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Perturbation Analysis and Neural Network-Based Initial Condition Estimation for the Sine-Gordon Equation

Junhong Ha, Sudeok Shon

TL;DR

This work addresses the stability of soliton solutions in the sine-Gordon equation under external perturbations by establishing a rigorous Hilbert-space variational framework that yields well-posedness and energy-based bounds for perturbations. It decomposes solutions around a soliton profile and derives the perturbation equation, proving existence, uniqueness, and regularity of weak solutions via coercive bilinear forms and Lipschitz nonlinearities. Complementing the analysis, the authors introduce a fully data-driven neural-network approach to estimate unknown initial conditions from limited observations, using synthetic forward-PDE data to train a network that maps (space, frequency, time) to initial-state estimates. Numerical experiments validate the perturbation theory and demonstrate effective initial-condition recovery even with noisy and sparse data, highlighting the potential of combining mathematical analysis with machine learning for nonlinear wave phenomena. The findings have implications for soliton dynamics in Josephson-junction-based quantum computing and broader nonlinear wave contexts where accurate initialization is challenging.

Abstract

The sine-Gordon equation is a fundamental nonlinear partial differential equation that governs soliton dynamics and phase evolution in a variety of physical systems, including Josephson junctions and superconducting circuits. In this study, we analyze the effects of external perturbations such as damping and driving forces on the stability of soliton solutions. Using a rigorous Hilbert space framework, we establish well-posedness and derive regularity results for the perturbed equation. In particular, we provide sufficient conditions for the boundedness of the perturbation function, which plays a crucial role in determining the persistence of the soliton structure. Furthermore, we propose a neural network-based approach for solving the inverse problem of estimating unknown initial conditions. By training a data-driven model on simulated PDE solutions, we demonstrate that the network can accurately recover the initial states from limited or noisy observations. Numerical simulations validate the theoretical results and highlight the potential of combining mathematical analysis with machine learning techniques to study nonlinear wave phenomena. This approach offers valuable insights into soliton behavior and has potential applications in the design of quantum computing systems based on Josephson junctions.

Perturbation Analysis and Neural Network-Based Initial Condition Estimation for the Sine-Gordon Equation

TL;DR

This work addresses the stability of soliton solutions in the sine-Gordon equation under external perturbations by establishing a rigorous Hilbert-space variational framework that yields well-posedness and energy-based bounds for perturbations. It decomposes solutions around a soliton profile and derives the perturbation equation, proving existence, uniqueness, and regularity of weak solutions via coercive bilinear forms and Lipschitz nonlinearities. Complementing the analysis, the authors introduce a fully data-driven neural-network approach to estimate unknown initial conditions from limited observations, using synthetic forward-PDE data to train a network that maps (space, frequency, time) to initial-state estimates. Numerical experiments validate the perturbation theory and demonstrate effective initial-condition recovery even with noisy and sparse data, highlighting the potential of combining mathematical analysis with machine learning for nonlinear wave phenomena. The findings have implications for soliton dynamics in Josephson-junction-based quantum computing and broader nonlinear wave contexts where accurate initialization is challenging.

Abstract

The sine-Gordon equation is a fundamental nonlinear partial differential equation that governs soliton dynamics and phase evolution in a variety of physical systems, including Josephson junctions and superconducting circuits. In this study, we analyze the effects of external perturbations such as damping and driving forces on the stability of soliton solutions. Using a rigorous Hilbert space framework, we establish well-posedness and derive regularity results for the perturbed equation. In particular, we provide sufficient conditions for the boundedness of the perturbation function, which plays a crucial role in determining the persistence of the soliton structure. Furthermore, we propose a neural network-based approach for solving the inverse problem of estimating unknown initial conditions. By training a data-driven model on simulated PDE solutions, we demonstrate that the network can accurately recover the initial states from limited or noisy observations. Numerical simulations validate the theoretical results and highlight the potential of combining mathematical analysis with machine learning techniques to study nonlinear wave phenomena. This approach offers valuable insights into soliton behavior and has potential applications in the design of quantum computing systems based on Josephson junctions.

Paper Structure

This paper contains 5 sections, 7 theorems, 74 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Perturbation Analysis
  3. Numerical solutions
  4. Neural Network-Based Estimation of Initial Conditions
  5. Conclusion

Key Result

Lemma 2.1

Suppose that $u\in W(0,T)$, and $u"+\mathcal{A}u\in L^2(0,T;H)$. Then, after a modification on a set of measure zero, $u\in C([0,T];V)$, $u'\in C([0,T];H)$ and, in the sense of distributions on $(0,T)$ one has

Figures (5)

  • Figure 1: Graphs of the original sine-G equation
  • Figure 2: Graphs of the perturbed sine-G equation
  • Figure 3: Graphs of the linearized sine-G equation
  • Figure 4: Graphs of the perturbed sine-G equation
  • Figure 5: The estimated initial conditions $u_0(x)$ and $v_0(x)$obtained from data sampled at $N(t) = 3$ time points, under a noise level of 0.05.

Theorems & Definitions (12)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.1
  • proof
  • Lemma 2.4
  • Theorem 2.2
  • ...and 2 more