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Long-Time Asymptotics for the Camassa-Holm Equation
Anne Boutet de Monvel, Aleksey Kostenko, Dmitry Shepelsky, Gerald Teschl
TL;DR
The paper addresses the long-time asymptotic behavior of decaying solutions of the Camassa–Holm equation. It applies nonlinear steepest descent through a vector Riemann–Hilbert formulation derived from scattering theory, including soliton effects and associated error analysis. The resulting framework gives the stated asymptotic solution description, while specific deformation cases require further analysis and the assumptions include w(x,0)>0.
Problem
The paper studies the long-time asymptotics of the Camassa–Holm equation for decaying initial data.
Method
The authors apply nonlinear steepest descent to a vector Riemann–Hilbert problem derived directly from scattering theory.
Results
The analysis gives long-time asymptotic formulas across four sectors and includes soliton effects in decaying oscillatory regions.
Takeaways & Limitations
The paper supplies complete asymptotic proofs beyond the solitonless oscillatory case and relates error estimates to initial-condition decay.
Takeaways & Limitations
The analysis assumes w(x,0)>0, and in cases (ii) and (iii) the nondecaying jump requires continuation in the subsequent section.
Abstract
from arXiv · showhide
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Camassa-Holm equation for decaying initial data, completing previous results by A. Boutet de Monvel and D. Shepelsky.
1. Introduction
The paper studies long-time asymptotics for rapidly decaying solutions of the Camassa–Holm equation using nonlinear steepest descent and a directly derived vector Riemann–Hilbert problem. It extends prior analysis by incorporating soliton effects in oscillatory sectors, complete proofs, and error estimates tied to initial-data decay.
- Equation and setting: The Camassa–Holm equation models shallow water waves, with u representing fluid velocity and κ related to the critical shallow-water wave speed.The equation is completely integrable and can be treated by inverse scattering.
- Equation and setting: The paper analyzes long-time asymptotics for real-valued classical solutions with rapidly decaying initial data and w(x,0)>0.The stated solution class is global for t>0 with w(x,t)>0.
- Approach: The authors use nonlinear steepest descent, beginning with a representation of the nonlinear solution through an associated Riemann–Hilbert problem.This approach follows the Deift–Zhou method and its earlier inverse-scattering developments.
- Approach: The work simplifies the prior approach by deriving the vector Riemann–Hilbert problem directly from scattering theory for the underlying Sturm–Liouville operator.It also addresses uniqueness for the vector problem in detail.
- Contributions: The analysis provides complete proofs that include soliton effects in decaying oscillatory sectors and error estimates in terms of initial-condition decay.Earlier oscillatory formulas applied only in the solitonless case.
2. Main result
The main-result section begins by recalling the prerequisites needed to state the paper’s asymptotic theorem.
- 2. Main result: The authors first recall several preliminary facts before stating the main result.This prepares the formulation of the subsequent theorem.
- 2. Main result: The recalled material supplies background for the paper’s statement of long-time solution asymptotics.The passage introduces the preparation immediately before the main result.
- 2. Main result: The section uses these preliminaries to organize the presentation of the main theorem.The passage explicitly places the recall before the result statement.
Scattering data.
The paper characterizes scattering data and organizes Camassa–Holm long-time behavior into four sectors of the (x,t)-half-plane. These sectors include solitons, modulated oscillations, and rapid decay, with Painlevé-described transition zones.
- The scattering data comprise a reflection coefficient and norming constants associated with the continuous spectrum and eigenvalues.
- Four sectors have qualitatively different leading asymptotics: a solitonic sector, two slowly decaying oscillatory sectors, and a fast-decay sector.
- Pure soliton solutions asymptotically split into single solitons with associated phase shifts.
- The oscillatory sectors match at x = 0, while transitions near x/κt = 2 and x/κt = −1/4 involve Painlevé transcendents.
- Unlike KdV, the Camassa–Holm asymptotic form is implicit and includes additional phase-shift terms.
3. The Inverse scattering transform and the Riemann–Hilbert problem
The inverse scattering transform uses a Liouville-transformed Sturm–Liouville problem and its scattering data to encode Camassa–Holm solutions. Jost solutions, transmission and reflection coefficients, and norming constants provide the ingredients for the inverse formulation.
- The Sturm–Liouville operator is mapped by a unitary Liouville transform to a self-adjoint Schrödinger operator.
- The decaying-data assumption implies q(y,t) belongs to L1(R,(1+|y|)^(l+1)dy).
- Two Jost solutions are analytic in the upper half-plane and continuous on its closure.
- The transmission coefficient is meromorphic in the upper half-plane with simple poles at iκ1,...,iκN.
- The time evolution of the reflection coefficient and norming constants supplies the evolving scattering data for inverse reconstruction.
Vector Riemann–Hilbert problem.
The paper constructs a vector Riemann–Hilbert problem directly from scattering theory. Its solution is specified by analyticity, jumps on the real axis, pole conditions, symmetry, and normalization, with uniqueness established for the vector formulation.
- A vector Riemann–Hilbert problem is derived directly from the scattering theory of the differential operator.
- The jump condition is posed on the real k-axis, oriented from negative to positive.
- The problem requires meromorphicity away from the real axis, with simple poles at ±iκj.
- Symmetry and normalization complement the jump and pole conditions in defining the Riemann–Hilbert problem.
- The pole condition at iκj suffices because the condition at −iκj follows by symmetry, and uniqueness is addressed for the vector problem.
Regular Riemann–Hilbert problem.
The meromorphic Riemann–Hilbert problem is converted into a holomorphic regular problem by replacing poles with jumps on small circles. This augmented contour is then prepared for nonlinear steepest descent.
- Pole conditions are rewritten as jump conditions, converting the meromorphic problem into a holomorphic Riemann–Hilbert problem.
- Small disjoint circles around ±iκj carry the additional jump conditions replacing the poles.
- The regular problem retains the real-axis jump, symmetry, and normalization conditions.
Uniqueness result.
The paper establishes uniqueness for the vector Riemann–Hilbert problem by analyzing possible zeros and using the symmetry condition. It also gives the unique one-soliton solution when the reflection coefficient vanishes.
- Uniqueness result.: The function m(k,x,t) is therefore the only solution of the vector Riemann–Hilbert problem.This conclusion is stated as Corollary 3.9.
- Uniqueness result.: The symmetry condition is essential because, without it, a family m(k)+ϑn(k) of solutions can occur.The paper notes that these solutions violate symmetry unless ϑ=0.
- Uniqueness result.: For a reflectionless one-soliton with one eigenvalue κ and norming constant γ(t), the unique solution is obtained from a meromorphic ansatz whose constants are fixed by pole conditions and normalization.The real-axis jump disappears, and the symmetry condition determines the solution form.
- Uniqueness result.: Any problematic zero can only occur at k1=0, and at least one component has a simple zero there.The argument excludes other locations after considering band edges and eigenvalues.
- Uniqueness result.: At an eigenvalue k1=iκj, the Jost solutions have at most simple zeros, while the transmission coefficient has a simple pole, preventing simultaneous vanishing.This contradiction rules out the corresponding nontrivial vanishing solution.
- Uniqueness result.: At k1=0, the Wronskian has a simple zero, so both Jost functions cannot have zeros of order greater than one.The proof uses the resulting contradiction between the Wronskian order and the orders of the two functions.
4. Conjugation and Deformation
The paper conjugates and deforms the Riemann–Hilbert problem so oscillatory jumps decay away from stationary points, with contour geometry determined by four c-regimes.
- Conjugation: The conjugation preserves the vector Riemann–Hilbert symmetry and normalization when the scalar factor satisfies d(−k)=d(k)^−1 and tends to one at infinity.This allows the transformed problem to retain the original symmetry structure.
- Contour deformation: Analytic continuation of R(k) permits contour deformation that separates exp(tΦ) and exp(−tΦ) into regions where they decay exponentially.Without analytic continuation, the reflection coefficient is split into an analytic part and a small remainder.
- Phase geometry: The phase has stationary points at ±k0 and ±k1, and the signs of Re(Φ) determine the deformation in four c-regimes.The regimes are c>2, 0<c<2, −1/4<c<0, and c<−1/4.
- Conjugation: The partial transmission coefficient T(k,c) is meromorphic off Σ(c), with simple poles at iκj and simple zeros at −iκj for κ0<κj.It also satisfies the symmetry T(−k,c)=T(k,c)^−1.
- Pole contributions: Near eigenvalue poles, all corresponding jumps are exponentially close to the identity except possibly one associated with κj=κ0, whose pole condition is retained.This isolates the potentially non-decaying pole contribution before the oscillatory analysis.
- Contour deformation: After deformation, the jump along the real axis disappears and the jumps on Σ± are exponentially close to the identity as t→∞.In the 0<c<2 and −1/4<c<0 cases, the contours are chosen according to the corresponding deformed geometries.
5. Reduction to a Riemann–Hilbert problem on a small cross
The oscillatory-region analysis localizes the Riemann–Hilbert problem near stationary phase points and solves each local problem by reducing it to a model problem on a cross.
- Local reduction: For −1/4<c<2, the jumps are exponentially close to the identity except in small neighborhoods of ±k0 and ±k1.The remaining analysis therefore focuses on these stationary phase neighborhoods.
- Local model: The problems on the small crosses Σc(kℓ) and Σc(−kℓ) are reduced to Theorem A.3, then combined through Theorem A.2 to solve the original problem.The opposite crosses are treated using an ansatz motivated by the vector problem's symmetry.
- Local model: A local coordinate change is introduced near each stationary point, together with the required behavior of the jump matrix and transmission factor.The local factors satisfy the regularity assumptions needed for the model analysis.
- Local solution: The local solution on Σc(kℓ) is obtained from the model problem using reflection-coefficient data at the stationary point.The construction also accounts for the conjugated problem near the corresponding negative stationary point.
Appendix A. Some results for Riemann–Hilbert problems
The appendix assumes a finite, smooth, transversely intersecting contour separated from the upper imaginary ray beyond a positive height.
- Contour assumptions: The contour Σ consists of finitely many smooth oriented finite curves with only finitely many transversal intersections.It is also assumed to remain a positive distance from {iy | y≥y0} for some y0>0.
A.1. Riemann–Hilbert problem for the soliton region.
The appendix supplies Riemann–Hilbert results for the soliton and oscillatory regions, including uniqueness, small-jump comparison, and a cross model for local analysis.
- Soliton-region problem: The soliton-region Riemann–Hilbert problem imposes meromorphicity, jumps, symmetry, and normalization conditions on m(k).The jump data must inherit symmetry under k↦−k.
- Soliton-region problem: When the jump matrix is close to the identity in L∞ and L2 norms, the problem has a unique large-time solution close to the one-soliton solution.The difference is O(ρ(t)) uniformly away from the contour and discrete poles, with ρ(t)→0.
- Oscillatory-region reduction: The oscillatory-region theorem reduces a global vector problem to local matrix problems near finitely many exceptional points when jumps are small away from those neighborhoods.The resulting error depends on the distance from k to the contour.
- Model problem on a cross: The local model is posed on a four-ray cross with jump matrices equal to the identity outside a fixed disk.The cross is oriented by increasing real part, and the model uses a phase with a quadratic stationary point at the origin.
- Model problem on a cross: The model solution has an explicit asymptotic coefficient involving r, Γ(iν), the phase, and ν=−(1/(2π))log(1−|r|^2).The coefficient is stated for |ζ|>ρ0.
- Model problem on a cross: The model error estimate is uniform in auxiliary parameters when r remains in a compact subset of D and the estimate constants are parameter-independent.This uniformity supports applying the local model across varying stationary-point data.