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The kinetic theory of $\mathrm{sech}^2 x$-type electron holes is studied. Thepotential of the electron holes is solved in the weak amplitude limit by thepseudo-potential method. We investigate the existence condition of the$\mathrm{sech}^2 x$ electron holes. It indicates that the derivatives oftrapped and untrapped distributions at the separatrix play significant roles indetermining the potential profile. The theory is then applied to theKappa-distributed plasmas. The amplitude and width of the $\mathrm{sech}^2 x$electron holes are analyzed. Finally, the theoretical results are verified bynumerical calculations.

Kinetic theory of $\mathrm{sech}^2x$ electron holes and applications to Kappa-distributed plasmas