5 results
Search Results
Now showing 1 - 5 of 5
- ItemGeneralized integrable evolution equations with an infinite number of free parameters(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Akhmediev, Nail; Ankiewicz, Adrian; Amiranashvili, Shalva; Bandelow, UweEvolution equations such as the nonlinear Schrödinger equation (NLSE) can be extended to include an infinite number of free parameters. The extensions are not unique. We give two examples that contain the NLSE as the lowest-order PDE of each set. Such representations provide the advantage of modelling a larger variety of physical problems due to the presence of an infinite number of higher-order terms in this equation with an infinite number of arbitrary parameters. An example of a rogue wave solution for one of these cases is presented, demonstrating the power of the technique.
- ItemSasa-Satsuma equation: Soliton on a background and its limiting cases(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Bandelow, Uwe; Akhmediev, NailWe present a multi-parameter family of a soliton on a background solutions to the Sasa-Satsuma equation. The solution is controlled by a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits a few nontrivial limiting cases that are considered in detail. Among these special cases is the NLSE limit and the limit of rogue wave solutions
- ItemPersistence of rouge waves in extended nonlinear Schrödinger equations : integrable Sasa-Satsuma case(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Bandelow, Uwe; Akhmediev, Nail N.We present the lowest order rogue wave solution of the Sasa-Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of Peregrine solution appears when the extension parameter of the SSE is reduced to zero.
- ItemGeneralized Sasa-Satsuma equation: Densities approach to new infinite hierarchy of integrable evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Ankiewicz, Adrian; Bandelow, Uwe; Akhmediev, NailWe derive the new infinite Sasa-Satsuma hierarchy of evolution equations using an invariant densities approach. Being significantly simpler than the Lax-pair technique, this approach does not involve ponderous 3 x3 matrices. Moreover, it allows us to explicitly obtain operators of many orders involved in the time evolution of the Sasa-Satsuma hierarchy functionals. All these operators are parts of a generalized Sasa-Satsuma equation of infinitely high order. They enter this equation with independent arbitrary real coefficients that govern the evolution pattern of this multi-parameter dynamical system.
- ItemSolitons on a background, rogue waves, and classical soliton solutions of Sasa-Satsuma equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bandelow, Uwe; Akhmediev, NailWe present the most general multi-parameter family of a soliton on a background solutions to the Sasa-Satsuma equation. The solution contains a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits nontrivial limiting cases, such as rogue waves and classical solitons, that are considered in detail.