[Paper Review] The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem

Anne Boutet de Monvel, Iryna Karpenko+1 more|arXiv (Cornell University)|Nov 26, 2020

Nonlinear Waves and Solitons27 references5 citations

TL;DR

This paper studies the large-time asymptotics of the modified Camassa-Holm (mCH) equation on a nonzero background using the nonlinear steepest descent method applied to a Riemann-Hilbert (RH) formulation. It derives explicit asymptotic expressions for the solution in the solitonless case, showing that the deviation from the background decays as $ t^{-1/2} $ and exhibits modulated, decaying trigonometric oscillations in the sectors $ \frac{3}{4} < \frac{x}{t} < 1 $ and $ 1 < \frac{x}{t} < 3 $, with phase and amplitude depending on the ratio $ \frac{x}{t} $. The results are uniform in compact subsectors and involve logarithmic corrections and spectral data from the scattering problem.

ABSTRACT

This paper deals with the Cauchy problem for the modified Camassa-Holm (mCH) equation \begin{alignat*}{4} &amp;m_t+\left((u^2-u_x^2)m ight)_x=0,&amp;\quad&amp;m:= u-u_{xx},&amp;\quad&amp;t&gt;0,&amp;\;&amp;-\infty0$), where the leading asymptotic term of the deviation of the solution from the background is nontrivial: this term is given by modulated (with parameters depending on $\frac{x}{t}$), decaying (as $t^{-1/2}$) trigonometric oscillations.

Motivation & Objective

• To analyze the long-time behavior of solutions to the modified Camassa-Holm (mCH) equation with initial data approaching a nonzero constant at spatial infinity.
• To extend the Riemann-Hilbert formalism developed previously to the large-time asymptotic regime for the mCH equation on a nonzero background.
• To apply the nonlinear steepest descent method to the RH problem to derive explicit asymptotic expressions for the solution in the solitonless case.
• To characterize the structure of the leading-order asymptotic term in the sectors $ \frac{3}{4} < \frac{x}{t} < 1 $ and $ 1 < \frac{x}{t} < 3 $, where the deviation from the background is nontrivial.

Proposed method

• The solution is represented via a Riemann-Hilbert (RH) factorization problem derived from the Lax pair of the mCH equation, enabling spectral analysis in the inverse scattering framework.
• The nonlinear steepest descent method is applied to the RH problem to deform contours and isolate contributions from stationary phase points in the spectral parameter $ \mu $.
• The asymptotic analysis focuses on the solitonless case, where the reflection coefficient $ r(\mu) $ is analytic and decaying, and the spectral data are encoded in the jump matrix of the RH problem.
• The method involves identifying stationary points of the phase function $ \theta(\mu, \xi) $, with $ \xi = \frac{x}{t} $, and performing local parametrix constructions near these points to extract leading-order behavior.
• The resulting asymptotics are expressed in terms of oscillatory terms with amplitude and frequency modulated by $ \xi = \frac{x}{t} $, including logarithmic corrections in the phase.
• The final asymptotic formula is derived through a transformation back to the physical $ (x,t) $-variables, incorporating spectral data and reflection coefficient information.

Experimental results

Research questions

• RQ1 How does the solution of the mCH equation behave as $ t \to \infty $ when the initial data approach a nonzero constant at spatial infinity?
• RQ2 What is the structure of the leading-order asymptotic term in the solitonless case for $ \frac{x}{t} \in \left(\frac{3}{4}, 1\right) \cup \left(1, 3\right) $?
• RQ3 How do the amplitude, frequency, and phase of the oscillations depend on the ratio $ \frac{x}{t} $ in the large-time regime?
• RQ4 What role does the reflection coefficient $ r(\mu) $ play in determining the asymptotic behavior, particularly in the absence of solitons?

Key findings

• In the sector $ \frac{3}{4} < \frac{x}{t} < 1 $, the solution asymptotically approaches $ u(x,t) = 1 + \sum_{j=0,1} \frac{C_1^{(j)}(\zeta-1)}{\sqrt{t}} \cos\left\{ C_2^{(j)}(\zeta-1)t + C_3^{(j)}(\zeta-1)\ln t + \tilde{C}_4^{(j)}(\zeta-1) \right\} + o(t^{-1/2}) $, with uniform error in compact subsectors.
• The amplitude coefficients $ C_1^{(j)}(\zeta-1) $ are proportional to $ \left( \frac{8h_j\kappa_j}{|3 - 4\kappa_j^2|} \right)^{1/2} $, where $ h_j = -\frac{1}{2\pi} \ln(1 - |r(\mu_j)|^2) $.
• The frequency $ C_2^{(j)}(\zeta-1) $ is given by $ \frac{(-1)^j 32\kappa_j^3}{(1 + 4\kappa_j^2)^2} $, with $ \kappa_j $ depending on $ \zeta $ via $ \kappa_j(\zeta) = \left( \frac{\sqrt{1+4\zeta} \pm (1+\zeta)}{4\zeta} \right)^{1/2} $.
• The phase correction $ \tilde{C}_4^{(j)}(\zeta-1) $ includes logarithmic terms from the reflection coefficient and the gamma function, reflecting the influence of spectral data on the oscillation phase.
• In the sector $ 1 < \frac{x}{t} < 3 $, the asymptotic structure is identical in form, with the same functional dependence on $ \zeta $, confirming the modulated oscillatory decay.
• The error term is uniform in any compact subsector $ \frac{3}{4} + \varepsilon < \zeta < 1 - \varepsilon $, ensuring robustness of the asymptotic description.