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For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set of N--soliton states, and their associated multisoliton solutions. We prove that (i) this set is a uniformly smooth manifold, and (ii) the$$\mathbf {M}_{N}$$ states are uniformly stable in H^s for each s>-1/2. One main tool in our analysis is an iterated Bäcklund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove a multisoliton from a multisoliton state (the soliton removal map). The properties and the regularity of these maps are extensively studied.
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Critical points of master functions and mKdV hierarchy of type $A^{(2)}_{2n}$, volume on Representations of Lie algebras, quantum groups and related topics, Contemp. Math., 713, Amer. Math. Soc., Providence, RI, 2018
It is considered the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra A(2)2n. The population is naturally partitioned into an infinite collection of complex cells C^m, where m are some positive integers. For each cell it is defined an injective rational map C^m→M(A^{(2)}_{2n}) of the cell to the space M(A^{(2)}_{2n}) of Miura opers of type A^{(2)}_{2n}. It is shown that the image of the map is invariant with respect to all mKdV flows on M(A^{(2)}_{2n}) and the image is point-wise fixed by all mKdV flows ∂/∂t_r with index r greater than 4m.
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- Award ID(s):
- 1665239
- PAR ID:
- 10087181
- Date Published:
- Journal Name:
- Contemporary mathematics - American Mathematical Society
- Volume:
- 713
- Issue:
- Representations of Lie algebras, quantum groups and related topics
- ISSN:
- 0271-4132
- Page Range / eLocation ID:
- 205 - 233
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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