Fractional integrable nonlinear systems

Abstract

Nonlinear equations and fractional calculus have become important mathematical descriptions of physical applications. We provide context for the intersection of these two mathematical theories with our discovery of integrable, i.e., exactly solvable, fractional nonlinear evolution equations. Using a general method which can be applied to any integrable system with sufficient structure, we derive the first known fractional integrable equations with nonlocal fractional operators, the fractional Korteweg-deVries and fractional nonlinear Schr\"odinger equations. For these equations, we find the general solution for decaying initial data in terms of a set of linear integral equations. Like other integrable equations, these fractional integrable equations have solitonic solutions, an infinite number of conservation laws, and elastic soliton-soliton interactions. The soliton solutions to these equations have velocities related to their amplitudes by a power law, a simple physical prediction common to these fractional integrable equations known as anomalous dispersion.

The method of finding these fractional integrable equations involves three key mathematical ingredients: power law dispersion, completeness of squared eigenfunctions, and the inverse scattering transform. All of these elements together allow us to define the nonlinear fractional operators underlying these fractional integrable equations through spectral theory, just as Fourier transforms can be used to define fractional derivatives. If any integrable system admits these three structural elements, it will have fractional integrable equations associated to it.

Once these fractional integrable equations are found, we solve them using the inverse scattering transform. The inverse scattering transform is an analytical solution method that linearizes certain nonlinear evolution equations; such equations are called integrable. The method associates an integrable equations to a scattering problem, e.g., the time-independent Schr\"odinger equation to solve the Korteweg-deVries equation. The solutions to this scattering problem are used to map the nonlinear equation into scattering space, where time evolution is simple. Then, recovering the solution to the nonlinear equation in physical space is performed by inverse scattering which involves solving a set of linear integral equations. The solutions to these integrable problems have radiation and $N$-soliton components where the solitons have elastic collisions. We linearize the fractional Korteweg-deVries by association to the time-independent Schr\"odinger equation and the nonlinear Schr\"odinger equations by the Ablowitz-Kaup-Newell-Segur scattering system.

We then apply this method to derive fractional extensions to the modified Korteweg-deVries, sine-Gordon, and sinh-Gordon equations, reviewing the inverse scattering theory in detail required to define and solve these equations. The scattering equation associated to these three fractional integrable equations is a scalar reduction of the Ablowitz-Kaup-Newell-Segur scattering system given a certain symmetry condition. We show how completeness for the Ablowitz-Kaup-Newell-Segur scattering system reduces to completeness for the scalar scattering problem and use this to define the relevant nonlinear fractional operators. Then, using this completeness relation, we verify explicitly that the one-soliton solutions to these equations are indeed solutions. As with the fractional Korteweg-deVries and nonlinear Schr\"odinger equations, these soliton solutions exhibit anomalous dispersion.

We then showed that this method can be applied to discrete systems by defining and solving the fractional integrable discrete nonlinear Schr\"odinger equation, whose solution is defined on a lattice, i.e., on discrete points on the real line, but still depends continuously on time. As with the other fractional equations, we demonstrated how the three key mathematical ingredients lead to an explicit form for the equation and how the inverse scattering transform can be used to linearize the problem. However, unlike the continuous fractional integrable equations, the one-soliton solution admits a peak velocity related to its amplitude in a much more complicated manner than in anomalous dispersion, allowing for potentially unexpected behavior. In particular, we demonstrate that the velocity can exhibit a turning point where it switches from increasing to decreasing with the fractional parameter at a certain value of the fractional parameter.

We also show how some of the characteristics of these fractional integrable equations reach beyond integrable equations by comparing this discrete integrable equation to the fractional averaged discrete nonlinear Schr\"odiner equation. This equation is closely related to its integrable counterpart, but has a much simpler mathematical form. Using a Fourier split step method, we numerically integrate the fractional averaged equation and find solitary wave solutions. By studying the emitted radiation, peak position, averaged amplitude, velocity, and form of the solitary waves, we demonstrate that these waves have similar characteristics to the integrable solitons and that these similarities are accentuated for positive fractional parameter and small amplitude waves.

This work invites further research into fractional integrable and nonintegrable nonlinear system and exploration of their many potential applications.