Orbital stability of periodic traveling-wave solutions for a dispersive equation
Tóm tắt
In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic dnoidal waves associated with the Kawahara equation.
Tài liệu tham khảo
Andrade, T.P., Pastor, A.: Orbital stability of periodic traveling-wave solutions for the BBM equation with fractional nonlinear term. Phys. D 317, 43–58 (2016)
Angulo, J., Natali, F.: Positivity properties of the Fourier transform and the stability of periodic travelling-wave solutions. SIAM J. Math. Anal. 40, 1123–1151 (2008)
Angulo, J., Bona, J.L., Scialom, M.: Stability of cnoidal waves. Adv. Differ. Equ. 11, 1321–1374 (2006)
Benjamin, T.B.: The stability of solitary waves. Proc. R. Soc. (Lond.) Ser. A 328, 153–183 (1972)
Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. Lond. Ser. A 411, 395–412 (1987)
Bona, J.L.: On the stability theory of solitary waves. Proc. R. Soc. Lond. Ser. A 344, 363–374 (1975)
Cristófani, F., Natali, F., Andrade, T.P.: Orbital stability of periodic traveling wave solutions for the Kawahara equation. J. Math. Phys. 58, 051504 (2017)
Frank, R.L., Lenzmann, E.: Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbb{R}\). Acta Math. 210, 261–318 (2013)
Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry I. J. Funct. Anal. 74, 160–197 (1987)
Hǎrǎguş, M., Kapitula, T.: On the spectra of periodic waves for infinite-dimensional Hamiltonian systems. Phys. D 237, 2649–2671 (2008)
Hǎrǎguş, M., Lombardi, E., Scheel, A.: Spectral stability of wave trains in the Kawahara equation. J. Math. Fluid Mech. 8, 482–509 (2006)
Hur, V.M., Johnson, M.: Stability of periodic traveling waves for nonlinear dispersive equations. SIAM J. Math. Anal. 47, 3528–3554 (2015)
Johnson, M.: Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg–de Vries equation. SIAM J. Math. Anal. 41, 1921–1947 (2009)
Kato, T.: Low regularity well-posedness for the periodic Kawahara equation. Differ. Int. Equ. 25, 1011–1036 (2012)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)
Kiper, A.: Fourier series coefficients for powers of the Jacobian elliptic functions. Math. Comput. 43, 247–259 (1984)
Natali, F., Neves, A.: Orbital stability of solitary waves. IMA J. Appl. Math. 79, 1161–1179 (2014)
Parkes, E.J., Duffy, B.R., Abbot, P.C.: The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations. Phys. Lett. A 295, 280–286 (2002)
Weinstein, M.I.: Modulation stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. 16, 472–490 (1985)
Weinstein, M.I.: Liapunov stability of ground states of nonlinear dispersive equations. Commun. Pure Appl. Math. 39, 51–68 (1986)