Finite-Length Soliton Solutions of the Local Homogeneous Nonlinear Schroedinger Equation

Abstract

We found a new kind of soliton solutions for the 5-parameter family of the potential-free Stenflo-Sabatier-Doebner-Goldin nonlinear modifications of the Schr\"odinger equation. In contradistinction to the "usual'' solitons like {\cosh[b(x-kt)]}^{-a}\exp[i(kx-ft)], the new {\em Finite-Length Solitons} (FLS) are nonanalytical functions with continuous first derivatives, which are different from zero only inside some finite regions of space. The simplest one-dimensional example is the function which is equal to {\cos[g(x-kt)]}^{1+d}\exp[i(kx-ft)] (with d>0) for |x-kt|<\pi/(2g), being identically equal to zero for |x-kt|>\pi/(2g). The FLS exist even in the case of a weak nonlinearity, whereas the ``usual'' solitons exist provided the nonlinearity parameters surpass some critical values.