In this paper, we consider the complex modified Korteweg–de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-, and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and nonstandard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order rogue wave of the complex mKdV and the nonlinear Schrödinger equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultrashort pulse technology.

Recently, Fokas presented a nonlocal Davey–Stewartson I (DSI) equation (Fokas 2016 Nonlinearity 29 319–24), which is a two-spatial dimensional analogue of the nonlocal nonlinear Schrödinger (NLS) equation (Ablowitz and Musslimani 2013 Phys. Rev. Lett . 110 064105), involving a self-induced parity-time-symmetric potential. For this equation, high-order periodic line waves and line breathers are derived by employing the bilinear method. The long wave limit of these periodic solutions yields two kinds of fundamental rogue waves, namely, kink-shaped and W-shaped line rogue waves. The interaction of fundamental line rogue waves generate higher-order rogue waves, which have several interesting patterns with different curvy profiles. Furthermore, semi-rational solutions are constructed, which are line rogue waves on a background of periodic line waves. Finally, two particular solutions of the nonlocal NLS equation, namely, a first-order rogue wave and a semi-rational solution are obtained as reductions of the corresponding solutions of the nonlocal DSI equation.

In this paper, we derive a Darboux transformation of the Hirota and the Maxwell-Bloch (H-MB) system which is governed by femtosecond pulse propagation through an erbium doped fiber and further generalize it to the matrix form of the $n$-fold Darboux transformation of this system. This $n$-fold Darboux transformation implies the determinant representation of $n$th solutions of $({E}^{[n]},{p}^{[n]},{$\eta${}}^{[n]})$ generated from the known solution of $(E,p,$\eta${})$. The determinant representation of $({E}^{[n]},{p}^{[n]},{$\eta${}}^{[n]})$ provides soliton solutions, positon solutions, and breather solutions (both bright and dark breathers) of the H-MB system. From the breather solutions, we also construct a bright and dark rogue wave solution for the H-MB system, which is currently one of the hottest topics in mathematics and physics. Surprisingly, the rogue wave solution for $p$ and $$\eta${}$ has two peaks because of the order of the numerator and denominator of them. Meanwhile, after fixing the time and spatial parameters and changing two other unknown parameters $$\alpha${}$ and $$\beta${}$, we generate a rogue wave shape.

A supertrace identity on Lie superalgebras is established. It provides a tool for constructing super-Hamiltonian structures of zero curvature equations associated with Lie superalgebras. Applications in the case of the Lie superalgebra B(0,1) present super-Hamiltonian structures of a super-AKNS soliton hierarchy and a super-Dirac soliton hierarchy.