{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T16:07:11Z","timestamp":1762272431704,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,8,23]],"date-time":"2019-08-23T00:00:00Z","timestamp":1566518400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We derive new kinetic and a porous medium equations from the nonlinear Schr\u00f6dinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread with time, and this fact answers the \u201cweak turbulence\u201d question for the nonlinear Schr\u00f6dinger equation with random potentials. We also derive Ohm\u2019s law for the porous medium equation.<\/jats:p>","DOI":"10.3390\/e21090823","type":"journal-article","created":{"date-parts":[[2019,8,26]],"date-time":"2019-08-26T04:38:23Z","timestamp":1566794303000},"page":"823","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["On the Wave Turbulence Theory for the Nonlinear Schr\u00f6dinger Equation with Random Potentials"],"prefix":"10.3390","volume":"21","author":[{"given":"Sergey","family":"Nazarenko","sequence":"first","affiliation":[{"name":"Institut de Physique de Nice, Universit\u00e9 C\u00f4te d\u2019Azur, Centre national de la recherche scientifique (CNRS), Parc Valrose, 06108 Nice, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Avy","family":"Soffer","sequence":"additional","affiliation":[{"name":"Mathematics Department, Rutgers University, New Brunswick, NJ 08903 USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Minh-Binh","family":"Tran","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,23]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Bourgain, J., Kenig, C.E., and Klainerman, S. 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Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"056211","DOI":"10.1103\/PhysRevE.79.056211","article-title":"Delocalization of wave packets in disordered nonlinear chains","volume":"79","author":"Skokos","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1007\/s10955-008-9649-1","article-title":"Long time Anderson localization for the nonlinear random Schr\u00f6dinger equation","volume":"134","author":"Wang","year":"2009","journal-title":"J. Stat. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"015104","DOI":"10.1063\/1.1855036","article-title":"The fermi\u2013pasta\u2013ulam problem: Fifty years of progress","volume":"15","author":"Berman","year":"2005","journal-title":"Chaos"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"015101","DOI":"10.1063\/1.1889345","article-title":"Introduction: The fermi\u2013pasta\u2013ulam problem: The first fifty years","volume":"15","author":"Campbell","year":"2005","journal-title":"Chaos"},{"key":"ref_14","first-page":"45","article-title":"Spreading, nonergodicity, and selftrapping: a puzzle of interacting disordered lattice waves","volume":"Volume 173","author":"Flach","year":"2016","journal-title":"Nonlinear Dynamics: Materials, Theory and Experiments"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"184101","DOI":"10.1103\/PhysRevLett.120.184101","article-title":"Weakly nonergodic dynamics in the Gross-Pitaevskii lattice","volume":"120","author":"Mithun","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"040501","DOI":"10.1103\/PhysRevLett.122.040501","article-title":"Wave packet spreading with disordered nonlinear discrete-time quantum walks","volume":"122","author":"Vakulchyk","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","first-page":"1031","article-title":"Physical realizability of anisotropic weak-turbulence Kolmogorov spectra","volume":"70","author":"Balk","year":"1990","journal-title":"Sov. Phys. JETP"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1017\/S0022377899008284","article-title":"A weak turbulence theory for incompressible magnetohydrodynamics","volume":"63","author":"Galtier","year":"2000","journal-title":"J. Plasma Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"066608","DOI":"10.1103\/PhysRevE.69.066608","article-title":"Noisy spectra, long correlations, and intermittency in wave turbulence","volume":"69","author":"Lvov","year":"2004","journal-title":"Phys. Rev. E"},{"doi-asserted-by":"crossref","unstructured":"Nazarenko, S. (2011). Wave turbulence. Lecture Notes in Physics, Springer.","key":"ref_20","DOI":"10.1007\/978-3-642-15942-8"},{"unstructured":"Zakharov, V.E., L\u2019vov, V.S., and Falkovich, G. (2012). Kolmogorov Spectra of Turbulence I: Wave Turbulence, Springer Science & Business Media.","key":"ref_21"},{"unstructured":"Craciun, G., and Tran, M.B. (2016). A reaction network approach to the convergence to equilibrium of quantum boltzmann equations for bose gases. arXiv.","key":"ref_22"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.physd.2018.06.003","article-title":"Quantum hydrodynamic approximations to the finite temperature trapped bose gases","volume":"380","author":"Jin","year":"2018","journal-title":"Physica D"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/S0167-2789(01)00192-0","article-title":"Wave turbulence and intermittency","volume":"152","author":"Newell","year":"2001","journal-title":"Physica D"},{"doi-asserted-by":"crossref","unstructured":"Pomeau, Y., and Tran, M.B. (2019). Statistical physics of non equilibrium quantum phenomena. Lecture Notes in Physics, Springer.","key":"ref_25","DOI":"10.1007\/978-3-030-34394-1"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"063001","DOI":"10.1088\/1751-8121\/aaf7b3","article-title":"A kinetic equation for ultra-low temperature bose\u2013einstein condensates","volume":"52","author":"Reichl","year":"2019","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2243","DOI":"10.1016\/j.jde.2018.04.031","article-title":"On coupling kinetic and Schrodinger equations","volume":"265","author":"Soffer","year":"2018","journal-title":"J. Differ. Equ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1016\/j.aim.2017.12.007","article-title":"On the dynamics of finite temperature trapped bose gases","volume":"325","author":"Soffer","year":"2018","journal-title":"Adv. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.physleta.2004.09.062","article-title":"Probability densities and preservation of randomness in wave turbulence","volume":"332","author":"Choi","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.physd.2004.11.016","article-title":"Joint statistics of amplitudes and phases in wave turbulence","volume":"201","author":"Choi","year":"2005","journal-title":"Physica D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/j.physleta.2005.02.072","article-title":"Anomalous probability of large amplitudes in wave turbulence","volume":"339","author":"Choi","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"305206","DOI":"10.1088\/1751-8113\/44\/30\/305206","article-title":"Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model","volume":"44","author":"Rivkind","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"unstructured":"Germain, P., Ionescu, A.D., and Tran, M.-B. (2017). Optimal local well-posedness theory for the kinetic wave equation. arXiv.","key":"ref_33"},{"unstructured":"V\u00e1zquez, J.L. (2007). The Porous Medium Equation: Mathematical Theory, Oxford University Press.","key":"ref_34"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"025603","DOI":"10.1103\/PhysRevA.81.025603","article-title":"Bose-einstein condensates on tilted lattices: Coherent, chaotic, and subdiffusive dynamics","volume":"81","author":"Kolovsky","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1559","DOI":"10.1088\/0022-3727\/9\/11\/005","article-title":"Some explicit solutions to the non-linear diffusion equation","volume":"9","author":"Tuck","year":"1976","journal-title":"J. Phys. 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