KAM TORI FOR DEFOCUSING KDV-MKDV EQUATION
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摘要
In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u~2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.
In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u~2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.
In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u~2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.
引文
[1] Bourgain J. Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. International Mathematics Research Notices, 1994, 1994(11):475–497
[2] Bourgain J. Recent progress on quasi-periodic lattice Schro¨dinger operators and Hamiltonian PDEs. Russian Math Surveys Russian Mathematical Surveys, 2004, 59(2):37–52
[3] Baldi P, Berti M, Montalto R. KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation. Mathematische Annalen, 2014, 359(1/2):471–536
[4] Baldi P, Berti M, Montalto R. KAM for quasi-linear KdV. Comptes Rendus Mathematique, 2014, 352(7/8):603–607
[5] Berti M, Biasco L, Procesi M. KAM theory for the Hamiltonian derivative wave equations. Annales Scientifiques De L′ecole Normale Sup′erieure, 2013, 46(2):301–373
[6] Craig W, Wayne C E. Newton’s method and periodic solutions of nonlinear wave equations. Communications on Pure&Applied Mathematics, 1993, 46(11):1409–1498
[7] Fan E. Uniformly constructing a series of exact solutions to nonlinear equations in mathematical physics.Chaos Solitons&Fractals, 2003, 16(5):819–839
[8] Huang Y, Wu Y, Meng F, Yuan W. All exact traveling wave solutions of the combined KdV-mKdV equation.Advances in Difference Equations, 2014, 2014(1):261
[9] Kuksin S B. Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum.Functional Analysis&Its Applications, 1987, 21(3):192–205
[10] Kuksin S B. On small-denominators equations with large variable coefficients. Zeitschrift Fu¨r Angewandte Mathematik Und Physik Zamp, 1997, 48(2):262–271
[11] Kuksin S B. Analysis of Hamiltonian PDEs. Oxford:Oxford Univ Press, 2000
[12] Kuksin S B. Fifteen years of KAM for PDE. Geometry Topology&Mathematical Physics, 2002:237–258
[13] Kappeler T, P¨oschel J. KdV&KAM. Berlin, Heidelberg:Springer-Verlag, 2003
[14] Lu D, Shi Q. New solitary wave solutions for the combined KdV-MKdv equation. Journal of Information&Computational Science, 2010, 8(8):1733–1737
[15] Liu J, Yuan X. A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations. Communications in Mathematical Physics, 2011, 307(3):629–673
[16] Liu J, Yuan X. Spectrum for quantum duffing oscillator and small-divisor equation with large-variable coefficient. Communications on Pure&Applied Mathematics, 2010, 63(9):1145–1172
[17] Liu J, Yuan X. KAM for the derivative nonlinear Schr¨odinger equation with periodic boundary conditions.Journal of Differential Equations, 2014, 256:1627–1652
[18] Mohamad MNB. Exact Solutions to the combined KdV and MKdV equation. Mathematical Methods in the Applied Sciences, 1992, 15(2):73–78
[19] Yan K K, Axelsen J, Maslov S. Wave propatation in nonlinear lattice. Wave Propagation in nonlinear lattice II. Journal of the Physical Society of Japan, 1975, 38(3):673
[20] Lax P D. Development of singularities of solutions of nonlinear hyperbolic partial differential equations.Journal of Mathematical Physics, 2004, 5(5):611–613
[21] Yang X, Tang J. New travelling wave solutions for combined KdV-mKdV equation and(2+1)-dimensional Broer-Kaup-Kupershmidt systemNew travelling wave solutions for combined KdV-mKdV equation and(2+1)-dimensional Broer-Kaup-Kupershmidt system. Chinese Phy, Series B, 2007, 16(2):310–317
[22] Shi Y, Xu J. KAM tori for defocusing modified KdV equation. Journal of Geometry&Physics, 2015, 90:1–10
[23] Wayne C E. Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Communications in Mathematical Physics, 1990, 127(3):479–528
[24] Xu S, Yan D. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure&Applied Analysis, 2016, 15(5):1857–1869
[25] Zhang J, Gao M, Yuan X. KAM tori of reversible partial differential equation. Nonlinearity, 2011, 24:1189–1228
[26] Zhang J. New solitary wave solution of the combined KdV and mKdV equation. Int J Theor Phys, 1998,37:1541–1546
[2] Bourgain J. Recent progress on quasi-periodic lattice Schro¨dinger operators and Hamiltonian PDEs. Russian Math Surveys Russian Mathematical Surveys, 2004, 59(2):37–52
[3] Baldi P, Berti M, Montalto R. KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation. Mathematische Annalen, 2014, 359(1/2):471–536
[4] Baldi P, Berti M, Montalto R. KAM for quasi-linear KdV. Comptes Rendus Mathematique, 2014, 352(7/8):603–607
[5] Berti M, Biasco L, Procesi M. KAM theory for the Hamiltonian derivative wave equations. Annales Scientifiques De L′ecole Normale Sup′erieure, 2013, 46(2):301–373
[6] Craig W, Wayne C E. Newton’s method and periodic solutions of nonlinear wave equations. Communications on Pure&Applied Mathematics, 1993, 46(11):1409–1498
[7] Fan E. Uniformly constructing a series of exact solutions to nonlinear equations in mathematical physics.Chaos Solitons&Fractals, 2003, 16(5):819–839
[8] Huang Y, Wu Y, Meng F, Yuan W. All exact traveling wave solutions of the combined KdV-mKdV equation.Advances in Difference Equations, 2014, 2014(1):261
[9] Kuksin S B. Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum.Functional Analysis&Its Applications, 1987, 21(3):192–205
[10] Kuksin S B. On small-denominators equations with large variable coefficients. Zeitschrift Fu¨r Angewandte Mathematik Und Physik Zamp, 1997, 48(2):262–271
[11] Kuksin S B. Analysis of Hamiltonian PDEs. Oxford:Oxford Univ Press, 2000
[12] Kuksin S B. Fifteen years of KAM for PDE. Geometry Topology&Mathematical Physics, 2002:237–258
[13] Kappeler T, P¨oschel J. KdV&KAM. Berlin, Heidelberg:Springer-Verlag, 2003
[14] Lu D, Shi Q. New solitary wave solutions for the combined KdV-MKdv equation. Journal of Information&Computational Science, 2010, 8(8):1733–1737
[15] Liu J, Yuan X. A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations. Communications in Mathematical Physics, 2011, 307(3):629–673
[16] Liu J, Yuan X. Spectrum for quantum duffing oscillator and small-divisor equation with large-variable coefficient. Communications on Pure&Applied Mathematics, 2010, 63(9):1145–1172
[17] Liu J, Yuan X. KAM for the derivative nonlinear Schr¨odinger equation with periodic boundary conditions.Journal of Differential Equations, 2014, 256:1627–1652
[18] Mohamad MNB. Exact Solutions to the combined KdV and MKdV equation. Mathematical Methods in the Applied Sciences, 1992, 15(2):73–78
[19] Yan K K, Axelsen J, Maslov S. Wave propatation in nonlinear lattice. Wave Propagation in nonlinear lattice II. Journal of the Physical Society of Japan, 1975, 38(3):673
[20] Lax P D. Development of singularities of solutions of nonlinear hyperbolic partial differential equations.Journal of Mathematical Physics, 2004, 5(5):611–613
[21] Yang X, Tang J. New travelling wave solutions for combined KdV-mKdV equation and(2+1)-dimensional Broer-Kaup-Kupershmidt systemNew travelling wave solutions for combined KdV-mKdV equation and(2+1)-dimensional Broer-Kaup-Kupershmidt system. Chinese Phy, Series B, 2007, 16(2):310–317
[22] Shi Y, Xu J. KAM tori for defocusing modified KdV equation. Journal of Geometry&Physics, 2015, 90:1–10
[23] Wayne C E. Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Communications in Mathematical Physics, 1990, 127(3):479–528
[24] Xu S, Yan D. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure&Applied Analysis, 2016, 15(5):1857–1869
[25] Zhang J, Gao M, Yuan X. KAM tori of reversible partial differential equation. Nonlinearity, 2011, 24:1189–1228
[26] Zhang J. New solitary wave solution of the combined KdV and mKdV equation. Int J Theor Phys, 1998,37:1541–1546