt + ² (Φt ∆Φ + 2∇Φ · ∇Φt ) = 0. (1.3) It was shown in Pego and Quintero, (1999) (see also Benney and Luke , (1964); Mileswki and Keller, (1996)) that the evolution of three-dimensional water waves with surface tension can be 1 Departamento Matem´ aticas, Universidad del Valle, A.A. 25360. Cali-Colombia. South America. Phone: 572–3393227
[email protected]
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´ RAUL ´ QUINTERO1 JOSE
reduced to studying the solution Φ(x, y, t) of the isotropic equation (1.3), where ² represents the amplitude parameter or nonlinear coefficient, µ represents the long-wave parameter or dispersion coefficient and a − b = σ − 1/3 with a, b > 0. We note that the unscaled equation (1.3) reduces in a suitable limit to the (KdV) equation and the (KP) equation (see Pego and Quintero, (1999)). In the one–dimensional case, the unscaled equation (1.3) reduces formally in a suitable limit to the (KdV) equation. ¶ µ 1 ητ − σ − ηXXX + 3ηηX = 0, (1.4) 3 where η represents the elevation. On the other hand, it is possible to establish that the one– dimensional version of equation (1.3) has physically meaningful solitary wave solutions by showing that in an appropriate scaling as ² → 0 there are solutions that converge to a solitary wave solution of the KdV equation. Acknowledgments: This work is supported by the Universidad del Valle, Cali-Colombia and partly by Colciencias under the project No 1106-05-10097. REFERENCES Alexander, J., and Sachs, R. Linear stability of solitary waves for Boussinesq–type equation. A computer assisted computation, Preprint. Benney, D. J., and Luke, J. C. (1964) Interactions of permanent waves of finite amplitude. J. Math. Phys. 43, 309–313. Bona, J., and Sachs, R. (1988) Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm. Math, Phys. 188, 15–29. Liu, Y. (1993) Instability of solitary waves for generalized Boussinesq equations. J. Dynamics Diff. Eq. 5, 537–558. Liu, Y. (1995) Instability and blow–up of solutions to a generalized Boussinesq equation. SIAM J. Math. Anal. 26, 1527–1546. Grillakis, M., Shatah, J., and Strauss, W. (1987) Stability Theory of Solitary Waves in Presence of Symmetry, I. Functional Analysis. 74, 160–197. Milewski, P.A., and Keller, J. B. (1996) Three dimensional water waves. Studies Appl. Math. 37, 149–166. Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York. Pego, R. L., and Quintero, J.L. (1999) Two–dimensional solitary Waves for a Benney-Luke equation. Physica D. 132, 476–496. Pego, R. L., and Weinstein, M. I. (1997) Convective linear stability of solitary waves for Boussinesq equations. Studies Appl. Math. 99, 311–375. Pego, R. L., Smereka, P., and Weinstein, M. (1995) Oscillatory instability of solitary waves in a continuum model of lattice vibrations Nonlinearity 8, 921–941. Smereka, P. (1992) A remark on the solitary waves stability for a Boussinesq equation. In L. Debnath.(ed). In nonlinear dispersive wave systems,Word scientific, Singapure. pp 255–263. Struwe, W. (1990) Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer-Verlag, New York.
