On the Integrability of the Poisson Driven Stochastic Nonlinear Schr ¨ odinger Equations By Javier Villarroel and Miquel Montero We consider the Cauchy problem for the dissipative nonlinear Schr¨ odinger equations driven by a Poisson noise, namely iu t + u xx + 2|u | 2 u = i  −Ŵu +  n (e −γ n − 1)δ (t − t n )u (t − n , x )  (1) where γ n > 0 and 0 < t 1 < ··· < t n < ··· are certain sequences of random numbers and Ŵ ∈ R + is the deterministic loss coefficient. This perturbation incorporates the possibility of sudden changes in the field that occur randomly. If Ŵ = 0, we prove that the resulting equation can be piece-wise related to the unperturbed NLS equation and show how to solve the initial value problem. We also determine a complete set of conserved quantities. When Ŵ = 0 the equation is nonintegrable. Nevertheless, we determine the random evolution of physically relevant quantities like the field’s Energy E(t) ≡  dx|u| 2 (t, x) and momentum. By considering a joint z-Laplace transform we obtain the mean Energy decay. A naturally related quantity is the “half-life”, or the time before the Energy degrades below a given value E 1 . We show that the mean of this random quantity satisfies an integral equation and solve it by Laplace transformation. In particular cases we also determine the complete probability distribution of Energy and half life. Address for correspondence: Javier Villarroel, Universidad de Salamanca, Facultad de Ciencias, Plaza Merced 37008 Salamanca, Spain; e-mail: javier@usal.es DOI: 10.1111/j.1467-9590.2011.00526.x 372 STUDIES IN APPLIED MATHEMATICS 127:372–393 C  2011 by the Massachusetts Institute of Technology