The short pulse equation (SP equation, or SPE)

was derived by Schäfer and Wayne [19] as a model equation for the propagation of ultra-short optical pulses in non-linear media. In this equation, u = u(x, t) is a real-valued function that represents the magnitude of the electric field. The short pulse equation is an alternative model to the non-linear Schrödinger (NLS) equation, the latter being used for describing the slow modulation of the amplitude of a weakly non-linear wave packet in a moving medium. NLS is used in non-linear optics with great success to describe slowly varying wave trains whose spectra are narrowly localized around the carrier frequency or to describe the propagation of sufficiently broad pulses. In the regime of ultra-short pulses where the width of optical pulse is in order of femtosecond, the SP equation is supposed to provide better approximation to the corresponding solution of the Maxwell equation while the NLS equation becomes less accurate. In [10], with the help of numerical simulations, it was shown that the SP equation can indeed be used to describe pulses with broad spectrum.

In [17, 18], it was shown that the SP equation is completely integrable, in the sense that it is the compatibility condition of a pair of linear, matrix-valued ordinary differential equations involving an external (spectral) parameter; such pair of equations is called the Lax pair. In the case of the SP equation, the associated Lax pair is as follows:

where U and V are 2 × 2 matrices dependent on the spectral parameter λ:

The Riemann–Hilbert approach to the study of solutions of the SP equation was presented in [8].

The modified short pulse (mSP) equation

was proposed by Sakovich [16], who studied integrable non-linear equations having the form

When ab=12, Equation 7 reduces to Equation 1 whereas the case ab=1 reduces to Equation 6, both cases being integrable. The mSP equation (6) was studied by Guo and Liu, who constructed soliton solutions by the Riemann–Hilbert method [14]. Matsuno [15] proposed the N- component generalization of Equation 6, which in the case N = 2 reads

Matsuno constructed the soliton solutions by solving the associated bilinear equations and constructed the local and non-local conservation laws of Equation 8.

Obviously, if v = u, then Equation 8 reduces to Equation 6. On the other hand, if v = ū, where the bar stands for the complex conjugation, the system (8) reduces [20] to

which will be called in what follows the modified focusing complex short pulse equation (mfcSP equation or mfcSPE). Notice that the reduction v = −ū gives rise to a defocusing version of Equation 9, having the minus sign at the place of the plus. In [20], some multiple smooth soliton, cuspon soliton, loop soliton, breather, and rogue wave solutions are constructed by N-fold Darboux transformation.

From the point of view of possible applications in optics, the mfcSP equation, being formulated for a complex-valued function, appears to be more informative: Similarly to the NLS equation, a complex-valued function can contain not only the information about the amplitude but also about the phase of the associated electromagnetic wave. On the other hand, the mfcSP equation is integrable: Its Lax pair is Equation 2, where [20]

Motivated by the above, in the present study, we develop a Riemann–Hilbert (RH) problem formalism for the inverse scattering transform to the initial value problem for the mfcSPE:

We assume that u₀(x) decays sufficiently fast at ±∞:

and we seek a solution u(x, t) that decays as x → ±∞ for all t > 0:

Notice that the RH approach for solving initial value problems for integrable non-linear PDE can be viewed as a version of the inverse scattering transform (IST) method for such problems, the more traditional realization of which is based on deriving and solving the Marchenko integral equation for the corresponding inverse problems, see, for example, [1] and references therein. Since the latter approach requires the representation of special solutions of the x-equation of the corresponding Lax pair in terms of so-called transformation operators, its application to cases where the dependence of the Lax equations on the spectral parameter is more involved (comparing, for example, with the case of the Korteweg-de Vries equation and its modified versions) is not straightforward because the very existence of the corresponding transformation operators is questionable. On the other hand, as we will show in the next section, the formalism of the RH problem allows us to establish an algorithmic procedure providing special solutions of the Lax pair equations with the necessary analytic properties.

In Section 2, we present a version of the Lax pair associated with the mfcSP equation, which is more convenient for controlling analytical properties of its special solutions, also known as the Jost solutions. They are then used in Section 3 to formulate a matrix Riemann–Hilbert problem suitable for solving the Cauchy problem (12). In this way, we give a representation of the solution u(x, t) of the problem (12) in terms of the solution of this RH problem. Then, in Section 4, we show that a solution of the RH problem with any appropriate jump matrix (ensuring the unique solvability of the RH problem) gives rise to a solution of the mfcSPE. In Section 5, we discuss the construction of soliton solutions using the formalism of the RH problem, which is illustrated numerically in Section 6.

The RH formalism for integrable non-linear equations utilizes the possibility of constructing special solutions of linear equations from the associated Lax pair, which are well controlled as functions of the spectral parameter, in the whole extended complex plane. For this purpose, it is useful to have the Lax pair equations in the form suitable for establishing analytic properties of solutions near the singular points with respect to spectral parameter of the Lax pair equations. For different domains in the complex plane, these solutions are defined differently and are related to each other at the boundaries between these domains.

To construct such special solutions of the differential equations from the Lax pair, it is convenient to pass to integral equations, whose solutions are particular solutions to the Lax pair equation.

Notice the coefficients U and V of the Lax pair are traceless matrices. Consequently, the determinant of a matrix solution to Equation 10 (composed of two vector solutions) is independent of x and t.

To obtain a RH problem with the jump condition on the real axis, as in the case of other Camassa–Holm-type equations [see [3–9]], we redefine the spectral parameter introducing k: = iλ.

Notice that U and V have singularities (in the extended complex k-plane) at k = 0 and at k = ∞. Namely, since U is singular at k = ∞ only, for dealing with the problem on the whole x-line it is important to control the behavior of special solutions of the Lax pair equations for large k. Assume that u(·, t) ∈ W^{2, 1}(ℝ) and transform the Lax pair to the following form [cf. [2–4, 8]]:

where the coefficients Q(x, t, k), Û(x, t, k), and V^(x,t,k) have the following properties:

To transform the Lax pair, we introduce Φ^:=GΦ with G = G(x, t) to be defined. Then, the Lax pair (10) takes form

Since U is a product of the spectral parameter and a matrix independent of it, we can define G so as Qx:=-GUG-1 is a diagonal matrix function satisfying item (i). Then, the degree of freedom in the determination of G (multiplication of G by a diagonal matrix from the left) can be used to provide us with Û satisfying (iii). Namely, introducing

we have

with the inverse

where m is not specified for the moment. Then,

where σ₃ is the Pauli matrix σ3=(1   00   -1).

To satisfy item (iii) for Û, we use the freedom of choice of m to make the diagonal part of U^=GxG-1 to be identically equal to zero. Complemented by a norming condition m(+∞, t) = 0, this leads to

which finally gives

Notice that m is purely imaginary and thus m̄=-m and |e^m| = 1.

As for the t–equation (17), we have:

Now, we can determine Q(x, t, k) by integrating Equation 21 w.r.t. x and taking into account that we want V^ in Equation 15 to vanish at x = ±∞ for all t. This gives

where

is normalized in such a way that x^-x→0 as x → +∞. Then, we have

where we have used the equality qt=(|u|2q)x which is actually the mfcSPE (9) rewritten as a conservation law. Correspondingly,

Remark 2.1. The dependence of the diagonal matrix Q on variables x^ and t, see Equation 25, is the same as in the case of the SP equation [see [8]]. This justifies the name of the mfcSPE as the modified SP equation: The same property holds for the pair consisting of the famous Korteweg–de Vries equation u_t + 6uu_x + u_xxx = 0 and the modified Korteweg–de Vries equation ut+6u2ux+uxxx=0.

Introducing

Equations 14 can be rewritten as

where [·, ·] denotes the matrix commutator. Now, we determine the special (Jost) solutions Φ~±(x,t,k) of Equation 29 as the 2 × 2 matrix-valued solutions of the associated Volterra integral equations:

where I is the identity matrix. Taking into account the definition of Q (25) and (26), we get

Respectively, Φ^±:=Φ~±e-Q are the Jost solutions of the Lax pair equations (14).

In what follows, the columns of a 2 × 2 matrix μ=(μ(1)μ(2)) are denoted by μ⁽¹⁾ and μ⁽²⁾. Since q is positive, the exponentials in Equation 32 as functions of y either decay to 0 or grow to ∞ as y goes to +∞ or to −∞, depending on the sign of the imaginary part of k (for real k, all exponentials are oscillating functions). Moreover, if we consider Equation 32 columnwise, the corresponding integral equation involves the exponentials of only one sign: either eik∫xyq(ξ,t)dξ or e-ik∫xyq(ξ,t)dξ. Consequently, we can determine the columns of Equation 32 via Neumann series for the corresponding integral equation, which converge if k belongs to the corresponding half–plane: the upper half–plane {k|Imk ≥ 0} or the lower half–plane {k|Imk ≤ 0}. The obtained Jost solutions satisfy the following properties [cf. [8]] for all (x, t):

The last property is due to the symmetry of the matrix Ǔ: = Û−ikqσ₃:

Remark 2.2. Introducing the new variable x^ as in Equation 26, Equation 14 reduces to the (non-self-adjoint) Dirac equation for Φ⌣(x^,t,k):=Φ^(x(x^,t),t,k):

where

Equation 36 is the spatial equation from the Lax pair associated with the focusing non-linear Schrödinger (fNLS) equation, see [12]. Therefore, the analytic properties of Φ~± stated above are the same as in the case of the fNLS equation considered in [12].

Now, we introduce the scattering matrix s(k) as the matrix relating the Jost solutions Φ^+ and Φ^- for those values of k where all their columns are determined (i.e., for real k):

or, in terms of Φ~±,

Notice that since Φ^+ and Φ^- are solutions of the same differential equations (16), the matrix s(k) does not depend on x^ and t. Consequently, s(k) can be determined by q(x, 0) only, by

Indeed, Φ~±(x^,0,k) are determined [see Equation 32] by Û(x, 0) and q(x, 0) which, in turn, are determined by q(x, 0) alone.

Due to the symmetry (34) and the fact that e^{Q(x, t, k)} satisfies the same symmetry as well, the scattering matrix can be rewritten with the help of two scalar spectral functions, a(k) and b(k), as follows:

Taking into account Remark 2.2, the spectral functions have properties, which are similar to those in case of the fNLS equation in [12]: