Universality Near the Gradient Catastrophe Point in the Semiclassical Sine‐Gordon Equation

Abstract

We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava, and Klein, we found that in a O. 4=5 / neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painlevé I tritronquée solution. A linear map can be explicitly made from the tritronquée solution to this neighborhood. Under this map: away from the tritronquée poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquée solution; localized defects appear at locations mapped from the poles of the tritronquée solution; the defects are proved universally to be a two-parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann-Hilbert problems, substantially generalizing [5] to establish universality beyond the context of solutions of a single equation. © 2021 Wiley Periodicals LLC. B.-Y. LU AND P. D. MILLER 3.2. Expression for g and determination of 3.3. The modulated librational wave region 3.4. Proof of Proposition 1.3 3.4.1. Asymptotically trivial jump conditions for O.w/ 3.4.2. Construction of a parametrix for O.w/ 3.4.3. Error analysis 3.5. The boundary of the modulated librational wave region and points of gradient catastrophe 3.6. Symmetries for x h 0 4. Proof of Theorem 1.6 4.1. Simplification of jump conditions near w h 4.2. Modified g-function near the gradient catastrophe 4.2.1. A more singular ansatz for g 4.2.2. Redetermination of .x; t/ and £ .x; t/ via construction of a conformal mapping W U 3 C 4.2.3. Values of the partial derivatives of s.x; t/ at the gradient catastrophe 4.3. The phase-linearized outer parametrix 4.4. Use of the modified g-function and inner parametrix for O.w/ near w h 4.5. Characterization of T.s / 4.5.1. Riemann-Hilbert Problem 4.1 and the Painlevé-I equation 4.5.2. Identification of the real tritronquée solution 4.5.3. Laurent expansions and additional identities 4.6. Global parametrix for O.w/, error analysis, and expressions for the potentials 4.6.1. Global parametrix definition 4.6.2. Error analysis 4.6.3. Asymptotic formulae for cos. 1 2 u N .x; t// and sin. 1 2 u N .x; t// 5. Proof of Theorem 1.10 5.1. Modifying the inner parametrix 5.2. Initial global parametrix, corresponding error matrix, and parametrix for the error 5.3. Corrected error matrix, small-norm problem, and formulae for the potentials 5.4. Differential equations satisfied by the leading terms 5.4.1. Relating C and S to sine-Gordon 5.4.2. E.w/ as a Darboux transformation matrix for K.ws X; T /

Localized structures on librational and rotational travelling waves in the sine-Gordon equation

“…The wave patterns are very similar to the solutions from appendix D of [3] or the rogue wave observed on figure 1 a (left). This confirms that rogue waves on a background of librational waves model defects in the fluxon condensate obtained in [3] from the Riemann–Hilbert problem.

Figure 2Surface plots of sin ( u ) versus ( x , t ) for rogue waves on the background of librational waves for k=sin⁡false(π6false) ( a ) and k=sin⁡false(11π24false) ( b ).

…”

“…1 in [20]. Another representation of the same solutions follows from the Riemann–Hilbert problem, as in appendix C of [3].…”

“…The sine-Gordon equation in the semi-classical limit can be written in the form: ϵ2uTT−ϵ2uXX+sin⁡false(ufalse)=0, where the subscripts denote partial derivatives of u = u ( X , T ) and the parameter ϵ is small. By using the initial data with zero displacement and large velocity, u ( X , 0) = 0 and ϵu T ( X , 0) = G ( X ), the authors of [1–3] studied the sequence falsefalse{ϵNfalsefalse}N∈N with ϵ N → 0 as N → ∞, where ϵ N is defined from the N -soliton (reflectionless) potential associated with the ϵ -independent velocity profile G ( X ). The sequence of solutions was termed as the fluxon condensate .…”

“…The sequence of solutions was termed as the fluxon condensate . The regime of rotational waves with ||G|false|Lnormal∞>2 was studied in [1], whereas the regime of librational waves with ||G|false|Lnormal∞<2 was studied recently in [3], the classification corresponds to the dynamics of a pendulum with an angle θ = θ ( t ) satisfying θ″false(tfalse)+sin⁡false(θfalse(tfalse)false)=0. …”

Higher order Airy and Painlevé asymptotics for the mKdV hierarchy

“…As we will show later, the role played by the Ablowitz-Segur solution and Airy function in the mKdV equation will be replaced by their higher-order generalizations. In the literatures, we note that the Painlevé transcendents and their higher-order analogues are crucial in asymptotic analysis of many integrable nonlinear differential equations, as can be seen from their appearances in the focusing nonlinear Schrödinger equation [4,5], in critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systems [10], in the Camassa-Holm equation [7], in the Sasa-Satsuma equation [24], in an extended mKdV equation [30,31] and in the sine-Gordon equation [32]. The higher order asymptotics in similarity region for other integrable equations can be found in [26,37].…”