[Paper Review] Integrable viscous conservation laws

Alessandro Arsie, Paolo Lorenzoni+1 more|arXiv (Cornell University)|Jan 5, 2013

Nonlinear Waves and Solitons7 references3 citations

TL;DR

This paper extends Dubrovin's perturbative approach to non-Hamiltonian, viscous scalar conservation laws by introducing a viscous central invariant that parameterizes integrable deformations. It proves integrability of the resulting equations via Miura transformations mapping them to the heat and Klein-Gordon hierarchies, and derives the viscous Painlevé I2 equation governing gradient catastrophe behavior at critical points.

ABSTRACT

We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamiltonian integrable systems. We write explicitly the negative and positive hierarchy associated with this equation and prove the integrability showing that they can be mapped respectively into the heat hierarchy and its negative counterpart, named the Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial data is also proven. We show how transport equations can be used to effectively construct asymptotic solutions via an extension of the quasi-Miura map that preserves the initial datum. The method is alternative to the method of the string equation for Hamiltonian conservation laws and naturally extends to the viscous case. Using these tools we derive the viscous analog of the Painlevé I2 equation that describes the universal behaviour of the solution at the critical point of gradient catastrophe.

Motivation & Objective

• To extend Dubrovin's perturbative method for Hamiltonian integrable systems to non-Hamiltonian, viscous scalar conservation laws.
• To classify integrable viscous conservation laws up to fifth order in the deformation parameter using Miura transformations preserving the equation form.
• To identify a single functional parameter, the viscous central invariant, as the unique invariant governing all such deformations.
• To prove integrability of the resulting equations by showing equivalence to known integrable hierarchies (heat and Klein-Gordon).
• To derive the viscous analog of the Painlevé I2 equation describing universal behavior at gradient catastrophe points.

Proposed method

• Adopt a perturbative expansion of the conservation law in a small parameter ε, with viscous corrections involving a(u)u_x and higher-order terms.
• Introduce the viscous central invariant a(u) as the sole functional parameter determining the integrable hierarchy.
• Use Miura transformations preserving the form of the equation to eliminate inessential functional parameters and classify deformations.
• Map the positive and negative hierarchies of the viscous Camassa-Holm equation to the heat and Klein-Gordon hierarchies via Miura-type transformations.
• Apply transport equations and an extended quasi-Miura map to construct asymptotic solutions preserving initial data.
• Derive the viscous Painlevé I2 equation via scaling and limiting analysis near the gradient catastrophe point, using rescaling and Cole-Hopf transformation to linearize the equation.

Experimental results

Research questions

• RQ1 Can Dubrovin’s perturbative approach be generalized to non-Hamiltonian, viscous scalar conservation laws?
• RQ2 Is the space of integrable viscous conservation laws parameterized by a single functional invariant, and what is its physical and geometric meaning?
• RQ3 What is the integrability structure of the viscous Camassa-Holm equation, and how does it relate to known integrable hierarchies?
• RQ4 How does the solution behave near the gradient catastrophe point, and what universal equation describes this behavior?
• RQ5 Can asymptotic solutions be effectively constructed using transport equations and extended Miura maps in the viscous setting?

Key findings

• All integrable viscous conservation laws are uniquely parameterized by the viscous central invariant a(u), with constant a(u) corresponding to the Burgers hierarchy.
• The case of linear a(u) yields a viscous analog of the Camassa-Holm equation, which is integrable and maps to the heat and Klein-Gordon hierarchies via Miura transformations.
• The negative and positive hierarchies of the viscous Camassa-Holm equation are isomorphic to the negative and positive heat hierarchies, respectively.
• A local well-posedness theorem is established for periodic initial data in the viscous case.
• Near the gradient catastrophe point, the solution behavior is governed by the viscous Painlevé I2 equation: $ U_{XX} + 3UU_X + U^3 - UT = X $, derived via scaling and limiting analysis.
• The viscous Painlevé I2 equation admits a linearization via Cole-Hopf transformation into $ w_{XXX} - T w_X = X w $, with general solution in terms of generalized hypergeometric functions $_0F_2$.