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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Based on modified version of the Pennes' bio-heat transfer equation, a simplified onedimensional bio-heat transfer model of the living tissues in the steady state has been applied on whole body heat transfer studies, and by using the Weierstrass' elliptic function, its corresponding analytic periodic and non-periodic solutions have been derived in this paper. Using the obtained analytic solutions, the effects of the thermal diffusivity, the temperature-independent perfusion component, and the temperature-dependent perfusion component in living tissues are analyzed numerically. The results show that the derived analytic solution is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to applications such as parameter measurement, temperature field reconstruction and clinical treatment.

We present a new family of analytic solutions of the nonlinear Schrödinger equation with an imaginary potential that describes the dynamics of a bright soliton in a Bose-Einstein condensate with an expulsive parabolic and complex potential. We found that under a safe range of parameters, the bright soliton can be compressed into very high local matter densities by increasing the value of the feeding parameter.

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We investigate the modulational instability of Stokes wave solutions on a system of coupled nonlinear electrical transmission lines with dispersive elements. In the continuum limit, and in suitable scaled coordinates, the voltage on the system is described by the two-dimensional coupled nonlinear Schrödinger equations. The set of coupled nonlinear Schrödinger equations obtained is analyzed via a perturbation approach. No assumption is made on the signs of the relevant coefficients such as the coefficients of nonlinearity and the coupling coefficients. A set of explicit criteria of modulational stability and modulational instability is derived and analyzed. It is numerically shown that the effect of the dispersive elements in the line is to decrease the instability region and the instability growth rate.