Abstract
In this article we present local well-posedness results in the classical Sobolev space \({H^{s}(\mathbb{R})}\) with s > 1/4 for the Cauchy problem of the Gardner equation, overcoming the problem of the loss of the scaling property of this equation. We also cover the energy space \({H^{1}(\mathbb{R})}\) where global well-posedness follows from the conservation laws of the system. Moreover, we construct solitons of the Gardner equation explicitly and prove that, under certain conditions, this family is orbitally stable in the energy space.
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Alejo, M.A.: Geometric breathers of the mKdV equation (submitted)
Alejo, M.A., Muñoz, C., Vega, L.: The Gardner equation and the L 2-stability of the N-soliton solution of the Korteweg–de Vries equation. Trans. Am. Math. Soc. (to appear)
Angulo J.: Stability of Cnoidal waves. Adv. Differ. Equ. 11(12), 1312–1374 (2006)
Angulo J.: Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg de Vries equations. J. Differ. Equ. 235, 1–30 (2007)
Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geom. Funct. Anal. 3, 107–156, 209–262 (1993)
Christ M., Colliander J., Tao T.: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Am. J. Math. 125, 1235–1293 (2003)
Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on \({\mathbb{R}}\) and \({\mathbb{T}}\) . JAMS 16, 705–749 (2003)
Gradshtein, I.S., Ryzhik, I.M.: Tables of integrals, series and products, Academic Press, New York (1969)
Kenig C., Ponce G., Vega L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46(4), 527–620 (1993)
Kenig C., Ponce G., Vega L.: The Cauchy problem for the KdV equation in Sobolev spaces of negative indices. Duke Math. J. 71(1), 1–20 (1993)
Kenig C., Ponce G., Vega L.: A bilinear estimate with applications to the KdV equation. J. Am. Math. Soc. 8(2), 573–603 (1996)
Linares F., Ponce G.: Introduction to nonlinear dispersive equations. Springer, Berlin (2000)
Tao T.: Multilinear weighted convolution of L 2 functions, and applications to non-linear dispersive equations. Am. J. Math. 123, 839–908 (2001)
Weinstein M.I.: Lyapunov stability of ground states of nonlinear dispersive equations. Commun. Pure Appl. Math. 39, 51–68 (1986)
Zhidkov, P.E.: Korteweg–de Vries and Nonlinear Schrödinger Equations: Qualitative Theory. In: Lecture Notes in Mathematics, vol. 1756. Springer, Berlin (2001)
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Alejo, M.A. Well-posedness and stability results for the Gardner equation. Nonlinear Differ. Equ. Appl. 19, 503–520 (2012). https://doi.org/10.1007/s00030-011-0140-3
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DOI: https://doi.org/10.1007/s00030-011-0140-3