2,944 research outputs found
Multi-soliton solutions for the supercritical gKdV equations
For the L^2 subcritical and critical (gKdV) equations, Martel proved the
existence and uniqueness of multi-solitons. Recall that for any N given
solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum
of these N solitons asymptotically as time goes to infinity. More recently, for
the L^2 supercritical case, Cote, Martel and Merle proved the existence of at
least one multi-soliton. In the present paper, as suggested by a previous work
concerning the one soliton case, we first construct an N-parameter family of
multi-solitons for the supercritical (gKdV) equation, for N arbitrarily given
solitons, and then prove that any multi-soliton belongs to this family. In
other words, we obtain a complete classification of multi-solitons for (gKdV).Comment: 32 pages, submitted, v2: hyperref links adde
Construction and characterization of solutions converging to solitons for supercritical gKdV equations
We consider the generalized Korteweg-de Vries equation in the supercritical
case, and we are interested in solutions which converge to a soliton in large
time in H^1. In the subcritical case, such solutions are forced to be exactly
solitons by variational characterization, but no such result exists in the
supercritical case. In this paper, we first construct a "special solution" in
this case by a compactness argument, i.e. a solution which converges to a
soliton without being a soliton. Secondly, using a description of the spectrum
of the linearized operator around a soliton due to Pego and Weinstein, we
construct a one parameter family of special solutions which characterizes all
such special solutions.Comment: 38 pages ; submitted ; v2: margins modifie
Calibration and Internal no-Regret with Partial Monitoring
Calibrated strategies can be obtained by performing strategies that have no
internal regret in some auxiliary game. Such strategies can be constructed
explicitly with the use of Blackwell's approachability theorem, in an other
auxiliary game. We establish the converse: a strategy that approaches a convex
-set can be derived from the construction of a calibrated strategy. We
develop these tools in the framework of a game with partial monitoring, where
players do not observe the actions of their opponents but receive random
signals, to define a notion of internal regret and construct strategies that
have no such regret
Approachability of Convex Sets in Games with Partial Monitoring
We provide a necessary and sufficient condition under which a convex set is
approachable in a game with partial monitoring, i.e.\ where players do not
observe their opponents' moves but receive random signals. This condition is an
extension of Blackwell's Criterion in the full monitoring framework, where
players observe at least their payoffs. When our condition is fulfilled, we
construct explicitly an approachability strategy, derived from a strategy
satisfying some internal consistency property in an auxiliary game. We also
provide an example of a convex set, that is neither (weakly)-approachable nor
(weakly)-excludable, a situation that cannot occur in the full monitoring case.
We finally apply our result to describe an -optimal strategy of the
uninformed player in a zero-sum repeated game with incomplete information on
one side
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