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We investigate the influence of quenched disorder on the steady states ofdriven systems of the elastic interface with non-local hydrodynamicinteractions. The generalized elastic model (GEM), which has been used tocharacterize numerous physical systems such as polymers, membranes, single-filesystems, rough interfaces, and fluctuating surfaces, is a standard approach tostudying the dynamics of elastic interfaces with non-local hydrodynamicinteractions. The criticality and phase transition of the quenched generalizedelastic model (qGEM) are investigated numerically, and the results arepresented in a phase diagram spanned by two tuning parameters. We demonstratethat in 1-d disordered driven GEM, three qualitatively different behaviorregimes are possible with a proper specification of the order parameter (meanvelocity) for this system. In the vanishing order parameter regime, thesteady-state order parameter approaches zero in the thermodynamic limit. Asystem with a non-zero mean velocity can be in either the continuous regime,which is characterized by a second-order phase transition, or the discontinuousregime, which is characterized by a first-order phase transition. The focus ofthis research was to investigate at the critical scaling features near thepinning-depinning threshold. The behavior of the quenched generalized elasticmodel at the critical depinning force is explored. Near the depinningthreshold, the critical exponent obtained numerically.

Anomalous phase diagram of the elastic interface with non-local hydrodynamic interactions in the presence of quench disorder