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ON THE MASSCRITICAL GENERALIZED KDV EQUATION
, 907
"... Abstract. We consider the masscritical generalized Korteweg–de Vries equation (∂t + ∂xxx)u = ±∂x(u 5) for realvalued functions u(t, x). We prove that if the global wellposedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global wellposedness ..."
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Abstract. We consider the masscritical generalized Korteweg–de Vries equation (∂t + ∂xxx)u = ±∂x(u 5) for realvalued functions u(t, x). We prove that if the global wellposedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global wellposedness and scattering conjecture for the masscritical nonlinear Schrödinger equation (−i∂t +∂xx)u = ±(u  4 u), there exists a minimalmass blowup solution to the masscritical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimalmass blowup solution is either a selfsimilar solution, a solitonlike solution, or a double hightolow frequency cascade solution. 1.
THE LOW REGULARITY GLOBAL SOLUTIONS FOR THE CRITICAL GENERALIZED KDV EQUATION
, 908
"... Abstract. In this paper, we establish the global wellposedness for the critical generalized KdV equation with the low regularity data. To be precise, we show that a unique and global solution exists for initial data in the Sobolev space Hs`R ´ with s> 1. Of course, we require that the mass is st ..."
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Abstract. In this paper, we establish the global wellposedness for the critical generalized KdV equation with the low regularity data. To be precise, we show that a unique and global solution exists for initial data in the Sobolev space Hs`R ´ with s> 1. Of course, we require that the mass is strictly less than 2 that of the ground state in the focusing case. This follows from “Imethod”, which was introduced by Colliander, Keel, Staffilani, Takaoka and Tao, and improves the result in [20]. 1.
POLYNOMIALINTIME UPPER BOUNDS FOR THE ORBITAL INSTABILITY OF SUBCRITICAL GENERALIZED KORTEWEGDE VRIES EQUATIONS
"... (Communicated by Gigliola Staffilani) Abstract. We prove polynomialintime upper bounds for the orbital instability of solitons for subcritical generalized Kortewegde Vries equations in H s x (R) with s < 1. By combining coercivity estimates of Weinstein with the Imethod as developed by Collia ..."
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(Communicated by Gigliola Staffilani) Abstract. We prove polynomialintime upper bounds for the orbital instability of solitons for subcritical generalized Kortewegde Vries equations in H s x (R) with s < 1. By combining coercivity estimates of Weinstein with the Imethod as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.
0 THE SUPERCRITICAL GENERALIZED KDV EQUATION: GLOBAL WELLPOSEDNESS IN THE ENERGY SPACE AND BELOW
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GLOBAL WELLPOSEDNESS FOR PERIODIC GENERALIZED KORTEWEGDE VRIES EQUATION
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