Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity

Abstract

We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, the one which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too.Periodic (alias lattice) potentials is a well-known ingredient of diverse physical settings represented by the nonlinear Schrödinger/Gross-Pitaevskii equations. The lattice potentials help to create self-trapped modes (solitons) which do not exist otherwise, or stabilize those solitons which are definitely unstable in free space. In particular, the lattice potentials generate the bandgap spectrum in the linearized version of the equation, and adding local cubic nonlinearity gives rise to a great variety of gap solitons and their bound complexes residing in the spectral gaps. On the other hand, an essential extension of the concept of lattice potentials is the introduction of nonlinear pseudopotentials, which are induced by spatially periodic modulation of the coefficient in front of the cubic term. While single-peak fundamental solitons (FSs) in nonlinear potentials were studied in detail, more sophisticated ones, such as narrow antisymmetric dipole solitons (DSs), which essentially reside in a single cell of the nonlinear lattice, were not previously considered in this setting. Their shape is similar to that of the so-called subfundamental species of gap solitons in linear lattices, which have a small stability region. In this work, we first develop a general classification of a potentially infinite number of different types of soliton complexes supported by the nonlinear lattice. For physical applications, the most significant finding is the existence of two branches of the DS family, one of which is entirely stable. Its stability is readily predicted by the celebrated Vakhitov-Kolokolov criterion, while the shape of a) Electronic

Global search for localised modes in scalar and vector nonlinear Schrödinger-type equations

“…where P (x) is a real function with alternating sign, develop a singularity at a finite point in R [40,41]. With regard to the system (5)- (6), this suggests that our method may remain effective even for a much wider class of nonlinearities.…”

“…where P (x) is a real function with alternating sign, develop a singularity at a finite point in R [40,41].…”