[Paper Review] A continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system

Katrin Grunert, Helge Holden+1 more|arXiv (Cornell University)|Feb 5, 2014

Nonlinear Waves and Solitons40 references30 citations

TL;DR

This paper introduces α-dissipative solutions as a continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system. By modifying the energy measure at wave breaking in Lagrangian coordinates using a parameter α ∈ [0,1], the method unifies global weak solutions with controlled dissipation, where α = 0 gives full energy conservation and α = 1 gives full dissipation, with intermediate values allowing fractional energy loss. The key contribution is a continuous family of solutions bridging the long-standing dichotomy in wave-breaking behavior.

ABSTRACT

We introduce a novel solution concept, denoted $\alpha$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa-Holm system on the line with vanishing asymptotics. All the $\alpha$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

Motivation & Objective

• To resolve the long-standing dichotomy between conservative and dissipative solutions in the two-component Camassa-Holm system.
• To construct a continuous family of global weak solutions parameterized by α ∈ [0,1] that interpolate between full energy conservation (α=0) and full dissipation (α=1).
• To unify the treatment of wave-breaking dynamics by introducing a single framework that handles both conservative and dissipative cases through a single parameter α.
• To ensure global existence and well-posedness of solutions by modifying the energy measure h in Lagrangian variables at collision times.

Proposed method

• Transform the 2CH system into Lagrangian coordinates using characteristics y(t,ξ), velocity U(t,ξ), and energy-related variables h and ¯h.
• Define α-dissipative solutions by discontinuously resetting the effective energy measure ¯h at each wave-breaking time τj(ξ) via ¯h(τj(ξ),ξ) = (1−α) limₜ↑τj(ξ) ¯h(t,ξ), where α controls dissipation.
• Maintain continuity of the full energy variable h across collisions while allowing ¯h to jump, ensuring consistent energy tracking.
• Use the system of ODEs in Lagrangian variables: yt = U, Ut = −Q, yt,ξ = Uξ, Ut,ξ = ½¯h + (U²−P)yξ, ht = 2(U²−P)Uξ, with P and Q defined via integral operators.
• Construct global solutions via an iteration argument in a carefully restricted initial data class G, ensuring existence and continuity in time.
• Map solutions back to Eulerian coordinates using pushforward measures µ = y#(¯h dξ) and ν = y#(h dξ), recovering weak solutions in H¹(R) × L²(R).

Experimental results

Research questions

• RQ1 Can a continuous family of solutions be constructed that interpolates between conservative and dissipative behaviors in the two-component Camassa-Holm system?
• RQ2 How can wave-breaking be consistently treated in a unified framework that allows for partial energy dissipation?
• RQ3 What is the role of the Lagrangian variable ¯h in tracking the effective energy after collisions, and how does α control its evolution?
• RQ4 How does the choice of α affect the long-time behavior and structure of the solution, particularly in the peakon-antipeakon case?
• RQ5 Can global weak solutions be constructed for general H¹ × L² initial data using this α-dissipative framework?

Key findings

• For α ∈ [0,1], α-dissipative solutions are global weak solutions of the 2CH system in Eulerian coordinates, with u(t) ∈ H¹(R) and µ(t) ∈ L²(R) for all t > 0.
• At wave breaking, the energy measure ¯h is reduced by a factor of (1−α), allowing continuous interpolation between full conservation (α=0) and full dissipation (α=1).
• In the peakon-antipeakon case, after collision at time t₀, the solution for t > t₀ corresponds to a new peakon-antipeakon solution with energy ˜E = √(1−α) E.
• For α = 1 (fully dissipative), the solution becomes identically zero after collision, with µ(t) = 0 and ν(t) = E²δ₀ for t > t₀.
• The solution structure for t > t₀ is explicitly computed: u(t,x) = sgn(x) ˜A(t)e⁻|x| for |x| > ˜γ(t), and u(t,x) = ˜B(t) sinh(x) for |x| ≤ ˜γ(t), with ˜A(t) = ˜E/(2 sinh(˜E/2 (t−t₀))), ˜B(t) = ˜E sinh⁻¹(˜E/2 (t−t₀)), and ˜γ(t) = ln(cosh(˜E/2 (t−t₀))).
• The energy measure ν(t) in Eulerian coordinates includes an additional term αE²/4 tanh⁻²(˜E/4 (t−t₀)) (1−tanh²(x/2)) for |x| ≤ ˜γ(t), reflecting the non-dissipated portion of the energy.