Ground-based, equal-arm-length laser interferometers are being built to measure high-frequency astrophysical gravitational waves. Because of the arm-length equality, laser light experiences the same delay in each arm and thus phase or frequency noise from the laser itself precisely cancels at the photodetector. This laser noise cancellation is crucial. Raw laser noise is orders of magnitude larger than other noises and the desired sensitivity to gravitational waves cannot be achieved without very precise cancellation. Laser interferometers in space, e.g., the proposed three-spacecraft LISA detector, will have much longer arm lengths and will be sensitive to much lower frequency gravitational radiation. In contrast with ground-based interferometers, it is impossible to maintain equal distances between spacecraft pairs ; thus laser noise cannot be cancelled by direct di †erencing of the beams. We analyze here an unequal-arm three-spacecraft gravitational wave detector in which each spacecraft has one free-running laser used both as a transmitter (to send to the other two spacecraft) and as a local oscillator (to monitor the frequencies of beams received from the other two spacecraft). This produces six data streams, two received time series generated at each of the three spacecraft. We describe the apparatus in terms of Doppler transfer functions of signals and noises on these one-way transits between pairs of test masses. Accounting for time-delays of the laser light and gravitational waves propagating through the apparatus, we discuss several simple and potentially useful combinations of the six data streams, each of which exactly cancels the noise from all three lasers while retaining the gravitational wave signal. Three of these combinations are equivalent to unequal-arm interferometers, previously analyzed by Tinto & Armstrong. The other combinations are new and may provide design and operational advantages for space-based detectors. Since at most three laser-noiseÈfree data streams can be independent, we provide equations relating the combinations reported here. We give the response functions of these laser-noiseÈcanceling data combinations for both a gravity wave signal and for the remaining noncancelled noise sources. Finally, using spacecraft separations and noise spectra appropriate for the LISA mission, we calculate the expected gravitational wave sensitivities for each laser-noiseÈcanceling data combination.

LISA ͑Laser Interferometer Space Antenna͒ is a proposed mission which will use coherent laser beams exchanged between three remote spacecraft to detect and study low-frequency cosmic gravitational radiation. Modeling each spacecraft as moving almost inertially, rigidly carrying a laser, beam splitters and photodetectors, we previously showed how the measured time series of Doppler shifts of the six one-way laser beams between spacecraft pairs could be combined, with suitable time delays, to cancel exactly the otherwise overwhelming phase noise of the lasers. Three of the combinations synthesized data that could in principle be obtained if the spacecraft separations were very precisely equal, as in Michelson interferometry; seven other combinations offered possible design advantages and useful redundancy. Here we extend those results by presenting time-delay equations for Doppler data from the actual drag-free configuration envisaged for the LISA mission. Each spacecraft will carry two proof-masses, shielded within two non-inertial optical benches carrying lasers and photodetectors. In this full drag-free configuration there are now twelve Doppler data streams, six measured with beams between the three vertex spacecraft and two with beams between each of the optical bench pairs on the three spacecraft. We show that generalizations of our previous linear data combinations, now using these twelve one-way Doppler measurements, can cancel the noises of all six lasers and also remove Doppler shifts due to the non-inertial motions of the six optical benches. It is noteworthy that adjacent optical benches need not be rigidly connected and that no phase locking of their lasers is required. From the latest LISA estimates for power spectra of remaining Doppler noises ͑very-low-level proof-mass ''acceleration'' noise, photodetector shot noise, and beam pointing noise͒ we compute the sensitivities of the generalized data combinations X and P. In the Appendix we give defining equations and sensitivity results for two additional data combinations, denoted E and U. Like X and P, these combinations only require data from four one-way laser links between the LISA spacecraft. LISA can achieve the desired gravitational wave strain performance of ϳ10 Ϫ23 with any of these combinations.

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A technique is developed for systematically deriving a ’’prolongation structure’’—a set of interrelated potentials and pseudopotentials—for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg−de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear ’’inverse scattering’’ equations for auxiliary functions, Bäcklund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.

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