- Premise:
The time-dependent Gross–Pitaevskii equation (GPE) is (https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation) $$ i \partial_t \psi = -\nabla^2 \psi + g |\psi|^2 \psi $$ plus other terms. In (https://arxiv.org/abs/quant-ph/9801064), the authors give a prescription to model dissipation of fluctuations in $\psi$: their prescription is $$ i \partial_t \psi = (1+i \Lambda)(-\nabla^2 \psi + g |\psi|^2 \psi) \qquad \qquad \Lambda<0 $$ They claim that this equation is, in fact, a rather general equation of motion which describes evolution towards equilibrium, that in this case is $\psi =0$.
- Question:
In analogy with what is done for the GPE, I wonder which is the effect of multiplying an ODE by a certain factor $1-i \lambda$, where $\lambda>0$. For example, consider
$$ i\dot \psi = v+g|\psi|^2\psi \qquad \rightarrow \qquad i\dot \psi = (1-i \lambda)(v+g|\psi|^2 \psi) $$
where $g$ and $v$ are real constants. For both equations we have a static solution $\psi_s$ such that $g|\psi_s|^2 \psi_s+v=0$. If we start at $t=0$ with $\psi(0)\neq \psi_s$, is it possible to conclude that $\psi(t)\rightarrow \psi_s$ for $t\rightarrow \infty$? Alternatively, how to understand which is the qualitative behaviour of the "velocity" $\dot\psi$ for $t\rightarrow \infty$? Can we conclude that for $\lambda>0$ we have
$$ \lim_{t\rightarrow\infty} |\dot{\psi}| \rightarrow 0 \quad ? $$
I expect that for a very small $\lambda$, so that $ (1-i \lambda)^{-1}\approx (1+i \lambda)$ we can redefine the time as $t\rightarrow (1+i\lambda)t$. This changes the Fourier, or Laplace transform of $\psi$ and $\dot\psi$, but I don't know how to formalise this to show that there is a "relaxation behavior".
- A final thought:
From the philosophical point of view, something similar happens also when one performs a "Wigner rotation" $t\mapsto it$ and maps the Schrodinger equation into the heat equation (e.g. Schrödinger versus heat equations or Schrödinger's Equation). The Schrodinger equation "oscillates", the heat equation describes a "relaxation". Is this concept useful for the above ODE case?
This isn't going to answer all of your questions, mainly because there's no single cut-and-dry answer. Your questions basically cut to the heart of how one tries to understand the behavior of nonlinear ODEs and PDEs from the ground up, and given the enormous variety of nonlinear ODEs and PDEs there is no way to describe how one does that in complete generality.
A common way that a mathematician comes up with heuristics for the behavior of ODEs/PDEs goes by the name of the method of dominant balance. The basic principle is that for a differential equation with at least three terms, generically two of the terms will be comparable and the third will be negligible. Thus you can split the study of the equation into regimes based on which terms of the equation are comparable.
For instance, in the GP equation $i\psi_t = -\nabla^2\psi + g|\psi|^2\psi$, we have a three-term equation, and consequently there are $3$ ways to choose two of the terms to be comparable and one term to be negligible. The first way gives us the regime $$ i\psi_t \approx -\nabla^2\psi, $$ the dispersive or linear regime. If the $\approx$ is replaced by $=$, then this is the free Schrodinger equation. So on this regime, the time evolution of $\psi$ is dominated by the behavior of the free Schroinger evolution. The second way gives the nonlinear regime $$ i\psi_t \approx g|\psi|^2\psi, $$ in which the evolution is dominated by the effects of the nonlinearity. To understand the effects, you can just solve the equation in this case (it's an ODE), and the ODE solution gives you oscillation. The last way is the stationary, or soliton regime $$ -\nabla^2\psi + g|\psi|^2\psi \approx 0. $$ As it turns out, whether or not one can expect this regime to be significant depends on the sign of $g$: if the nonlinearity is focusing, there exist regular and spatially localized solutions to $-\nabla^2\psi + g|\psi|^2\psi = 0$, while there are no such solutions if the nonlinearity is defocusing. One interprets this regime as the one where the linear effects (corresponding to $-\nabla^2\psi$) and the nonlinear effects ($g|\psi|^2\psi$) are of equal strength, which is precisely how soliton behavior arises. Indeed, GP solitons are constructed by taking solutions to this equation, then applying the symmetry group of GP to these stationary solutions; since it is only possible in the focusing case to solve the stationary equation, this explains why GP only exhibits soliton behavior for focusing nonlinearities.
When you multiply the equation by a complex number $a+ib$, this merely adds extra terms to the equation and you can study them pairwise as before using dominant balance. The equation would be $$ i\psi_t = -a\nabla^2\psi - ib\nabla^2\psi + ag|\psi|^2\psi + ibg|\psi|^2\psi. $$ Comparing pairs, some of the pairs are not significantly different from before; e.g. $$ i\psi_t \approx -a\nabla^2\psi $$ is still basically a free Schrodinger equation, though the factor of $a$ does affect how strong this effect is. There are some new effects: $$ i\psi_t \approx -ib\nabla^2\psi $$ is a heat equation after cancelling the $i$-s, and consequently this term contributes a dissipative effect to the evolution. This should contribute to a tendency for the solution to vanish at infinite time. Another effect is $$ i\psi_t \approx igb|\psi|^2\psi. $$ This ODE exhibits finite blowup, and thus the $igb|\psi|^2\psi$ term could contribute to singularity formation. And so on and so forth with the other terms. The overall behavior of $\psi$ is basically determined by how these disparate effects on the evolution compare and interact with each other, which is in general a complicated thing to analyze (especially for GP, which remains a subject of much research today).
With your ODE one can perform a similar analysis: examine the terms in your equation pairwise, and deduce what effects you will observe in various regimes. The various regimes are usually distinct timescales for your problem: for instance, in the original GP equation, the $-\nabla^2\psi$ term usually dominates the evolution for short times (i.e. the evolution is primarily linear for short times, basically the timescale of local wellposedness), and the nonlinear effect from $g|\psi|^2\psi$ tends to take effect over longer timescales.
Once one knows the observable effects of the various terms of the equation on various regimes, it is also possible to piece things together. There are lots of techniques that go into this and they depend on the equation. They generally fall into the realm of asymptotic analysis overall, and some techniques in the area are multiple-scale analysis and the method of matched asymptotic expansions. For a good reference on these topics, see the classic book of Bender and Orszag.