Loop-type geometric folding of exact solutions of shifted nonlocal NLS and MKdV equations

Abstract

Based on the notion of foldon, we introduce a geometric framework for constructing folded parametric wave representations of exact solutions of some shifted nonlocal nonlinear Schr\"{o}dinger and modified Korteweg-de Vries equations. Unlike the method of constructing loops in $(2+1)$-dimensional integrable models based on universal variable separation approach or hodograph transformation, we consider a simplified geometric approach of constructing loop-type folded profiles via non-monotonic parametrization of the spatial coordinate associated with the exact solution of the $(1+1)$-dimensional shifted nonlocal equations. A sufficient condition under which folding takes place is provided in the form of sign change of the derivative of folding map. Applying one- and two-soliton solutions of various shifted nonlocal nonlinear Schr\"{o}dinger and modified Korteweg-de Vries equations found earlier, we show how different folding maps generate different loop-type folded profiles. In particular, we analyze the influence of deformation parameters and solution parameters on the geometry of folded waves. We show that the effect of the folding leads only to the modification of the spatial parametrization and generates various geometric structures like regular loop-type, oscillating-type, and singular-type folded profiles for certain values of parameters.