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Functional relations in renormalization group methods for a class of ordinary differential equations

Atsuo Kuniba, Rurika Motohashi

Abstract

We develop a renormalization group (RG)-based perturbation scheme for a class of ordinary differential equations, including first-order systems with semisimple or nilpotent linear parts, as well as scalar higher-order equations. The key observation is that the secular coefficients arising in naive perturbation theory satisfy an exact functional relation. This yields, in a unified manner, several fundamental features of the RG method: the renormalized amplitudes satisfy a closed functional relation with a group-like structure, the RG equation governing their slow dynamics is obtained directly, the absence of secular terms is ensured to all orders, and the relation between bare and renormalized amplitudes admits an explicit inversion. The results extend earlier ones for second-order scalar equations.

Functional relations in renormalization group methods for a class of ordinary differential equations

Abstract

We develop a renormalization group (RG)-based perturbation scheme for a class of ordinary differential equations, including first-order systems with semisimple or nilpotent linear parts, as well as scalar higher-order equations. The key observation is that the secular coefficients arising in naive perturbation theory satisfy an exact functional relation. This yields, in a unified manner, several fundamental features of the RG method: the renormalized amplitudes satisfy a closed functional relation with a group-like structure, the RG equation governing their slow dynamics is obtained directly, the absence of secular terms is ensured to all orders, and the relation between bare and renormalized amplitudes admits an explicit inversion. The results extend earlier ones for second-order scalar equations.

Paper Structure

This paper contains 15 sections, 11 theorems, 132 equations, 3 figures.

Table of Contents

  1. Introduction
  2. First-Order ODE System with a Semisimple Coefficient Matrix
  3. Naive perturbation
  4. Functional equation for secular coefficients
  5. Autonomous case
  6. Examples
  7. First-Order ODE System with a Nilpotent Coefficient Matrix
  8. Naive perturbation
  9. Functional equation for secular coefficients
  10. Scalar $N$th-Order ODE
  11. Naive perturbation
  12. Functional equation for secular coefficients
  13. A difference equation
  14. Conclusion
  15. Numerical plots for Example \ref{['ex:CD']}

Key Result

Lemma 1

Let $s$ and $A=(A_1,\ldots, A_n)$ be arbitrary parameters. The formal power series $y=y(\varepsilon,t)$ of the form (yf3) that satisfies the conditions (i), (ii), and (iii) below is unique, and coincides with $Y(\varepsilon,t,A)$ in (Yv3). $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: The real part (red) and the imaginary part (blue) of $y_1(t)=y^*_2(t)$.
  • Figure 2: Numerical solutions of the RG equation \ref{['rteq']}.
  • Figure 3: Red: $\mathrm{Re}(y_1(t))$ obtained by direct numerical integration of \ref{['ex1:eq']} as given in Figure \ref{['fig:1']}. Black: $\mathrm{Re}(Y_1(t))$ obtained from the renormalized expansion \ref{['ex1:yy']}, using $R(t)$ and $\theta(t)$ in Figure \ref{['fig:RT']}.

Theorems & Definitions (21)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • Example 5
  • Example 6
  • Lemma 7
  • Lemma 8
  • ...and 11 more