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 $g>0$, we construct the unique solution of the Tsarev system, which matches the genus $g+1$ and $g-1$ solutions on the transition boundaries. Next we characterize initial data such that the solution of the Whitham equations has genus $g\leq N$, $N>0$, for all real $t\geq 0$ and $x$.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