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An analytical weakly nonlinear model of the Benjamin–Feir instability of aStokes wave on nonuniform unidirectional current is presented. The modeldescribes evolution of a Stokes wave and its two main sidebands propagatingon a slowly varying steady current. In contrast to the models based onversions of the cubic Schrödinger equation, the current variations could bestrong, which allows us to examine the blockage and consider substantialvariations of the wave numbers and frequencies of interacting waves. Thespatial scale of the current variation is assumed to have the same order asthe spatial scale of the Benjamin–Feir (BF) instability. The model includes waveaction conservation law and nonlinear dispersion relation for each of thewave's triad. The effect of nonuniform current, apart from lineartransformation, is in the detuning of the resonant interactions, whichstrongly affects the nonlinear evolution of the system.  The modulation instability of Stokes waves in nonuniform moving media hasspecial properties. Interaction with countercurrent accelerates the growthof sideband modes on a short spatial scale. An increase in initial wavesteepness intensifies the wave energy exchange accompanied by wave breakingdissipation, resulting in asymmetry of sideband modes and a frequencydownshift with an energy transfer jump to the lower sideband mode, anddepresses the higher sideband and carrier wave. Nonlinear waves may evenoverpass the blocking barrier produced by strong adverse current. Thefrequency downshift of the energy peak is permanent and the system does notrevert to its initial state. We find reasonable correspondence between theresults of model simulations and available experimental results for waveinteraction with blocking opposing current. Large transient or freak waveswith amplitude and steepness several times those of normal waves may formduring temporal nonlinear focusing of the waves accompanied by energy incomefrom sufficiently strong opposing current. We employ the model for theestimation of the maximum amplification of wave amplitudes as a function ofopposing current value and compare the result obtained with recentlypublished experimental results and modeling results obtained with thenonlinear Schrödinger equation

An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current