| Full text | |
| Author(s): |
Figueira, Renata O.
;
Panthee, Mahendra
Total Authors: 2
|
| Document type: | Journal article |
| Source: | NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS; v. 31, n. 4, p. 23-pg., 2024-07-01. |
| Abstract | |
We consider the initial value problems (IVPs) for the modified Korteweg-de Vries (mKdV) equation partial derivative(t)u+partial derivative(3)(x)u+mu u(2)partial derivative(x)u=0,x is an element of R,t is an element of R, u(x,0) =u0(x) where u is a real valued function and mu=+/- 1, and the cubic nonlinear Schrodinger equation with third order dispersion (tNLS equation in short) partial derivative(t)v+i alpha partial derivative(3)(x)v+beta delta u(2)partial derivative(xv)=0,x is an element of R,t is an element of R, v(x,0) =v0(x) where alpha, beta and gamma are real constants and v is a complex valued function. In both problems, the initial data u0 and v0 are analytic on R and have uniform radius of analyticity sigma 0in the space variable. We prove that the both IVPs are locally well-posed for such data by establishing an analytic version of the trilinear estimates, and showed that the radius of spatial analyticity of the solution remains the same sigma 0till some lifespan0<T0 <= 1. We also consider the evolution of the radius of spatialanalyticity sigma(t) when the local solution extends globally in time and prove that for any timeT >= T0it is bounded from below bycT-43, for them KdV equation in the defocusing case (mu=-1) and by cT-(4+epsilon),epsilon>0,for the tNLS equation. The result for the mKdV equation improves the one obtained in Bona et al. (Ann Inst Henri Poincare 22:783-797, 2005)and, as far as we know, the result for the tNLS equation is the new one. (AU) | |
| FAPESP's process: | 23/06416-6 - Nonlinear phenomena and dispersion |
| Grantee: | Mahendra Prasad Panthee |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 21/04999-9 - Evolution of the radius of analyticity for dispersive equations and systems involving them |
| Grantee: | Renata de Oliveira Figueira |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |