Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “TIME FUNCTION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,711 records · Page 95
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

End-to-End Simulations of a 3.4-Meter Detector Wall for Neutron-Diagnosed Subcritical Experiments

The Nevada National Security Site, together with Los Alamos National Laboratory and Lawrence Livermore National Laboratory, is developing a novel diagnostic to measure the reactivity of subcritical experiments. This capability is known as neutron-diagnosed subcritical experiments. The decay of the fission gamma rays from the neutron-interrogated subcritical experiment is measured as a function of time with a large (~3-meter diameter) detector wall consisting of 151 individual detector pixels. The data from this current mode measurement inform the neutron multiplication factor, keff, and thus the relative reactivity of the subcritical experiment. The Nevada National Security Site developed the Gamma Array Simulation Toolkit initially to help inform the design of the individual detector pixels and the 3.4-meter diameter detector wall to be fielded as part of neutron-diagnosed subcritical experiments. This toolkit is now being used to simulate and predict the performance of the final design of the individual detector pixels and the aggregate detector wall. Additionally, key detector characteristics evaluated from these simulations include impulse response, pulse height spectrum, number of photoelectrons per MeV, detector efficiency, and cross talk between detector pixels. Collectively, the results of these simulations inform how well the fission gamma ray die-off distribution from a neutron-diagnosed subcritical experiment measurement can be resolved. This is critical to determining the relative reactivity of the experiment. Furthermore, the Gamma Array Simulation Toolkit can be used to aid in the analysis of the experimental data from a neutron-diagnosed subcritical experiment measurement once the simulation has been benchmarked.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2015

Unit Commitment With Continuous-Time Generation and Ramping Trajectory Models

There is increasing evidence of shortage of ramping resources in the real-time operation of power systems. To explain and remedy this problem systematically, we take a novel look at the way the day-ahead unit commitment (UC) problem represents the information about load, generation and ramping constraints. We specifically investigate the approximation error made in mapping of the original problem, that would decide the continuous-time generation and ramping trajectories of the committed generating units, onto the discrete-time problem that is solved in practice. We first show that current practice amounts to approximating the trajectories with linear splines. We then offer a different representation through cubic splines that provides physically feasible schedules and increases the accuracy of the continuous-time generation and ramping trajectories by capturing sub-hourly variations and ramping of load in the day-ahead power system operation. The corresponding day-ahead UC model is formulated as an instance of mixed-integer linear programming (MILP), with the same number of binary variables as the traditional UC formulation. Numerical simulation over real load data from the California ISO demonstrate that the proposed UC model reduces the total day-ahead and real-time operation cost, and the number of events of ramping scarcity in the real-time operations.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2021

Modeling and Impact of Hyperloop Technology on the Electricity Grid

This paper provides an overview of hyperloop technologies, including a brief discussion of key components of a hyperloop system that determine the electric power requirements as a function of time. At-scale hyperloop systems do not yet exist, so a model was used to generate a set of load profiles for conceptual hyperloop realizations at four locations in the United States: two systems in California and one each in Colorado and Ohio. In all four cases, the modeled hyperloop load profiles included pulses in both active and reactive power. Grid modeling was performed to estimate the grid impacts and for dis-cussing grid integration challenges. The paper discusses how energy storage systems might be used to eliminate the pulsating load characteristics or significantly reduce it to accommodate current grid planning guidelines

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2017

Guest Editorial High Performance Computing (HPC) Applications for a More Resilient and Efficient Power Grid

The power grid has been evolving over the last 120 years, but it is seeing more changes in this decade and next than it has seen over the past century. In particular, the widespread deployment of intermittent renewable generation, smart loads and devices, hierarchical and distributed control technologies, phasor measurement units, energy storage, and widespread usage of electric vehicles will require fundamental changes in methods and tools for the operation and planning of the power grid. The resulting new dynamic and stochastic behaviors will demand the inclusion of more complexity in modeling the power grid. Solving such complex models in the traditional computing environment will be a major challenge. Along with the increasing complexity of power system models, the increasing complexity of smart grid data further adds to the prevailing challenges. In this environment, the myriad of smart sensors and meters in the power grid increase by multiple orders of magnitude, so do the volume and speed of the data. The information infrastructure will need to drastically change to support the exchange of enormous amounts of data as smart grid applications will need the capability to collect, assimilate, analyze and process the data, to meet real-time grid functions. High performance computing (HPC) holds the promise to enhance these functions, but it is a great resource that has not been fully explored and adopted for the power grid domain.

Huang, Zhenyu Henry↗
DOE OSTI2016

The escape of high explosive products: An exact-solution problem for verification of hydrodynamics codes

This paper documents the escape of high explosive (HE) products problem. The problem, first presented by Fickett & Rivard, tests the implementation and numerical behavior of a high explosive detonation and energy release model and its interaction with an associated compressible hydrodynamics simulation code. The problem simulates the detonation of a finite-length, one-dimensional piece of HE that is driven by a piston from one end and adjacent to a void at the other end. The HE equation of state is modeled as a polytropic ideal gas. The HE detonation is assumed to be instantaneous with an infinitesimal reaction zone. Via judicious selection of the material specific heat ratio, the problem has an exact solution with linear characteristics, enabling a straightforward calculation of the physical variables as a function of time and space. Lastly, implementation of the exact solution in the Python code ExactPack is discussed, as are verification cases for the exact solution code.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Flame Edge Dynamics and Interaction in a Multi-Nozzle Can Combustor with Fuel Staging

The characterization and mitigation of thermoacoustic combustion instabilities in gas turbine engines is necessary to reduce pollutant emissions, premature wear, and component failure associated with unstable flames. Fuel staging, a technique in which the fuel flow to a multi-nozzle combustor is unevenly distributed between the nozzles, has been shown to mitigate the intensity of self-excited combustion instabilities in multiple nozzle combustors. In our previous work, we hypothesized that staging suppresses instability through a phase-cancellation effect in which the heat release rate from the staged nozzle oscillates out of phase with that of the other nozzles, leading to destructive interference that suppresses the instability. This previous theory, however, was based on chemiluminescence imaging, which is a line-of-sight integrated technique. In this work, we use high-speed laser-induced fluorescence to further investigate instability suppression in two staging configurations: center-nozzle and outer-nozzle staging. An edge-tracking algorithm is used to compute local flame edge displacement as a function of time, allowing instability-driven edge oscillation phase coherence and other instantaneous flame dynamics to be spectrally and spatially resolved. Analysis of flame edge oscillations shows the presence of convecting coherent fluctuations of the flame edge caused by periodic vortex shedding. When the system is unstable, these two flame edges oscillate together as a result of high-intensity longitudinal-mode acoustic oscillations in the combustor that drive periodic vortex shedding at each of the nozzle exits. In the stable cases, however, the phase between the oscillations of the center and outer flame edges is greater than 90 degrees (~114 degrees), suggesting that the phase-cancellation hypothesis may be valid. This analysis allows a better understanding of the instantaneous flame dynamics behind flame edge oscillation phase offset and fuel staging-based instability suppression.

Doleiden, Daniel↗
DOE OSTI2018

Study of plasma induced nanostructure formation and surface morphology changes on tungsten and stainless steel at atmospheric pressure

The formation of nanostructures such as “fuzz,” holes, and flakes was observed on tungsten and stainless steel anodes in a 1 atm glow discharge with helium as the feed gas. The structures exhibited morphology similar to the tungsten fuzz growth observed in experiments that simulate wall conditions in fusion reactors, suggesting that similar physics may be present in both scenarios. The anode surface temperature ranged between 1000 and 1300 K with an ion fluence rate of order 10 24 s -1 m -2 . In this work, scanning electron microscope and transmission electron microscope were used to characterize the evolution of nanostructure formation as a function of time. Observed nanostructures were found to vary with the morphology of the discharge plasma attachment. The composition of the structures was found to be predominantly the native metal by using energy dispersive x-ray spectroscopy. This work gives some insight into the formation of nanostructures at atmospheric pressure, which not only has applications for material processing, but also provides potentially another avenue to study fuzz formation processes relevant to fusion plasma-wall interactions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2013

Low dose tomographic fluoroscopy: 4D intervention guidance with running prior

Purpose: Today's standard imaging technique in interventional radiology is the single- or biplane x-ray fluoroscopy which delivers 2D projection images as a function of time (2D+T). This state-of-the-art technology, however, suffers from its projective nature and is limited by the superposition of the patient's anatomy. Temporally resolved tomographic volumes (3D+T) would significantly improve the visualization of complex structures. A continuous tomographic data acquisition, if carried out with today's technology, would yield an excessive patient dose. Recently the authors proposed a method that enables tomographic fluoroscopy at the same dose level as projective fluoroscopy which means that if scanning time of an intervention guided by projective fluoroscopy is the same as that of an intervention guided by tomographic fluoroscopy, almost the same dose is administered to the patient. The purpose of this work is to extend authors' previous work and allow for patient motion during the intervention.Methods: The authors propose the running prior technique for adaptation of a prior image. This adaptation is realized by a combination of registration and projection replacement. In a first step the prior is deformed to the current position via affine and deformable registration. Then the information from outdated projections is replaced by newly acquired projections using forward and backprojection steps. The thus adapted volume is the running prior. The proposed method is validated by simulated as well as measured data. To investigate motion during intervention a moving head phantom was simulated. Real in vivo data of a pig are acquired by a prototype CT system consisting of a flat detector and a continuously rotating clinical gantry.Results: With the running prior technique it is possible to correct for motion without additional dose. For an application in intervention guidance both steps of the running prior technique, registration and replacement, are necessary. Reconstructed volumes based on the running prior show high image quality without introducing new artifacts and the interventional materials are displayed at the correct position.Conclusions: The running prior improves the robustness of low dose 3D+T intervention guidance toward intended or unintended patient motion.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

Poster — Thur Eve — 27: Flattening Filter Free VMAT Quality Assurance: Dose Rate Considerations for Detector Response

Introduction: Radiation detector responses can be affected by dose rate. Due to higher dose per pulse and wider range of mu rates in FFF beams, detector responses should be characterized prior to implementation of QA protocols for FFF beams. During VMAT delivery, the MU rate may also vary dramatically within a treatment fraction. This study looks at the dose per pulse variation throughout a 3D volume for typical VMAT plans and the response characteristics for a variety of detectors, and makes recommendations on the design of QA protocols for FFF VMAT QA. Materials and Methods: Linac log file data and a simplified dose calculation algorithm are used to calculate dose per pulse for a variety of clinical VMAT plans, on a voxel by voxel basis, as a function of time in a cylindrical phantom. Diode and ion chamber array responses are characterized over the relevant range of dose per pulse and dose rate. Results: Dose per pulse ranges from <0.1 mGy/pulse to 1.5 mGy/pulse in a typical VMAT treatment delivery using the 10XFFF beam. Diode detector arrays demonstrate increased sensitivity to dose (+./− 3%) with increasing dose per pulse over this range. Ion chamber arrays demonstrate decreased sensitivity to dose (+/− 1%) with increasing dose rate over this range. Conclusions: QA protocols should be designed taking into consideration inherent changes in detector sensitivity with dose rate. Neglecting to account for changes in detector response with dose per pulse can lead to skewed QA results.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

SU-F-R-12: Prediction of TrueBeam Hardware Issues Using Trajectory Log Analysis

Purpose: To predict potential failures of hardware within the Varian TrueBeam linear accelerator in order to proactively replace parts and decrease machine downtime. Methods: Machine downtime is a problem for all radiation oncology departments and vendors. Most often it is the result of unexpected equipment failure, and increased due to lack of in-house clinical engineering support. Preventative maintenance attempts to assuage downtime, but often is ineffective at preemptively preventing many failure modes such as MLC motor failures, the need to tighten a gantry chain, or the replacement of a jaw motor, among other things. To attempt to alleviate downtime, software was developed in house that determines the maximum value of each axis enumerated in the Truebeam trajectory log files. After patient treatments, this data is stored in a SQL database. Microsoft Power BI is used to plot the average maximum error of each day of each machine as a function of time. The results are then correlated with actual faults that occurred at the machine with the help of Varian service engineers. Results: Over the course of six months, 76,312 trajectory logs have been written into the database and plotted in Power BI. Throughout the course of analysis MLC motors have been replaced on three machines due to the early warning of the trajectory log analysis. The service engineers have also been alerted to possible gantry issues on one occasion due to the aforementioned analysis. Conclusion: Analyzing the trajectory log data is a viable and effective early warning system for potential failures of the TrueBeam linear accelerator. With further analysis and tightening of the tolerance values used to determine a possible imminent failure, it should be possible to pinpoint future issues more thoroughly and for more axes of motion.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2016

SU-F-T-566: Absolute Film Dosimetry for Stereotactic Radiosurgery and Stereotactic Body Radiotherapy Quality Assurance Using Gafchromic EBT3 Films

Purpose: To evaluate the dosimetric uncertainty associated with Gafchromic (EBT3) films and establish an absolute dosimetry protocol for Stereotactic Radiosurgery (SRS) and Stereotactic Body Radiotherapy (SBRT). Methods: EBT3 films were irradiated at each of seven different dose levels between 1 and 15 Gy with open fields, and standard deviations of dose maps were calculated at each color channel for evaluation. A scanner non-uniform response correction map was built by registering and comparing film doses to the reference diode array-based dose map delivered with the same doses. To determine the temporal dependence of EBT3 films, the average correction factors of different dose levels as a function of time were evaluated up to four days after irradiation. An integrated film dosimetry protocol was developed for dose calibration, calibration curve fitting, dose mapping, and profile/gamma analysis. Patient specific quality assurance (PSQA) was performed for 93 SRS/SBRT treatment plans. Results: The scanner response varied within 1% for the field sizes less than 5 × 5 cm{sup 2}, and up to 5% for the field sizes of 10 × 10 cm{sup 2}. The scanner correction method was able to remove visually evident, irregular detector responses found for larger field sizes. The dose response of the film changed rapidly (∼10%) in the first two hours and plateaued afterwards, ∼3% change between 2 and 24 hours. The mean uncertainties (mean of the standard deviations) were <0.5% over the dose range 1∼15Gy for all color channels for the OD response curves. The percentage of points passing the 3%/1mm gamma criteria based on absolute dose analysis, averaged over all tests, was 95.0 ± 4.2. Conclusion: We have developed an absolute film dose dosimetry protocol using EBT3 films. The overall uncertainty has been established to be approximately 1% for SRS and SBRT PSQA. The work was supported by a Research Scholar Grant, RSG-15-137-01-CCE from the American Cancer Society.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

SU-G-JeP4-10: Measurement of Prostate Motion Trajectories During Prostate SBRT Boost Treatments with a Rectafix

Purpose: To determine prostate motion during SBRT boost treatments with a Rectafix rectal sparing device in place using kV imaging during treatment. Methods: Patients each had three gold fiducial markers inserted into the prostate and received two VMAT boost fractions of 9.5–10 Gy under the PROMETHEUS clinical trial protocol with a Rectafix rectal retractor in place. Two-dimensional kilovoltage images of fiducial markers were acquired continuously during delivery. Three patients were treated on a Varian Clinac iX linear accelerator (6X, 600 MU/min), where kV images were acquired at 5 Hz during treatment. Seven patients were treated on a Varian Truebeam linear accelerator (10XFFF, 2400 MU/min) where kV images were acquired every 3 seconds. Images were processed off-line using the Kilovoltage Intrafraction Monitoring (KIM) software after treatment. KIM determines prostate position in three dimensions from 2D kV projections using a probability density model and a pre-treatment kV arc. The 3D displacement of the prostate was quantified as a function of time throughout each fraction. Results: From all fractions analyzed, it was found that the prostate had moved less than 1 mm in any direction from its initial position 84.6% of the time. The prostate was between 1 and 2 mm from its initial position 14.2% of the time, between 2 and 3 mm of its initial position 0.8% of the time and was greater than 3 mm from its initial position only 0.4% of the time. Conclusion: The amount of prostate motion observed during prostate SBRT boost treatments with a Rectafix device in place was minimal and lower than that observed in non-Rectafix studies. The Rectafix device reduces rectal dose as well as immobilizing the prostate. Kimberley Legge is the recipient of an Australian Postgraduate Award.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2014

Rheological signatures of gelation and effect of shear melting on aging colloidal suspension

Colloidal suspensions that are out of thermodynamic equilibrium undergo physical aging wherein their structure evolves to lower the free energy. In aqueous suspension of Laponite, physical aging accompanies increases of elastic and viscous moduli as a function of time. In this work, we study temporal evolution of elastic and viscous moduli at different frequencies and observe that freshly prepared aqueous suspension of Laponite demonstrates identical rheological behavior reported for the crosslinking polymeric materials undergoing chemical gelation. Consequently at a certain time, tan δ is observed to be independent of frequency. However, for samples preserved under rest condition for longer duration before applying the shear melting, the liquid to solid transition subsequent to shear melting shows greater deviation from classical gelation. We also obtain continuous relaxation time spectra from the frequency dependence of viscous modulus. We observe that, with an increase in the rest time, continuous relaxation time spectrum shows gradual variation from negative slope, describing dominance of fast relaxation modes to positive slope representing dominance of slow relaxation modes. We propose that the deviation from gelation behavior for the shear melted suspensions originates from inability of shear melting to completely break the percolated structure thereby creating unbroken aggregates. The volume fraction of such unbroken aggregates increases with the rest time. For small rest times presence of fewer number of unbroken aggregates cause deviation from the classical gelation. On the other hand, at high rest times presence of greater fraction of unbroken aggregates subsequent to shear melting demonstrate dynamic arrest leading to inversion of relaxation time spectra.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Genome-Wide Analysis of RNA Decay in the Cyanobacterium Synechococcus sp. Strain PCC 7002

ABSTRACT RNA degradation is an important process that influences the ultimate concentration of individual proteins inside cells. While the main enzymes that facilitate this process have been identified, global maps of RNA turnover are available for only a few species. Even in these cases, there are few sequence elements that are known to enhance or destabilize a native transcript; even fewer confer the same effect when added to a heterologous transcript. To address this knowledge gap, we assayed genome-wide RNA degradation in the cyanobacterium Synechococcus sp. strain PCC 7002 by collecting total RNA samples after stopping nascent transcription with rifampin. We quantified the abundance of each position in the transcriptome as a function of time using RNA-sequencing data and later analyzed the global mRNA decay map using machine learning principles. Half-lives, calculated on a per-ORF (open reading frame) basis, were extremely short, with a median half-life of only 0.97 min. Despite extremely rapid turnover of most mRNA, transcripts encoding proteins involved in photosynthesis were both highly expressed and highly stable. Upon inspection of these stable transcripts, we identified an enriched motif in the 3′ untranslated region (UTR) that had similarity to Rho-independent terminators. We built statistical models for half-life prediction and used them to systematically identify sequence motifs in both 5′ and 3′ UTRs that correlate with stabilized transcripts. We found that transcripts linked to a terminator containing a poly(U) tract had a longer half-life than both those without a poly(U) tract and those without a terminator. IMPORTANCE RNA degradation is an important process that affects the final concentration of individual mRNAs, affecting protein expression and cellular physiology. Studies of how RNA is degraded increase our knowledge of this fundamental process as well as enable the creation of genetic tools to manipulate RNA stability. By studying global transcript turnover, we searched for sequence elements that correlated with transcript (in)stability and used these sequences to guide tool design. This study probes global RNA turnover in a cyanobacterium, Synechococcus sp. strain PCC 7002, that both has a unique array of RNases that facilitate RNA degradation and is an industrially relevant strain that could be used to convert CO 2 and sunlight into useful products.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2023

Drivers of stability and transience in composition-functioning links during serial propagation of litter-decomposing microbial communities

ABSTRACT Biotic factors that influence the temporal stability of microbial community functioning are an emerging research focus for the control of natural and engineered systems. The discovery of common features within community ensembles that differ in functional stability over time is a starting point to explore biotic factors. We serially propagated a suite of soil microbial communities through five generations of 28-day microcosm incubations to examine microbial community compositional and functional stability during plant litter decomposition. Using dissolved organic carbon (DOC) abundance as a target function, we hypothesized that microbial diversity, compositional stability, and associated changes in interactions would explain the relative stability of the ecosystem function between generations. Communities with initially high DOC abundance tended to converge towards a “low DOC” phenotype within two generations, but across all microcosms, functional stability between generations was highly variable. By splitting communities into two cohorts based on their relative DOC functional stability, we found that compositional shifts, diversity, and interaction network complexity were associated with the stability of DOC abundance between generations. Further, our results showed that legacy effects were important in determining compositional and functional outcomes, and we identified taxa associated with high DOC abundance. In the context of litter decomposition, achieving functionally stable communities is required to utilize soil microbiomes to increase DOC abundance and long-term terrestrial DOC sequestration as one solution to reduce atmospheric carbon dioxide concentrations. Identifying factors that stabilize function for a community of interest may improve the success of microbiome engineering applications. IMPORTANCE Microbial community functioning can be highly dynamic over time. Identifying and understanding biotic factors that control functional stability is of significant interest for natural and engineered communities alike. Using plant litter–decomposing communities as a model system, this study examined the stability of ecosystem function over time following repeated community transfers. By identifying microbial community features that are associated with stable ecosystem functions, microbial communities can be manipulated in ways that promote the consistency and reliability of the desired function, improving outcomes and increasing the utility of microorganisms.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2013

Manifestation of dynamic chaos in spin-spin relaxation processes in a solid

The results of several experiments on the dynamics of paramagnetic spin systems are analyzed on the basis of the theory developed here. It is shown that the asymptotic expressions for time correlation functions of various (high) orders are mutually similar and have a characteristic form [18], which probably indicates the 'forgetting' of the initial conditions due to the development of Lyapunov instability in the dynamic system. The existence of exponential functions with close exponents, which describe the damping of correlations in the experimental observation of signals from free NMR precession and their buildup in experiments on multiquantum NMR, are manifestations of this instability because the Lyapunov exponents exist in pairs. The results suggest the development of 'deterministic chaos' in the systems under investigation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Collapse and revival of photoinduced oscillations in the optical reflectivity of bismuth

The average optical reflectivity of bismuth as a function of time t after irradiation by a short laser pulse has been calculated. The amplitude A of photoinduced oscillations in the average optical reflectivity is shown to have extrema under certain conditions. The time τ{sub j} (j is a natural number) at which the amplitude A reaches the jth extremum has been calculated. The calculated dependences of the times τ{sub 1} and τ{sub 2} at which, respectively, the first and second extrema (the first minimum and the first maximum) of the amplitude A are reached on the maximum laser pulse energy density Q are consistent with the experimental data from.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2018

Self-Similar Character of Generation of Superradiance Pulses in an Electron–Wave Backward Wave Oscillator

The phenomenon of superradiance of electron bunches provides a promising method for generating high-power ultrashort electromagnetic pulses. In the present paper, it is shown that the equations describing the interaction between an electron beam and an electromagnetic wave in a BWO-type oscillator admit a self-similar solution that describes an amplifying and compressing pulse. It is demonstrated by numerical simulation that the generation of short superradiance pulses is of self-similar character at the initial stage of the transient process. The specific features of the main characteristics of such pulses (the amplitude and width of a pulse and the coordinates of its maximum) are investigated as a function of time and control parameters. It is shown that, after a transient process, the solution reaches the self-similar stage, where the pulse amplitude and the inverse of its squared width grow linearly with time.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗