Lectures on the Energy Critical Nonlinear Wave Equation

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We numerically solved the NSE Eq. A probe beam is overlapped at the input with a small 0. For the sake of clarity, here we use a narrow probe, so that our the envelope of our initial condition has a width comparable to the wavelength. The initial condition splits in two parts propagating in opposite directions.

As previously discussed, the evolution after separation is governed by Bogoliubov dispersion relation 23 , 27 , 29 which allows us to interpret these waves as Bogoliubov particles. At long-wavelengths, dispersive effects quantum pressure are negligible and density waves move at constant velocity the sound speed with no distorsion of their shape.

An example of these dynamics is shown in Fig. For higher probe powers the density waves enter into the nonlinear regime.


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After the separation, the density profiles steepen on the front edges and then breaks into higher frequency ripples, Fig. This is due to the fact that the propagation velocity is now a function of the density, and therefore is different for different points in the profile. Since points of higher density propagate faster than points at lower densities, the profile increasingly steepens in the course of time. These results well reproduce the typical evolution of a density wave in ideal compressible, non viscous fluid up to the formation of flow discontinuities Figure 1 d shows lineouts of the sound speed c s and background fluid velocity v corresponding to the wave profiles displayed in Fig.

As can be seen, both speeds are significantly modified due to high amplitude of the propagating waves. In particular, the density wave induces locally a nonzero flow velocity. We recall that in linear acoustic analogues, the spacetime curvature is determined by the fluid velocity and the sound speed profiles spatially homogeneous flows provide a flat spacetime.

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Here the background flow is induced by the wave itself. In other words, the initially flat spacetime geometry is curved by the waves, which are then in turn distorted by the spacetime metric. This suggests the possibility to achieve a geometrical description of the dynamics, as a kind of backreaction on the wave by its own effective metric.

The self-steepening of waves can be experimentally observed by using a similar setup to ref. The propagation dynamics occurs along the major axis and our system can be considered as one-dimensional. By controlling the relative angle and power of the two beams we created an interference pattern of the desired wavelength and modulation depth.

Methanol has a negative thermo-optic coefficient but a low absorption in the visible. The absorbed energy is released in the form of heat, which in turn provides the defocusing nonlinearity via thermo-optic effect After the cell, the probe beam profile i. As discussed in previous sections, a strong self-steepening effect can be observed if the quantum pressure and nonlocal terms play a marginal role, at least in the first stages of propagation.

Here, the probe beam is incident at a 0. The beam width has been chosen in order to observe several periods of the density wave. Typical profiles of the photon fluid density waves are shown in Fig. For low amplitude input the wave evolves as a linear perturbation of the background and preserves the initial sinusoidal shape blue curve. Given its long wavelength, the density wave evolves as a sound wave with a constant sound speed determined by the background density.

This regime and the complete dispersion curve have been exhaustively characterized in ref. Let us remark that even such a small-amplitude wave would eventually end up to steepen and produce a shock. This is simply due to the fact that nonlinear effects are cumulative as the wave propagates. However, the smaller is the amplitude of the wave, the larger are the propagation distances required to develop observable changes in the wave profiles. For the propagation distances in our experiments tens of cm and the above amplitudes few percent of the pump , waves display a linear behaviour. As expected, the self-steepening observed in the experiments Fig.

This clearly show how self-steepening is affected by a nonlocal nonlinearity, which has the effect of smoothening out any sharp features in the nonlinear response. Numerical simulations with the initial condition used in Fig. In terms of these variables we obtain two advection equations. As in linear acoustic analogues, the characteristic curves demarcate the region of causally connected events.

The acoustic line element can be written as. However, depending on the initial consitions, they may generate other background structures. This is a difference with respect to GR, where spacetime metrics are the only background structures possessed by the theory and necessary to describe the dynamics. Other trivial solutions of Eq. This would obviously modify the wave dynamics, while remaining globally unaffected waves propagating on it; ii waves propagating in a single direction. These particular solutions fails to provide a metric, since they would define an incomplete sound cone.

We could just define a single set of trajectories, the positive or the negative characteristics, by tracing the path in space-time of a hypothetical observer. In other words, we have not a causal structure that could be associated with a conformal class of Lorentzian metrics. However, for most of the initial conditions, Eq.

Nevertheless, it is possible to reconstruct an approximate solution which well describe the observed self-steepening dynamics up to the shock formation. After a short transient, the initial profile separates into two parts propagating in opposite directions, similarly to what observed in Fig. When such right-going and left-going profiles are sufficiently well separated, they can be considered as nearly independent , in the sense that the right-going wave is not affecting the evolution of the left-going wave and viceversa.

So, the complete solution is well approximated by the superposition of two components propagating in opposite directions. Such components, known as simple waves, can be thus independently calculated. Simple waves are particular solutions of Eq. The complete solution can be reconstructed via linear superposition of these two elementary components.

We remark that such reconstruction cannot provide an exact solution of the nonlinear problem. On the other hand, when the wavepackets are sufficiently well separated their mutual coupling is negligible and then the solution is very well approximated by the superposition of the two independent simple waves calculated above.


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  7. Notice that any point in the wave profile i. Such waves can be conveniently regarded as a superposition of a density fluctuation propagating relative to the fluid, at the speed of sound and the movement of the fluid induced by the wave itself with velocity v. Since for right-going waves and viceversa for left-going waves , the propagation velocity of a given point in the wave profile increases with the density: points of higher density propagate faster than points of lower density leading to self-steepening of the wave front.

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    In a finite time the wave front will become vertical, implying an unphysical multivalued solution. As the gradient of the solution profile becomes increasingly steep, two characteristics become closer and closer. At the point of intersection, the solution is multivalued and the gradients are infinite gradient catastophe The resulting spacetime is not null complete.

    In fact, the family of half null geodesics, i. Consequently, propagating waves reach a singularity in a finite time. The appearence of a singularity can be explicitly shown by direct calculation of the Ricci scalar R , which in two dimensions fully determines the spacetime curvature. Substituting into Eq. In two dimensions the Ricci scalar is twice the Gaussian curvature , which for the above diagonal metric is given by.

    Invariant Measure for a Three Dimensional Nonlinear Wave Equation

    The fact that a multi-evaluated density or flow velocity would imply the divergence of R can be deduced from the definition of the acoustic metric, as already introduced in linear acoustic analogues. However, in linear models a singular spacetime can only be externally imposed, e. Here instead, a curvature singularity emerges spontaneously in a finite time, starting from a constant, homogeneous background.

    This is due to the nonlinear interplay between the propagating wave and the underlying metric, whose curvature is generated by the wave itself. This is an interesting point that warrants further investigation. In linear analogue models, the quantum pressure is usually neglected in the derivation of the acoustic metric and, when is taken into account, is responsible for the high-energy breaking of Lorentz invariance.

    In a similar fashion, our geometrical analoguey has been established in the ideal pure nonlinear system quantum pressure neglected.

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    As a result, the nonlinear interaction self-steepening is the manifestation of the curvature of a dynamic spacetime metric generated by the wave. As time evolves, the wave density profile changes, thus modifying the curvature of the corresponding metric which, in turn, will affect the density profile.

    It is known that in the presence of quantum pressure the singularity will never form: when the curvature becomes sufficiently strong, the quantum pressure comes into play and will eventually counterbalance the curvature. In this context, the fact that R has the same spatial and temporal dependence as the quantum pressure is not at all a coincidence. The perfect spatiotemporal matching between curvature nonlinearity and dispersive term quantum pressure is the unavoidable condition to have the compensation between the two effects, necessary to prevent the formation of the singularity.

    The resulting shock waves can be thus seen as spacetime structures of maximal —though finite— curvature. In Fig. Although the presence of quantum pressure and nonlocal effects prevent the formation of a singularity, the convergence of the characteristics is a clear indication of an increasing spacetime curvature, in agreement with the ideal picture based on the emergent metric. Trajectories of points with the same density generated by the right-moving high amplitude density wave shown in Fig. Quantum fluids such as BECs, polariton fluids and photon fluids have been proposed as platforms to study analogue gravity effects.

    To date these have focused on the propagation of weak amplitude density waves on top of a given background configuration. This has led to a series of important kinematic studies, including e. It is possible to extend these experimental models into the nonlinear regime where the background curved geometry determining the propagation of the waves is generated by the waves themselves. This self-interaction can thus be interpreted as kind of gravitational influence on the wave by its own effective metric. Such analogue nonlinear models are truer in spirit to general relativity, where mass distributions evolve in a spacetime metric that is modified by mass itself.

    In spite of this fact, in the presence of particular symmetries there is a precise correspondence between the gravitational field equations and the fluid dynamics. Therefore, suitably constrained photon-flows could be exploited to mimic the dynamics of gravitation black holes, spacetime singularities included 22 , and cosmological solutions 37 , The characterization measurements are reported in ref. In this limit, using Eq. By using Eq. Therefore for long wavelengths, the shift is observed to saturate to this limit value.

    Example problems: Flanged and unflanged open-ended pipes. Example problem: Standing wave ratio. Example problems: Quality factor and resonance frequency in Helmholtz resonators. Powered by PmWiki. Part I: Oscillations and Wave Equation. Lecture: Ch-1b: Damped Harmonic Oscillator. Lecture: Ch-2b: Properties of the wave equation: speed of sound, wave number, input impedance, etc. Example problems: Standing waves, boundary conditions pptx. Lecture: Ch-2c: General considerations: Various coordinate systems pdf. Lecture: Ch Suggested problems involve important mathematical questions such as existence and regularity of solutions to PDEs that describe various wave phenomena.

    For instance, the NLS and their combinations with the Korteweg-de-Vries and wave equations have been proposed as models for many basic wave phenomena. Due to their physical significance, it is essential to develop tools to understand behavior of solutions to these nonlinear equations and the investigator plans to work in that direction via adapting some tools from her earlier work on equations of fluid motion such as Navier-Stokes equations that describe fundamental properties of viscous fluids to the context of dispersive equations.

    On the other hand, the investigator plans to continue her recent work on physically inspired questions related to derivation of dispersive PDEs from many body Boson systems.

    The proposed activity contains an interdisciplinary approach in the sense that it has potential to bring dispersive PDE methods to the level of many body quantum dynamics and vise versa. In particular, the long term goal is to try to adapt some of the recent advances from dispersive PDEs to the many body systems, where one has physically relevant questions that are beyond the reach of known mathematical methods.

    Some full text articles may not yet be available without a charge during the embargo administrative interval. Some links on this page may take you to non-federal websites. Their policies may differ from this site. Andrea R. Nahmod, Natasa Pavlovic and Gigliola Staffilani. Thomas Chen and Natasa Pavlovic. Pure Appl.



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