A singular limit problem for the Ibragimov-Shabat equation

We consider the Ibragimov-Shabat equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L^p setting


Introduction
Bäcklund transformations have been useful in the calculation of soliton solutions of certain nonlinear evolution equations of physical significance [7,18,23,24] restricted to one space variable x and a time coordinate t. The classical treatment of the surface transformations, which provide the origin of Bäcklund theory, was developed in [9]. Bäcklund transformations are local geometric transformations, which construct from a given surface of constant Gaussian curvature −1 a two parameter family of such surfaces. To find such transformations, one needs to solve a system of compatible ordinary differential equations [8].
In [12,13], the authors used the notion of differential equation for a function u(t, x) that describes a pseudo-spherical surface, and they derived some Bäcklund transformations for nonlinear evolution equations which are the integrability condition sl(2, R)− valued linear problems [11,10,15,16,24].
In [17], the authors had derived some Bäcklund transformations for nonlinear evolution equations of the AKNS class. These transformations explicitly express the new solutions in terms of the known solutions of the nonlinear evolution equations and corresponding wave functions which are solutions of the associated Ablowitz-Kaup-Newell-Segur (AKNS) system [1,26].
In [14], the authors used Bäcklund transformations derived in [12,13] in the construction of exact soliton solutions for some nonlinear evolution equations describing pseudospherical surfaces which are beyond the AKNS class. In particular, they analyzed the following equation [2]: where g(u) is any solution of the linear ordinary differential equation (1.2) g ′′ (u) + µg(u) = θ, µ, θ ∈ R.
In [22], Rabelo proved that the system of the equations (1.1) and (1.2) describes pseudospherical surfaces and possesses a zero-curvature representation with a parameter.
One more equation, that describes pseudo-spherical surface, is the following one [25]: which is the Ibraginov-Shabat equation. Following [5,6], we consider the following diffusive approximation of (1.3) We consider the initial value problem for (1.4), so we augment (1.4) with the initial condition on which we assume that We are interested in the no high frequency limit, i.e., we send β → 0 in (1.4). In this way, we pass from (1.4) to which is a scalar conservation law. We study the dispersion-diffusion limit for (1.4). Therefore, we fixe two small numbers 0 < ε, β < 1, and consider the following third order problem where u ε,β,0 is a C ∞ approximation of u 0 such that where C 0 is a constant independent on ε and β.
The main result of this paper is the following theorem.
The paper is organized in three sections. In Section 2, we prove some a priori estimates, while in Section 3 we prove Theorem 1.1.

A priori Estimates
This section is devoted to some a priori estimates on u ε,β . We denote with C 0 the constants which depend only on the initial data, and with C(T ) the constants which depend also on T .

Proof of Theorem 1.1
In this section, we prove Theorem 1.1. The following technical lemma is needed [20].
We conclude by proving that u is a distributional solution of (1.4). Let φ ∈ C ∞ (R 2 ) be a test function with compact support. We have to prove that We have that Let us show that Due to (1.10), (2.1) and (2.4), We prove that Thanks to (2.5) and the Hölder inequality, that is (3.5). Therefore, (3.3) follows from (1.9), (3.1), (3.4) and (3.5).