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Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems

Dmitry Shepelsky, Iryna Karpenko, Stepan Bogdanov, Jaroslaw E. Prilepsky

Abstract

We consider the Riemann--Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation. An RH problem for the solution of the finite-band problem has been recently derived via the Fokas method [1,2]. Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.

Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems

Abstract

We consider the Riemann--Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation. An RH problem for the solution of the finite-band problem has been recently derived via the Fokas method [1,2]. Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.
Paper Structure (18 sections, 93 equations, 2 figures, 1 table)

This paper contains 18 sections, 93 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The inverse problem by the RH approach
  3. Direct problem in the periodic case: a sketch
  4. Direct problem in the periodic case: details
  5. Analytic properties of $R(z)$
  6. RH problem associated with the periodic problem for the NLS
  7. Verifying the initial conditions
  8. Verifying the periodicity
  9. From the basic RH problem to a RH problem with structure \ref{['J_j']}-\ref{['RHnorm-Psi']}
  10. Step 1: Undressing the jump matrices on ${\mathbb R}\cup i{\mathbb R}$
  11. $R(z)$ in connection with the theory of periodic finite-band solutions of the NLS
  12. Step 2: getting rid of the jumps across ${\mathbb R}\cup i{\mathbb R}$ as well as of the singularities
  13. Step 3: Reducing the jump across $\Sigma^N$ to the form of \ref{['J_j']}
  14. Evolution
  15. Examples
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Three examples in the case with $N=1$ with common main spectrum $z_0=0.2780 + i$ and $z_1= 1.2780 + i$ and different phases $\phi_0$ and $\phi_1$. The phases are depicted as points $e^{\phi_j}$ on the unit circles around the corresponding $z_j$.
  • Figure 2: The example in the case with $N=2$ with main spectrum $z_0=-1+3 i$, $z_1=5 i$, $z_2=1+3 i$ and phases $\phi_0$, $\phi_1$, $\phi_2$.