955 research outputs found
From the solution of the Tsarev system to the solution of the Whitham equations
We study the Cauchy problem for the Whitham modulation equations for monotone
increasing smooth initial data. The Whitham equations are a collection of
one-dimensional quasi-linear hyperbolic systems. This collection of systems is
enumerated by the genus g=0,1,2,... of the corresponding hyperelliptic Riemann
surface. Each of these systems can be integrated by the so called hodograph
transform introduced by Tsarev. A key step in the integration process is the
solution of the Tsarev linear overdetermined system. For each , we
construct the unique solution of the Tsarev system, which matches the genus
and solutions on the transition boundaries. Next we characterize
initial data such that the solution of the Whitham equations has genus , , for all real and .Comment: Latex2e 41 pages, 5 figure
Critical asymptotic behavior for the Korteweg-de Vries equation and in random matrix theory
We discuss universality in random matrix theory and in the study of
Hamiltonian partial differential equations. We focus on universality of
critical behavior and we compare results in unitary random matrix ensembles
with their counterparts for the Korteweg-de Vries equation, emphasizing the
similarities between both subjects.Comment: review paper, 19 pages, to appear in the proceedings of the MSRI
semester on `Random matrices, interacting particle systems and integrable
systems
The KdV hierarchy: universality and a Painleve transcendent
We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the
small dispersion limit where \e\to 0. For negative analytic initial data with
a single negative hump, we prove that for small times, the solution is
approximated by the solution to the hyperbolic transport equation which
corresponds to \e=0. Near the time of gradient catastrophe for the transport
equation, we show that the solution to the KdV hierarchy is approximated by a
particular Painlev\'e transcendent. This supports Dubrovins universality
conjecture concerning the critical behavior of Hamiltonian perturbations of
hyperbolic equations. We use the Riemann-Hilbert approach to prove our results
Modulation of Camassa--Holm equation and reciprocal transformations
We derive the modulation equations or Whitham equations for the Camassa--Holm
(CH) equation. We show that the modulation equations are hyperbolic and admit
bi-Hamiltonian structure. Furthermore they are connected by a reciprocal
transformation to the modulation equations of the first negative flow of the
Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by
the Casimir of the second Poisson bracket of the KdV averaged flow. We show
that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation
equations is quite different: indeed the KdV averaged bi-Hamiltonian structure
can always be related to a semisimple Frobenius manifold while the CH one
cannot
Numerical study of a multiscale expansion of KdV and Camassa-Holm equation
We study numerically solutions to the Korteweg-de Vries and Camassa-Holm
equation close to the breakup of the corresponding solution to the
dispersionless equation. The solutions are compared with the properly rescaled
numerical solution to a fourth order ordinary differential equation, the second
member of the Painlev\'e I hierarchy. It is shown that this solution gives a
valid asymptotic description of the solutions close to breakup. We present a
detailed analysis of the situation and compare the Korteweg-de Vries solution
quantitatively with asymptotic solutions obtained via the solution of the Hopf
and the Whitham equations. We give a qualitative analysis for the Camassa-Holm
equationComment: 17 pages, 13 figure
Experimental results of a terrain relative navigation algorithm using a simulated lunar scenario
This paper deals with the problem of the navigation of a lunar lander based on the Terrain Relative Navigation approach. An algorithm is developed and tested on a scaled simulated lunar scenario, over which a tri-axial moving frame has been built to reproduce the landing trajectories. At the tip of the tri-axial moving frame, a long-range and a short-range infrared distance sensor are mounted to measure the altitude. The calibration of the distance sensors is of crucial importance to obtain good measurements. For this purpose, the sensors are calibrated by optimizing a nonlinear transfer function and a bias function using a least squares method. As a consequence, the covariance of the sensors is approximated with a second order function of the distance. The two sensors have two different operation ranges that overlap in a small region. A switch strategy is developed in order to obtain the best performances in the overlapping range. As a result, a single error model function of the distance is found after the evaluation of the switch strategy. Because of different environmental factors, such as temperature, a bias drift is evaluated for both the sensors and properly taken into account in the algorithm. In order to reflect information of the surface in the navigation algorithm, a Digital Elevation Model of the simulated lunar surface has been considered. The navigation algorithm is designed as an Extended Kalman Filter which uses the altitude measurements, the Digital Elevation Model and the accelerations measurements coming from the moving frame. The objective of the navigation algorithm is to estimate the position of the simulated space vehicle during the landing from an altitude of 3 km to a landing site in the proximity of a crater rim. Because the algorithm needs to be updated during the landing, a crater peak detector is conceived in order to reset the navigation filter with a new state vector and new state covariance. Experimental results of the navigation algorithm are presented in the paper
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