Screening and localization in the nonlinear Anderson problem

We study the spreading dynamics of an initially localized wave packet of finite norm in one-dimensional nonlinear Schrödinger lattices with random potential. The problem has gained considerable interest in the literature, and it continues to attract attention due to its connection with the general properties of behavior of systems with competition between nonlinearity, nonlocality, and randomness. It is shown that adding small dielectric coupling to the ambient random medium leads to asymptotic localization of the nonlinear field regardless of the Kerr nonlinearity strength. If the electric susceptibility is zero, then the nonlinear field undergoes sharp localization-delocalization transition above a certain critical value of the nonlinearity parameter. The nonlinear localization length is found to be <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi mathvariant="normal">Λ</a:mi><a:mi>loc</a:mi></a:msub><a:mo>≃</a:mo><a:mo form="prefix">exp</a:mo><a:mrow><a:mo>[</a:mo><a:mrow><a:mo>(</a:mo><a:mi>π</a:mi><a:mo>/</a:mo><a:msub><a:mi>ɛ</a:mi><a:mi>r</a:mi></a:msub><a:mo form="prefix">tan</a:mo><a:mi>δ</a:mi><a:mo>)</a:mo></a:mrow><a:mo form="prefix">ln</a:mo><a:mi>β</a:mi><a:mo>]</a:mo></a:mrow></a:mrow></a:math>, where <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mo form="prefix">tan</f:mo><f:mi>δ</f:mi></f:mrow></f:math> is the dielectric loss tangent, <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>ɛ</h:mi><h:mi>r</h:mi></h:msub></h:math> is the relative permeability of the medium, and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>β</i:mi></i:math> characterizes the Kerr nonlinearity. The model predicts a possibility of self-induced localization when the “medium” to which the wave field is dielectrically coupled is the wave function itself. The mathematical methods, stipulated here, pave the way towards understanding the wave processes in complex media with competition between randomness, dispersiveness, and nonlinearity, such as Anderson localization of a wave packet interacting with the environment.