Bilinear forms and dark soliton behaviors for a higher-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous alpha helical protein

In this paper, a higher-order variable-coefficient nonlinear Schrodinger equation is studied, which describes an inhomogeneous alpha helical protein with higher-order excitation and interaction under the continuum approximation. With the aid of auxiliary function, we obtain the variable-coefficient Hirota's bilinear equations under a set of integrable constraints. Using the Hirota's method and symbolic computation, we derive the dark one-, two- and N-soliton solutions. Influences of the variable coefficients on the soliton velocity, amplitude, and shape are analyzed. For instance, when the variable coefficients are the linear and quadratic functions of time, since the pharmacological efficacy in specific sites of the alpha helical protein diffuses linearly and quadratically as time goes on, we obtain a parabolic and cubic soliton. Interactions between/among the two, three, and four solitons with different values of variable coefficients are also discussed with the results including the parabolic, cubic, periodical, and stationary solitons.