We consider the derivative nonlinear Schrödinger equation with constant potential as a model for wave propagation on a discrete nonlinear transmission line. This equation can be derived in the small amplitude and long wavelength limit using the standard reductive perturbation method and complex expansion. We construct some exact soliton and elliptic solutions of the mentioned equation by perturbation of its Stokes wave solutions. We find that for some values of the coefficients of the equation and for some parameters of solutions, the graphical representations show some kinds of symmetries such as mirror symmetry and rotational symmetry.

• A review is focused on 1D, 2D, and 3D matter-wave solitons in Bose-Einstein condensates under the action of spatiotemporally modulated cubic nonlinearity and time-dependent trapping potentials • Most essential problems under the consideration is the shape and stability of solitons and other coherent structures, including stabilization against the critical collapse • Both analytical results (exact and approximate ones) and systematically produced numerical findings are summarized • The modulational instability in these models and its nonlinear development is addressed in detail • Stability and motion of multi-component solitons in binary and spinor (triple) solitons is considered

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