The short pulse equation by a Riemann–Hilbert approach

Abstract

We explore a new connection between Seiberg-Witten theory and quantum statistical systems by relating the dual partition function of SU (2) Super Yang-Mills theory in a self-dual Ω-background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painlevé III 3 τ function. In addition we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a mathematical proof of the conjecture for the local P 1 × P 1 geometry.

Initial value problem and soliton solutions of the single-cycle short pulse equation via the Riemann-Hilbert approach

“…First, different from [11] in which the two-soliton solutions of the SCP equation were obtained from the known soliton solutions of the sine-Gordon equation, in this paper, the two-soliton solutions of the SCP equation are obtained directly by the RH problem. Second, in the case when the scattering coefficient a(k) has a pair of negative conjugate zeros, unlike the two-soliton solutions of the SP equation given in [13], the two-soliton solutions we have obtained has the property: the partial derivative of the spatial variable ( ) x y t , to the variable y is always greater than or equal to 0. This means that the obtained twosoliton solutions do not appear to blow up about the space variable x.…”

“…And we have proved that when one of the zeros approaches another, the limit of the two-soliton solutions is one-soliton solutions. The previous literatures [11,[13][14][15][16][17][18][19][20][21] and so on have not discussed this situation.…”

“…Generalizations of the SP equation were studied in [10][11][12]. [13] solved the solution and studied the long-term asymptotic of the short pulse equation under the given initial value by a Riemann-Hilbert approach. Recently, [14][15][16][17][18][19][20][21] studied other nonlinear differential equations via the RH approach.…”

“…According to theorem 4.1, we get Finally, we obtain the two-soliton solutions of the initial value problem sin sinh cos cosh sin cosh , 7 2 2 2 1 2 1 2 2 2 1 2 2 2 Solving the partial derivative of x to y, we get From the above discussions, we know that the partial derivative of x to y is always greater than or equal to 0, which is different from the two-soliton solutions of the SP equation given in [13]. Equation (74) means that the two-soliton solution we have obtained will not blow up about space variable x.…”

Long-time asymptotics for the Hirota equation on the half-line

“…The nonlinear steepest descent method was first introduced in 1993 by Deift and Zhou [8], it turn out to be very successful for analysing the long-time asymptotics of initial-value problems for a large range of nonlinear integrable evolution equations in a rigorous and transparent form. Numerous new significant results about the asymptotics theory of initialvalue problems for different completely integrable nonlinear equations were obtained based on the analysis of the corresponding Riemann-Hilbert (RH) problems [3,[5][6][7]18,22,23]. After that, Fokas announced a new unified approach [11,12] to construct the matrix RH problems for the analysis of initial-boundary value (IBV) problems for linear and nonlinear integrable systems.…”