[Paper Review] On Integrability of Nonautonomous Nonlinear Schroedinger Equations

Sergeĭ K. Suslov|arXiv (Cornell University)|Dec 16, 2010

Nonlinear Waves and Solitons52 references8 citations

TL;DR

This paper presents a systematic method to transform nonautonomous, inhomogeneous nonlinear Schrödinger equations with time-dependent quadratic Hamiltonians into the standard autonomous nonlinear Schrödinger equation via explicit time-dependent transformations. The key contribution is a closed-form transformation using solutions to a Riccati-type system of ODEs, which reduces the problem to a completely integrable form solvable by inverse scattering, with explicit links to Green's functions of generalized harmonic oscillators.

ABSTRACT

We show, in general, how to transform the nonautonomous nonlinear Schroedinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is outlined. A few simple integrable systems are discussed.

Motivation & Objective

• To establish a general method for transforming nonautonomous, inhomogeneous nonlinear Schrödinger equations with quadratic Hamiltonians into the standard autonomous form.
• To identify the necessary and sufficient conditions under which such transformations yield complete integrability.
• To provide explicit, quadrature-based expressions for the transformation parameters using solutions to a Riccati-type system of ODEs.
• To connect the transformation to the Green's function of the corresponding linear problem, a link previously missing in the literature.
• To demonstrate the method on specific integrable systems, including soliton solutions and Lax pair constructions.

Proposed method

• The transformation is defined by a gauge and coordinate change: $\psi(x,t) = \frac{1}{\sqrt{\mu(t)}} e^{i(\alpha(t)x^2 + \delta(t)x + \kappa(t))} \chi(\xi,\tau)$, with $\xi = \beta(t)x + \varepsilon(t)$, $\tau = \gamma(t)$.
• The time-dependent functions $\alpha, \beta, \gamma, \delta, \varepsilon, \kappa$ are determined by solving a system of first-order ODEs (2.6)–(2.11), which form a Riccati-type system.
• The Riccati system is reduced to a second-order linear ODE for $\mu(t)$ via substitution $\alpha = \frac{1}{4a(t)} \frac{\mu'(t)}{\mu(t)} - \frac{d(t)}{2a(t)}$, yielding equation (2.13).
• The resulting autonomous equation $i\chi_\tau + h_0 |\chi|^2 \chi = \chi_{\xi\xi}$ is known to be completely integrable via inverse scattering and Lax pair methods.
• The method explicitly links the transformation to the Green's function of the underlying linear generalized harmonic oscillator problem.
• The approach is applied to derive soliton solutions and Lax pairs for specific nonautonomous systems, including those with time-dependent external potentials and nonlinearities.

Experimental results

Research questions

• RQ1 Can all nonautonomous nonlinear Schrödinger equations with quadratic Hamiltonians be transformed into the standard autonomous integrable form?
• RQ2 What is the explicit form of the transformation that maps such nonautonomous systems to the autonomous nonlinear Schrödinger equation?
• RQ3 How are the transformation parameters related to the Green's function of the corresponding linear problem?
• RQ4 What is the structure of the equivalent Lax pair for the nonautonomous system after transformation?
• RQ5 Can this method be systematically applied to derive soliton solutions and conservation laws for specific nonautonomous systems?

Key findings

• The nonautonomous nonlinear Schrödinger equation (2.2) with time-dependent coefficients is transformed into the standard autonomous form $i\chi_\tau + h_0 |\chi|^2 \chi = \chi_{\xi\xi}$ via the explicit transformation (2.4) under the conditions (2.6)–(2.11).
• The transformation parameters $\alpha, \beta, \gamma, \delta, \varepsilon, \kappa$ are determined by solving a Riccati-type system of ODEs, which reduces to a second-order linear ODE for $\mu(t)$ as shown in equation (2.13).
• The method establishes a direct connection between the transformation and the Green's function of the generalized harmonic oscillator, a link previously unemphasized in the literature.
• The resulting autonomous equation is completely integrable, admitting $N$-soliton solutions, conservation laws, and an equivalent Lax pair via standard inverse scattering techniques.
• Specific examples, including a soliton solution governed by the second Painlevé equation, are derived explicitly, with the profile expressed in terms of the nonlinear Airy function $A_{k_0}(\zeta)$ and its asymptotics.
• The method is general and applies to the broadest class of one-dimensional nonautonomous and inhomogeneous nonlinear Schrödinger equations with quadratic Hamiltonians, as formalized in equations (2.2) and (2.3).