Short Pulse Equation Research Articles

Soliton interactions and modulation instability for the N-coupled complex short pulse equations in an optical fiber

Under investigation in this paper are the N-coupled complex short pulse (N-CSP) equations, which describe the propagation of simultaneous N ultrashort pulses in a nonlinear fiber medium. Through the Hirota method, the m-bright-n-dark [Formula: see text] soliton solutions are obtained. Elastic and inelastic interactions between the mixed-type solitons are derived through the asymptotic analysis. As an example, the oblique interactions and the bound states of the 2-bright-1-dark and 1-bright-2-dark solitons for 3-CSP equations are analyzed graphically. Finally, the condition of the modulation instability of the plane-wave solutions for 2-CSP equations is given through the linear stability analysis.

Multiple complex and multiple real soliton solutions for the integrable sine-Gordon equation

In this work, we study the integrable sine-Gordon equation which is related to the short pulse equation (SPE). We show that this equation passes the Painlevé test to confirm its integrability. We introduce new complex forms of the simplified Hirota's method to formally derive multiple complex soliton solutions for this equation. Our results show that the complex simplified Hirota's method explicitly constructs new multiple complex soliton solutions for more integrable equations.

On the Hamiltonian–Krein index for a non-self-adjoint spectral problem

We investigate the instability index of the spectral problem \[ − c 2 y + b 2 y + V ( x ) y = − i z y ′ -c^2y + b^2y + V(x)y = -\mathrm {i} z y’ \] on the line R \mathbb {R} , where V ∈ L loc 1 ( R ) V\in L^1_\textrm {loc}(\mathbb {R}) is real valued and b , c > 0 b,c>0 are constants. This problem arises in the study of stability of solitons for certain nonlinear equations (e.g., the short pulse equation and the generalized Bullough–Dodd equation). We show how to apply the standard approach in the situation under consideration, and as a result we provide a formula for the instability index in terms of certain spectral characteristics of the 1-D Schrödinger operator H V = − c 2 d 2 d x 2 + b 2 + V ( x ) H_V=-c^2\frac {d^2}{dx^2}+b^2 +V(x) .

Nonlocal Short Pulse Equations

We introduce nonlocal integrable reductions of the Wadati-Konno-Ichikawa (WKI) coupled equations. Using the nonlocal reductions of Ablowitz and Musslimani, we obtain new integrable equations such as the nonlocal short pulse (SP) equations and the nonlocal elastic beam (EB) equations.

Long-time asymptotics for the short pulse equation

In this paper, we analyze the long-time behavior of the solution of the initial value problem (IVP) for the short pulse (SP) equation. As the SP equation is a completely integrable system, which posses a Wadati–Konno–Ichikawa (WKI)-type Lax pair, we formulate a 2×2 matrix Riemann–Hilbert problem to this IVP by using the inverse scattering method. Since the spectral variable k is the same order in the WKI-type Lax pair, we construct the solution of this IVP parametrically in the new scale (y,t), whereas the original scale (x,t) is given in terms of functions in the new scale, in terms of the solution of this Riemann–Hilbert problem. However, by employing the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems, we can get the explicit leading order asymptotic of the solution of the short pulse equation in the original scale (x,t) as time t goes to infinity.

Super extensions of the short pulse equation

From a super extension of the Wadati, Konno and Ichikawa scheme for integrable systems and using a osp(1,2) valued connection 1-form we obtain super generalizations for the Short Pulse equation as well for the Elastic Beam equation.

Stability and interaction of few-cycle pulses in a Kerr medium.

The different aspects of few-cycle pulse dynamics governed by the regularized short pulse equation (RSPE) are reported. It is shown that the RSPE provides an accurate description of the dynamics of the few-cycle pulse whose duration is larger than a single optical period when the few-cycle pulse's spectrum is in the medium's anomalous dispersion regime. The approximate solutions of the RSPE are constructed from the soliton solutions of the nonlinear Schrödinger (NLS) equation. We demonstrate numerically that the stability of these few-cycle pulses strongly depends on their pulse duration. Furthermore, the interactions of the two and three few-cycle pulses are studied. When pulse parameters are suitably chosen, we show the elastic collision, inelastic collision and repulsive interaction between these multi few-cycle pulses. It is revealed that the interactions of the multi few-cycle pulses rely heavily on their pulse duration.

Darboux Transformation for a Semidiscrete Short-Pulse Equation

We define a Darboux transformation in terms of a quasideterminant Darboux matrix on the solutions of a semidiscrete short-pulse equation. We also give a quasideterminant formula for N-loop soliton solutions and obtain a general expression for the multiloop solution expressed in terms of quasideterminants. Using quasideterminants properties, we find explicit solutions and as an example compute one- and two-loop soliton solutions in explicit form.

A Riemann–Hilbert approach for the modified short pulse equation

ABSTRACTWe present a Riemann–Hilbert approach for the modified short pulse equation on the line. This approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used in studying its long-time behavior, also to describe the soliton solutions.

On soliton solutions of multi-component semi-discrete short pulse equation

The short pulse (SP) equation is an integrable equation. Multi-component generalizations of the SP equation are important for describing the polarization or anisotropic effects in optical fibers. An integrable semi-discretization of multi-component SP equation via Lax pair and Darboux transformation (DT) has been presented. We derive a Lax pair representation for the multi-component semi-discrete short pulse (sdSP) equation in the form of a block matrices by generalizing the 2 × 2 Lax pair matrices to the case of . A DT is studied for the multi-component sdSP equation and is used to compute soliton solutions of the system. Further, by expanding quasideterminants, we compute cuspon-soliton, smooth-soliton and loop-soliton solutions of the complex sdSP equation.

Solution of the short pulse equation using lattice-Boltzmann and the tanh method

Solution of the short pulse equation using lattice-Boltzmann and the tanh method

Integrability of a Generalized Short Pulse Equation Revisited

We further generalize the generalized short pulse equation studied recently in [Commun. Nonlinear Sci. Numer. Simulat. 39 (2016) 21-28; arXiv:1510.08822], and find in this way two new integrable nonlinear wave equations which are transformable to linear Klein-Gordon equations.

Convergence of the regularized short pulse equation to the short pulse one

AbstractWe consider the regularized short‐pulse 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 the short‐pulse one. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the setting.

Transformations between Nonlocal and Local Integrable Equations

AbstractRecently, a number of nonlocal integrable equations, such as the ‐symmetric nonlinear Schrödinger (NLS) equation and ‐symmetric Davey–Stewartson equations, were proposed and studied. Here, we show that many of such nonlocal integrable equations can be converted to local integrable equations through simple variable transformations. Examples include these nonlocal NLS and Davey–Stewartson equations, a nonlocal derivative NLS equation, the reverse space‐time complex‐modified Korteweg–de Vries (CMKdV) equation, and many others. These transformations not only establish immediately the integrability of these nonlocal equations, but also allow us to construct their Lax pairs and analytical solutions from those of the local equations. These transformations can also be used to derive new nonlocal integrable equations. As applications of these transformations, we use them to derive rogue wave solutions for the partially ‐symmetric Davey–Stewartson equations and the nonlocal derivative NLS equation. In addition, we use them to derive multisoliton and quasi‐periodic solutions in the reverse space‐time CMKdV equation. Furthermore, we use them to construct many new nonlocal integrable equations such as nonlocal short pulse equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa–Satsuma equations.

Generalizations of the short pulse equation.

We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.

Lax representations for matrix short pulse equations

The Lax representation for different matrix generalizations of Short Pulse Equations (SPEs) is considered. The four-dimensional Lax representations of four-component Matsuno, Feng, and Dimakis-Müller-Hoissen-Matsuno equations are obtained. The four-component Feng system is defined by generalization of the two-dimensional Lax representation to the four-component case. This system reduces to the original Feng equation, to the two-component Matsuno equation, or to the Yao-Zang equation. The three-component version of the Feng equation is presented. The four-component version of the Matsuno equation with its Lax representation is given. This equation reduces the new two-component Feng system. The two-component Dimakis-Müller-Hoissen-Matsuno equations are generalized to the four-parameter family of the four-component SPE. The bi-Hamiltonian structure of this generalization, for special values of parameters, is defined. This four-component SPE in special cases reduces to the new two-component SPE.

Comment on "Defocusing complex short-pulse equation and its multi-dark-soliton solution".

In their recent paper, Feng etal. [Phys. Rev. E 93, 052227 (2016)PREHBM2470-004510.1103/PhysRevE.93.052227] proposed a complex short-pulse equation of both focusing and defocusing types. They studied in detail the defocusing case and derived its multi-dark-soliton solutions. Nonetheless, from a physical viewpoint in order to better and deeply understand their genuine implications, we find it useful to provide a real and proper background for the derivation of the previous evolution system while showing that the expression of the nonlinear electric polarization the above authors used in their scheme is not suitable for getting the defocusing complex short-pulse equation.

Reply to "Comment on 'Defocusing complex short-pulse equation and its multi-dark-soliton solution' ".

Our paper [Phys. Rev. E 93, 052227 (2016)PREHBM2470-004510.1103/PhysRevE.93.052227], proposing an integrable model for the propagation of ultrashort pulses, has recently received a Comment by Youssoufa etal. [Phys. Rev. E 96, 026201 (2017)10.1103/PhysRevE.96.026201] about a possible flaw in its derivation. We point out that their claim is incorrect since we have stated explicitly that a term is neglected to derive our model equation in our paper. Furthermore, the integrable model is validated by comparing with the normalized Maxwell equation and other known integrable models. Moreover, we show that a similar approximation has to be performed in deriving the same integrable equation as explained in the Comment.

Periodic solutions and solitons to two complex short pulse (CSP) equations in optical fiber

Two complex short pulse equations in optical fiber, namely, the focusing complex short pulse (FCSP) equation and the defocusing complex short pulse (DFCSP) equation, are investigated. By F-expansion method, a series of periodic traveling wave solutions are obtained, which involve Jacobian elliptic functions. When the modulus m→1, the solutions degenerate to the soliton solutions, respectively. Partial Jacobian elliptic solutions and soliton solutions are visualized to illustrate the propagation properties of the solutions.

Large time asymptotics of solutions to the short-pulse equation

We consider the long-time behavior of solutions to the short-pulse equation. Using the method of testing by wave packets, we prove small data global existence and modified scattering.