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We study theoretically Bose-Einstein condensates with polarized dipolarinteractions in anisotropic traps. We map the parameter space by varying thetrap frequencies and dipolar interaction strengths and find an irregular-shapedregion of parameter space in which density-oscillating condensate states occur,with maximum density away from the trap center. These density-oscillatingstates may be biconcave (red-blood-cell-shaped), or have two or four peaks. Forall trap frequencies, the condensate becomes unstable to collapse forsufficiently large dipole interaction strength. The collapse coincides with thesoftening of an elementary excitation. When the condensate mode isdensity-oscillating, the character of the softening excitation is related tothe structure of the condensate. We classify these excitations by linear andangular characteristics. We also find excited solutions to the Gross-Pitaevskiiequation, which are always unstable.

Stability and structure of an anisotropically trapped dipolar Bose-Einstein condensate: angular and linear rotons