Physica 16D (1985) 339-357 North-Holland, Amsterdam SIMILARITY STRUCTURE OF WAVE-COLLAPSE Kristoffer RYPDAL IMR, University of Tromsa, N-9001 Tromso, Norway Jens Juul RASMUSSEN and Knud THOMSEN Association EURA TOM, Riso National Laboratory, Phys. Dept. Risa, DK-4000 Roskilde, Denmark Received 14 November 1984 Similarity transformations of the cubic SchriSdinger equation (CSE) are investigated. The transformations are used to remove the explicit time variation in the CSE and reduce it to differential equations in the spatial variables only. Two different methods for similarity reduction are employed and the significance of similarity in the evolution of a collapsing wave packet is investigated. Numerical solutions in radial symmetry demonstrate that the similarity behaviour is local in space and time, and that some similarity solutions must be classified as improper solutions. The nature of the collapsing singularity is reexamined. 1. Introduction Certain nonlinear evolution equations of great physical significance exhibit solutions which de- velop a singularity within a finite time. In the mathematics literature this singular behaviour is most commonly called "blow-up", whereas in physics journals the same phenomenon is referred to as "wave-collapse". One analytical approach to these problems is the method of moments [1-2], or the "virial theory" of collapse, as termed by some western authors [3]. This method applies to equa- tions which possess a certain hierarchy of con- servation laws; O,W+ V.~'= O, (1) 0,Sa+ ~- T= 0. (2) By defining an average position (x)- N-lf.~xdx, and a mean square spatiaJ width of the wave packet ((Ax) 2) ------ N-lf..~ (x - (x))2dx, where N-f.#'dx, one easily deduces from (1) and (2) d 2 ((Ax) 2) = 2A, (3) dr2 where A - (2/N)fTr Tdx - (S/N) 2, S = fS+'dx, and Tr T is the trace of T. A particularly simple case occurs if the next conservation law takes the form 0 t Tr r + V" Q = 0, (4) from which it follows that A is a constant of motion, and eq. (3) can be integrated to yield ((Ax) 2) = At 2 + Bt + C. (5) If A<0, ((Ax) 2)~0 in a finite time, and a singularity develops at the average position (x). The virial theory presupposes finite integrals, N, S, A, i.e. for symmetric configurations the den- sities .#',SP, T must decay faster than r - ° as r ~ o0, where D is the number of spatial dimen- sions and r- I xl. Virial theory gives a sufficient criterion for singularity formation for localized (finite integrals) waveforms. It does not, however, provide a necessary criterion because, as will be demonstrated in the following, a singularity may form locally even though ((Ax) 2) remains finite or even increases. Another analytical approach is to search for self-similar substitutions. In most papers in the 0167-2789/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)