Description of the inelastic collision of two solitary waves for the BBM equation

We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves).Comment: submitted for publication. Corrected typo in Theorem 1.

Asymptotic stability of solitary waves

We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, In particular, we study the case where f(u)=u p+1 / (p+1) , p =1, 2, 3 (and 30, with f ∈ C 4 ). The same asymptotic stability result for KdV is also proved for the case p =2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values of p between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameter p increases beyond the critical value p =4.) The solution is decomposed into a modulating solitary wave, with time-varying speed c(t) and phase γ( t ) ( bound state part ), and an infinite dimensional perturbation ( radiating part ). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. As p →4 − , the local decay or radiation rate decreases due to the presence of a resonance pole associated with the linearized evolution equation for solitary wave perturbations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46489/1/220_2005_Article_BF02101705.pd

Retrospective analysis of energy use and conservation trends: 1972-1982. Appendix

This appendix contains the detailed documentation corresponding to the end-use sectoral analyses presented in the main report. The data and methods used to calculate alternative scenarios for estimating energy savings in four economic sectors are provided in this volume. Appendix A contains the detailed documentation for the residential sector analysis. The methodology used to prepare estimates of building energy savings in the commercial sector is provided in Appendix B. Finally, Appendices C and D discuss the data and explain the analytical techniques used to derive estimates of energy savings in the industrial and transportation sectors, respectively. 9 refs., 14 figs., 86 tabs

Retrospective analysis of energy use and conservation trends: 1972-1982

The primary objective of the research reported here is to analyze energy use trends for the entire economy and by end-use sector (residential, commercial, and industrial, and transportation). In particular, an examination of energy trends was undertaken for the period 1972 through 1982 to determine the magnitude of ''energy savings'' attributable to: (1) changes in economic activity; (2) efficiency improvements relative to the 1972 stock (including structures and capital equipment); and (3) efficiency improvements relative to 1960 to 1972 trends. In addition to identifying the measures of energy savings described above, the causes of efficiency improvement relative to 1972 are explored. For example, energy savings due to efficiency improvements in the residential sector are explained by such activities as changes in household size, migration, improved shell and appliance efficiencies, and increased wood use. For this research, Pacific Northwest Laboratory (PNL) developed a consistent methodology for analyzing energy use trends by end-use sector. Alternative measures of energy use trends (i.e., alternative base cases) were developed for the purpose of measuring energy savings. The energy use trends were calculated as the product of economic activity levels (number of households in the residential sector, square feet of floor space in the commercial sector, output in the industrial sector, and person-miles/ton-miles traveled in the transportation sector) and energy use intensities (energy use per household, energy use per square foot of commercial floor space, energy use per unit of industrial output, and energy use per mile traveled). Energy savings were then defined as differences between alternative estimates of energy use. Data and methods used to derive the alternative estimates are contained in a separate volume of this report (PNL-5026-App.)