Effective integration of the nonlinear vector Schrödinger equation

Abstract

Abstract. A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schrödinger equation (Manakov system) is developed. The allied spectral variety is a trigonal Riemann surface, which is described explicitly and the solutions of the equations are given in terms of θ-functions of the surface. The final formulae are effective in that sense that all entries like transcendental constants in exponentials, winding vectors etc. are expressed in terms of prime-form of the curve and well algorithmized operations on them. That made the result available for direct calculations in applied problems implementing the Manakov system. The simplest solutions in Jacobian ϑ-functions are given as particular case of general formulae and discussed in details.

New Degeneration of Fay's Identity and its Application to Integrable Systems

“…Consider the following expansion of the normalized holomorphic differentials ω j near a, 8) where V a,j , W a,j , U a,j ∈ C. Let us denote by D a the operator of directional derivative along the vector V a = (V a,1 , . .…”

“…From (A.2), one sees that any contour in H 1 (R g ) which is invariant under τ is a combination of A-cycles only. In particular, the closed contour τ + ∈ H 1 (R g ) can be written as 8) for some N ∈ Z g . This proves (A.6).…”

“…Denote by˜ an oriented contour connectingã andb. First, let us check that Using the action (A.6) of τ on the A-cycles in the homology group H 1 (R g \ {a, b}), we get the following action of τ on the normalized meromorphic differentials of third kind Ω b−a : 8) (notice that τ a = b). Hence, the last term in the right hand side of (B.7) is equal to Let us now consider cases (1) and (2) separatly.…”

The full positive flows of Manakov hierarchy, Hamiltonian structures and conservation laws

“…Morever, the periodic and quasi-periodic waves solutions for the CNLS equation of Manakov type were presented by using Lax pair method and the general method of reduction of Abelian functions to elliptic functions [30][31][32]. Elgin et al [33,34] constructed the finite gap solutions to the vector NLS equation using an algebro-geometric technique. There are some other articles refer to the periodic and quasi-periodic solutions with different methods [35][36][37][38][39][40][41].…”

Adaptive Wave Models for Sophisticated Option Pricing

“…It admits "bright" and "dark" soliton solutions. The simplest solution of (15)- (16), the so-called Manakov bright 2-soliton, has the form resembling that of the sech-solution (12) (see [34][35][36][37][38][39][40] …”