On the Cauchy problem of dispersive Burgers type equations

We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+\partial_x [T_u u]-T_{\frac{\partial_x u}{2}}u+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ which contains the main non linear "worst interaction" terms, i.e low-high interaction terms, of the usual weakly dispersive Burgers type equation: $$\partial_t u+u\partial_x u+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ with $u_0 \in H^s(\mathbb D)$, where $\mathbb D=\mathbb T \text{ or } \mathbb R$. Through a paradifferential complex Cole-Hopf type gauge transform we introduce for the study of the flow map regularity of Gravity-Capillary equation, we prove a new a priori estimate in $H^s(\mathbb D)$ under the control of $\left\Vert(1+\left\Vert u\right\Vert_{L^\infty_x})\left\Vert u \right\Vert_{W^{2-\alpha,\infty}_x}\right\Vert_{L^1_t}$, improving upon the usual hyperbolic control $\left\Vert \partial_x u\right\Vert_{L^1_tL^\infty_x}$. Thus we eliminate the "standard" wave breaking scenario in case of blow up as conjectured by J. C. Saut and C. Klein in their numerical study of the dispersive Burgers equation. For $\alpha\in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $$\partial_tu+ \partial_x |D|^{\alpha-1}u=R_\infty(u),\ \alpha \in ]2,3[,$$ where $R_\infty$ is a regularizing operator under the control of $\left\Vert u\right\Vert_{L^\infty_t C^{2-\alpha}_*}$

Numerical solutions of the MRLW equation by cubic B-spline Galerkin finite element method

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation has been obtained by a numerical technique based on a lumped Galerkin method using cubic B-spline finite elements. Solitary wave motion, interaction of two and three solitary waves have been studied to validate the proposed method. The three invariants ( 1 2 3 I ,I ,I ) of the motion have been calculated to determine the conservation properties of the scheme. Error norms L2 and L∞ have been used to measure the differences between the exact and numerical solutions. Also, a linear stability analysis of the scheme is proposed

The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter \a>0. The high-frequency (or: semi-classical) parameter is \eps>0. We let \eps and \a go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption. Under these assumptions, we prove that the solution u^\eps radiates in the outgoing direction, {\bf uniformly} in \eps. In particular, the function u^\eps, when conveniently rescaled at the scale \eps close to the origin, is shown to converge towards the {\bf outgoing} solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in \eps) of the limiting absorption principle. Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schr\"odinger propagator, our analysis relies on the following tools: (i) For very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schr\"odinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in \eps

Semiclassical and spectral analysis of oceanic waves

In this work we prove that the shallow water flow, subject to strong wind forcing and linearized around an adequate stationary profile, develops for large times closed trajectories due to the propagation of Rossby waves, while Poincar\'e waves are shown to disperse. The methods used in this paper involve semi-classical analysis and dynamical systems for the study of Rossby waves, while some refined spectral analysis is required for the study of Poincar\'e waves, due to the large time scale involved which is of diffractive type

Digital Signal Processing Research Program

Contains table of contents for Section 2, an introduction, reports on twenty research projects and a list of publications.Lockheed Sanders, Inc. Contract BZ4962U.S. Army Research Laboratory Grant QK-8819U.S. Navy - Office of Naval Research Grant N00014-93-1-0686National Science Foundation Grant MIP 95-02885U.S. Navy - Office of Naval Research Grant N00014-95-1-0834U.S. Navy - Office of Naval Research Grant N00014-96-1-0930U.S. Navy - Office of Naval Research Grant N00014-95-1-0362National Defense Science and Engineering FellowshipU.S. Air Force - Office of Scientific Research Grant F49620-96-1-0072National Science Foundation Graduate Research Fellowship Grant MIP 95-02885Lockheed Sanders, Inc. Grant N00014-93-1-0686National Science Foundation Graduate FellowshipU.S. Army Research Laboratory/ARL Advanced Sensors Federated Lab Program Contract DAAL01-96-2-000