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Abstract
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In this paper, we investigate the traveling soliton and the per iodic wave solutions of the nonlinear
Schr€ odinger equation (NLSE) wit h generalized nonlinear function ality. We also explore the
underlyin g close connection between the well-known KdV equation and the NLSE . It is remarked
that bot h one-dimension al KdV and NLSE mo dels shar e the same pseudoen ergy spectrum. We also
derive the traveling wave solutions for two cases of weakly nonlin ear mathematical models,
namely, the Helmholt z and the Du ffing oscillators’ pot entials. It is fo und that these models only
allow gray-type NLSE solitary propagations. It is also found that the pseudofrequ ency ratio for the
Helmholtz potential between the nonlinear per iodic carrier and the modulated sinusoida l waves is
always in the range 0.5 X/x 0.537285 regardless of the potential parameter values. The values
of X/x ¼ {0.5, 0.537285} correspond to the cnoidal waves modulus of m ¼ {0, 1} for soliton and
sinusoidal limits and m ¼ 0.5, respectively. Moreover, the current NLSE model is extended to fully
NLSE (FNLSE) situation for Sagdeev oscillator pseudopotential which can be derived using a
closed set of hydrodynamic fluid equations with a fully integrable Hamiltonian system. The generalized quasi-three-dimensional traveling wave solution is also derived. The current simple hydrodynamic plasma model may also be generalized to two dimensions and other complex situations
including different charged species and cases with magnetic or gravitational field effects.
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