The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space

Abstract The Boussinesq a b c d system is a 4-parameter set of equations posed in R t × R x , originally derived by Bona, Chen and Saut [11] , [12] as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation [17] . Among many particular regimes, depending each of them in terms of the value of the parameters ( a , b , c , d ) present in the equations, the generic regime is characterized by the setting b , d > 0 and a , c 0 . If additionally b = d , the a b c d system is Hamiltonian. The equations in this regime are globally well-posed in the energy space H 1 × H 1 , provided one works with small solutions [12] . In this paper, we investigate decay and the scattering problem in this regime, which is characterized as having (quadratic) long-range nonlinearities, very weak linear decay O ( t − 1 / 3 ) because of the one dimensional setting, and existence of non scattering solutions (solitary waves). We prove, among other results, that for a sufficiently dispersive a b c d systems (characterized only in terms of parameters a , b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone | x | ≤ | t | . We prove this result by constructing three suitable virial functionals in the spirit of works [27] , [28] , and more precisely [42] (valid for the simpler scalar “good Boussinesq” model), leading to global in time decay and control of all local H 1 × H 1 terms. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the energy space.

[1]  Jean-Claude Saut,et al.  Well-Posedness of Strongly Dispersive Two-Dimensional Surface Wave Boussinesq Systems , 2012, SIAM J. Math. Anal..

[2]  J. Bona,et al.  Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation , 1988 .

[3]  Y. Martel,et al.  Nonexistence of small, odd breathers for a class of nonlinear wave equations , 2016, 1607.06421.

[4]  Claudio Munoz,et al.  On the nonlinear stability of mKdV breathers , 2012, 1206.3151.

[5]  F. Linares,et al.  Asymptotic behavior of solutions of a generalized Boussinesq type equation , 1995 .

[6]  Miguel A. Alejo,et al.  Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear Electrodynamics , 2017, 1707.02595.

[7]  Claudio Muñoz,et al.  ON THE VARIATIONAL STRUCTURE OF BREATHER SOLUTIONS II: PERIODIC MKDV EQUATION , 2017 .

[8]  Thierry Colin,et al.  Long Wave Approximations for Water Waves , 2005 .

[9]  A note on the existence of traveling-wave solutions to a Boussinesq system , 2014, 1408.0494.

[10]  Frank Merle,et al.  A Liouville theorem for the critical generalized Korteweg–de Vries equation , 2000 .

[11]  Ellen Shiting Bao,et al.  Existence and Symmetry of Ground States to the Boussinesq abcd Systems , 2015 .

[12]  Frank Merle,et al.  Asymptotic stability of solitons of the subcritical gKdV equations revisited , 2005 .

[13]  Gideon Simpson,et al.  Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons , 2014, 1406.3196.

[14]  Cosmin Burtea,et al.  New long time existence results for a class of Boussinesq-type systems , 2015, 1509.07797.

[15]  Y. Martel,et al.  Stability of $N$ solitary waves for the generalized BBM equations , 2004 .

[16]  Chao Wang,et al.  The Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems II , 2015, SIAM J. Math. Anal..

[17]  Jean-Claude Saut,et al.  Asymptotic Models for Internal Waves , 2007, 0712.3920.

[18]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[19]  Claudio Munoz,et al.  Kink dynamics in the $\phi^4$ model: asymptotic stability for odd perturbations in the energy space , 2015, 1506.07420.

[20]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[21]  David Lannes,et al.  Large time existence for 3D water-waves and asymptotics , 2007, math/0702015.

[22]  Solitary-wave solutions to Boussinesq systems with large surface tension , 2009 .

[23]  Yvan Martel,et al.  Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations , 2001 .

[24]  Tohru Ozawa,et al.  On small amplitude solutions to the generalized Boussinesq equations , 2006 .

[25]  Juan C. Pozo,et al.  Scattering in the Energy Space for Boussinesq Equations , 2017, 1707.02616.

[26]  Pierre Raphaël,et al.  The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation , 2005 .

[27]  Jerry L. Bona,et al.  Sharp well-posedness results for the BBM equation , 2008 .

[28]  T. Tao,et al.  Asymptotic decay for a one-dimensional nonlinear wave equation , 2010, 1011.0949.

[29]  Claudio Muñoz Instability in nonlinear Schrödinger breathers. , 2017 .

[30]  C. Amick,et al.  Regularity and uniqueness of solutions to the Boussinesq system of equations , 1984 .

[31]  M. Schonbek,et al.  Existence of solutions for the boussinesq system of equations , 1981 .

[32]  Miguel A. Alejo,et al.  On the variational structure of breather solutions I: Sine-Gordon equation , 2017 .

[33]  Robin Ming Chen,et al.  On the ill-posedness of a weakly dispersive one-dimensional Boussinesq system , 2013 .

[34]  Miguel A. Alejo,et al.  Nonlinear stability of Gardner breathers , 2017, 1705.04206.

[35]  Yue Liu,et al.  Instability of solitary waves for generalized Boussinesq equations , 1993 .

[36]  Min Chen,et al.  Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory , 2002, J. Nonlinear Sci..

[37]  Li Xu,et al.  The Cauchy problem on large time for surface waves Boussinesq systems , 2012 .

[38]  Existence and asymptotic decay for a Boussinesq-type model , 2008 .

[39]  Atanas Stefanov,et al.  Spectral Stability for Subsonic Traveling Pulses of the Boussinesq "abc" System , 2013, SIAM J. Appl. Dyn. Syst..

[40]  J. Bona,et al.  Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory , 2004 .

[41]  Yue Liu,et al.  Decay and Scattering of Small Solutions of a Generalized Boussinesq Equation , 1997 .

[42]  F. Linares Global Existence of Small Solutions for a Generalized Boussinesq Equation , 1993 .

[43]  Stanley Snelson Asymptotic stability for odd perturbations of the stationary kink in the variable-speed 𝜙⁴ model , 2016, Transactions of the American Mathematical Society.

[44]  Nghiem V. Nguyen,et al.  Existence of traveling-wave solutions to Boussinesq systems , 2011, Differential and Integral Equations.

[45]  Miguel A. Alejo,et al.  Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers , 2013, 1308.0998.