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1. Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

   https://doi.org/10.1007/s00222-021-01039-z  

   Buckmaster, T.; Germain, P.; Hani, Z.; Shatah, J. (September 2021, Inventiones mathematicae)

   null (Ed.)

   Abstract Consider the cubic nonlinear Schrödinger equation set on a d -dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation , predicted in wave turbulence theory, on a nontrivial timescale. 

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2. On the derivation of the wave kinetic equation for NLS

   https://doi.org/10.1017/fmp.2021.6  

   Deng, Yu; Hani, Zaher (January 2021, Forum of Mathematics, Pi)

   Abstract A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $$T_{\mathrm {kin}} \gg 1$$ and in a limiting regime where the size L of the domain goes to infinity and the strength $$\alpha $$ of the nonlinearity goes to $$0$$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $$T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$$ and $$\alpha $$ is related to the conserved mass $$\lambda $$ of the solution via $$\alpha =\lambda ^2 L^{-d}$$ . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $$(\alpha , L)$$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $$\alpha $$ approaches $$0$$ like $$L^{-\varepsilon +}$$ or like $$L^{-1-\frac {\varepsilon }{2}+}$$ (for arbitrary small $$\varepsilon $$ ), we exhibit the wave kinetic equation up to time scales $$O(T_{\mathrm {kin}}L^{-\varepsilon })$$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $$T_*\ll T_{\mathrm {kin}}$$ and identify specific interactions that become very large for times beyond $$T_*$$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $$T_*$$ toward $$T_{\mathrm {kin}}$$ for such scaling laws seems to require new methods and ideas. 

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3. A note on cascade flux laws for the stochastically-driven nonlinear Schrödinger equation

   https://doi.org/10.1088/1361-6544/ad3794  

   Bedrossian, Jacob (April 2024, Nonlinearity)

   In this note we point out some simple sufficient (plausible) conditions for ‘turbulence’ cascades in suitable limits of damped, stochastically-driven nonlinear Schrödinger equation in ad-dimensional periodic box. Simple characterizations of dissipation anomalies for the wave action and kinetic energy in rough analogy with those that arise for fully developed turbulence in the 2D Navier–Stokes equations are given and sufficient conditions are given which differentiate between a ‘weak’ turbulence regime and a ‘strong’ turbulence regime. The proofs are relatively straightforward once the statements are identified, but we hope that it might be useful for thinking about mathematically precise formulations of the statistically-stationary wave turbulence problem. 

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4. The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics

   https://doi.org/10.3390/quantum5020024  

   Canfield, Jeremy; Galler, Anna; Freericks, James K. (June 2023, Quantum Reports)

   Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral. 

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5. Analytical construction of soliton families in one‐ and two‐dimensional nonlinear Schrödinger equations with nonparity‐time‐symmetric complex potentials

   https://doi.org/10.1111/sapm.12383  

   Yang, Jianke (April 2021, Studies in Applied Mathematics)

   Abstract The existence of soliton families in nonparity‐time‐symmetric complex potentials remains poorly understood, especially in two spatial dimensions. In this article, we analytically investigate the bifurcation of soliton families from linear modes in one‐ and two‐dimensional nonlinear Schrödinger equations with localized Wadati‐type nonparity‐time‐symmetric complex potentials. By utilizing the conservation law of the underlying non‐Hamiltonian wave system, we convert the complex soliton equation into a new real system. For this new real system, we perturbatively construct a continuous family of low‐amplitude solitons bifurcating from a linear eigenmode to all orders of the small soliton amplitude. Hence, the emergence of soliton families in these nonparity‐time‐symmetric complex potentials is analytically explained. We also compare these analytically constructed soliton solutions with high‐accuracy numerical solutions in both one and two dimensions, and the asymptotic accuracy of these perturbation solutions is confirmed. 

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