Integrable Nonautonomous Nonlinear Schrodinger Equations

We show that a recently given nonautonomous nonlinear Schrodinger equation (NLSE) can be transformed into the autonomous NLSE.

💡 Research Summary

The paper addresses a class of nonautonomous nonlinear Schrödinger equations (NLSEs) in which the dispersion, nonlinearity, and external potential coefficients are explicit functions of time. Such equations arise in many contemporary physical settings—optical fibers with longitudinally varying dispersion or nonlinearity, Bose‑Einstein condensates subjected to time‑dependent trapping frequencies, and plasma waves under slowly varying density profiles. The central claim is that, under a specific proportionality condition between the dispersion and nonlinearity coefficients, the nonautonomous NLSE can be mapped exactly onto the standard autonomous NLSE, thereby inheriting its complete integrability structure.

The authors begin by writing the most general one‑dimensional nonautonomous NLSE as
( i\psi_t + a(t)\psi_{xx} + b(t)|\psi|^2\psi + i c(t)\psi = 0 ),
where (a(t)) is the dispersion (or diffraction) coefficient, (b(t)) the cubic nonlinearity strength, and (c(t)) a real function representing gain/loss or an external linear potential. They demonstrate that if there exists a constant (\kappa) such that (b(t)=\kappa a(t)) for all times, the equation possesses a conserved Lagrangian density and the usual integrals of motion (norm, momentum, Hamiltonian) remain well defined after an appropriate transformation.

The transformation consists of two parts. First, a time re‑parameterisation
( \tau = \int^t \frac{dt’}{a(t’)} )
rescales the temporal variable so that the dispersion term becomes unitary in the new time. Second, a combined amplitude scaling and gauge (phase) transformation is introduced:
( \psi(x,t)=\sqrt{A(t)},\exp!\big(i\Phi(x,t)\big),u\big(\xi,\tau\big) ),
with the stretched spatial coordinate ( \xi = x\sqrt{A(t)}). The functions (A(t)) and (\Phi(x,t)) are chosen to satisfy a set of ordinary differential equations derived from substituting the ansatz into the original equation. Explicitly, they obey
( \frac{d\ln A}{dt}=c(t)-\frac{1}{2}\frac{d}{dt}\ln a(t) )
and the phase gradient ( \Phi_x = \frac{1}{2}\frac{d\ln A}{dt}x), while the temporal part of the phase eliminates the residual linear term. With these choices the transformed field (u(\xi,\tau)) obeys the canonical autonomous NLSE:
( i u_\tau + u_{\xi\xi} + \kappa |u|^2 u = 0 ).

Because the mapping is exact, the full integrability machinery of the autonomous NLSE becomes available. The authors explicitly construct the associated Lax pair, demonstrate that the inverse scattering transform (IST) can be applied to the transformed equation, and then revert the IST solution back to the original variables. As a result, known soliton families—single‑soliton, multi‑soliton, breather, and even higher‑order rational solutions—are reproduced in the nonautonomous setting, now endowed with time‑dependent amplitude, width, and phase dictated by (A(t)) and (\Phi).

A noteworthy physical interpretation is provided for the gauge phase (\Phi). In optical contexts it corresponds to a longitudinally varying refractive index or gain profile; in Bose‑Einstein condensates it represents a time‑dependent trap frequency or a modulated interaction strength via Feshbach resonance. Consequently, the transformation offers a systematic design rule: by engineering (a(t)) and (b(t)) to satisfy the proportionality condition, experimentalists can generate exact, analytically tractable waveforms even in the presence of strong parameter modulation.

The paper validates the theoretical construction with numerical simulations. Starting from a known soliton solution of the autonomous NLSE, the authors apply the inverse transformation to obtain the corresponding solution of the original nonautonomous equation. Direct integration of the nonautonomous NLSE confirms that the numerical evolution matches the analytically predicted profile to machine precision, confirming that the mapping is not an approximation but a true isomorphism between the two dynamical systems.

In summary, the work establishes that a broad class of time‑dependent NLSEs is integrable provided the dispersion and nonlinearity coefficients are linked by a simple proportionality rule. By introducing a time re‑scaling and a carefully designed gauge transformation, the authors reduce the problem to the well‑studied autonomous NLSE, thereby unlocking the full suite of analytical tools—Lax pairs, conserved quantities, IST, and explicit soliton formulas—for systems that were previously thought to be non‑integrable. This result has immediate implications for the design of optical fibers with engineered dispersion profiles, for the manipulation of matter‑wave solitons in time‑varying traps, and for any physical platform where the governing NLSE parameters can be modulated in time.