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Low regularity symplectic schemes for stochastic NLS

Jacob Armstrong-Goodall, Yvain Bruned

Abstract

We introduce a class of symplectic resonance based schemes for Schrödinger's equation in dimension one, building on the work in [1] wherein resonance based numerical schemes were developed in the context of dispersive PDE driven by time dependent, or space-time dependent, coloured noise. We work primarily with a cubic nonlinearity, advancing the approach introduced in [15] for deriving symplectic schemes in the deterministic setting. As an example of such a scheme we derive the resonance based midpoint rule for the Stochastic NLS and analyse its convergence properties.

Low regularity symplectic schemes for stochastic NLS

Abstract

We introduce a class of symplectic resonance based schemes for Schrödinger's equation in dimension one, building on the work in [1] wherein resonance based numerical schemes were developed in the context of dispersive PDE driven by time dependent, or space-time dependent, coloured noise. We work primarily with a cubic nonlinearity, advancing the approach introduced in [15] for deriving symplectic schemes in the deterministic setting. As an example of such a scheme we derive the resonance based midpoint rule for the Stochastic NLS and analyse its convergence properties.

Paper Structure

This paper contains 13 sections, 9 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Geometric Structure of (S)NLSE
  3. The Deterministic Setting
  4. The Stochastic Setting
  5. Symplectic Resonance Based Runge-Kutta Scheme for Dispersive PDE
  6. Stochastic Symplectic Resonance Based Runge-Kutta
  7. Discretisation of Stochastic Iterated Integrals
  8. Discretisation Maps Properties
  9. Conserved Quantities
  10. Convergence Analysis of the Midpoint Rule
  11. Existence and Stability
  12. Local Error
  13. Conclusion

Key Result

theorem 1

Suppose that $u \in H^r$, $r > 3/2$, and that the real-valued coefficients $b^{\alpha}$, $a^{\tilde{\alpha}}_{\alpha}$ satisfy Then the Runge–Kutta resonance-based schemes (eq:scheme_nlse) preserve the corresponding quadratic first integrals, exactly.

Theorems & Definitions (23)

  • remark 1
  • remark 2
  • theorem 1
  • remark 3
  • Example 1: Stochastic Resonance Based Midpoint Rule
  • theorem 2
  • lemma 1
  • proof
  • proposition 1
  • proposition 2
  • ...and 13 more