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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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Cited by 9 (5 self)
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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.
Global rough solutions to the critical generalized KdV
, 908
"... We prove that the initial value problem (IVP) for the critical generalized KdV equation ut +uxxx+(u 5)x = 0 on the real line is globally wellposed in H s (R) in s> 3/5 with the appropriate smallness assumption on the initial data. 1 ..."
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We prove that the initial value problem (IVP) for the critical generalized KdV equation ut +uxxx+(u 5)x = 0 on the real line is globally wellposed in H s (R) in s> 3/5 with the appropriate smallness assumption on the initial data. 1
Panthee; Sharp local wellposedness of KdV type equations with dissipative perturbations
, 2013
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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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