On the Integrability of the Poisson Driven Stochastic Nonlinear Schrödinger Equations

We consider the Cauchy problem for the dissipative nonlinear Schrödinger equations driven by a Poisson noise, namely image where γn > 0 and 0 < t1 < ⋯ < tn < ⋯ are certain sequences of random numbers and is the deterministic loss coefficient. This perturbation incorporates the possibility of sudden changes in the field that occur randomly. If Γ= 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 Γ≠ 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 E1. 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.