Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPerturbation of Riemann-Hilbert jump contours: smooth parametric dependence with application to semiclassical focusing NLS
309
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A perturbation of a class of scalar Riemann-Hilbert problems (RHPs) with the jump contour as a finite union of oriented simple arcs in the complex plane and the jump function with a $zlog z$ type singularity on the jump contour is considered. The jump function and the jump contour are assumed to depend on a vector of external parameters $vecbeta$. We prove that if the RHP has a solution at some value $vecbeta_0$ then the solution of the RHP is uniquely defined in a some neighborhood of $vecbeta_0$ and is smooth in $vecbeta$. This result is applied to the case of semiclassical focusing NLS.
rate research
Read More
We consider the one dimensional focusing (cubic) Nonlinear Schrodinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth dependence of the asymptotic solution on the parameter. Numerical results supporting our estimates of important quantities are presented.
The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated $N$-phase nonlinear wave solutions to the focusing nonlinear Schrodinger (fNLS) equation, and b) the Riemann-Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated $N$-phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fNLS.
We develop the Riemann-Hilbert problem approach to inverse scattering for the two-dimensional Schrodinger equation at fixed energy. We obtain global or gener
We consider the mass-critical focusing nonlinear Schrodinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrodinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations.
Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution includes the complete preliminary group classification of the class and the construction of singular Lie-symmetry extensions, which are not related to subalgebras of the equivalence algebra. The complete preliminary group classification is based on classifying appropriate subalgebras of the entire infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel invariance algebra. The results obtained can be used to construct exact solutions of nonlinear wave and elliptic equations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
