Models of few optical cycle solitons beyond the slowly varying envelope approximation

An exact nonsingular solitary wave solution of the Schafer–Wayne short pulse equation is derived from the breather solution of the sine-Gordon equation by means of a transformation between these two integrable equations.

Solitary wave solutions of the short pulse equation

In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.

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Wave-Breaking and Peakons for a Modified Camassa-Holm Equation

We study a new hierarchy of equations containing the short pulse equation, which describes the evolution of very short pulses in nonlinear media, and the elastic beam equation, which describes nonlinear transverse oscillations of elastic beams under tension. We show that the hierarchy of equations is integrable. We obtain the two compatible Hamiltonian structures. We construct an infinite series of both local and nonlocal conserved charges. A Lax description is presented for both systems. For the elastic beam equations we also obtain a nonstandard Lax representation.

The short pulse hierarchy

Geometry of jet spaces and integrable systems

We prove that the Schafer–Wayne short pulse equation (SPE), which describes the propagation of ultrashort optical pulses in nonlinear media, is integrable. First, we discover a Lax pair of the SPE that is found to be of the Wadati–Konno–Ichikawa type. Second, we construct a chain of transformations that relates the SPE with the sine-Gordon equation.

The Short Pulse Equation Is Integrable

A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg–de Vries equation with a source. A Lax representation and an auto-Backlund transformation are found for the new equation, and its traveling wave solutions and generalized symmetries are studied.

A new integrable generalization of the Korteweg–de Vries equation

A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found of the new equation, and its travelling wave solutions and generalized symmetries are studied.

A new integrable generalization of the Korteweg - de Vries equation

Sufficient conditions for wave breaking are found for the short- pulse equation describing wave packets of few cycles on the ultra-short pulse scale. The analysis relies on the method of characteristics and conserved quan- tities of the short-pulse equation and holds both on an infinite line and in a periodic domain. Numerical illustrations of the finite-time wave breaking are given in a periodic domain.

https://www.intlpress.com/site/pub/files/_fulltext/journals/dpde/2009/0006/0004/DPDE-2009-0006-0004-a001.pdf

Anton Sakovich

Papers

The Short Pulse Equation Is Integrable

Solitary wave solutions of the short pulse equation

A new integrable generalization of the Korteweg–de Vries equation

A new integrable generalization of the Korteweg - de Vries equation

Wave breaking in the short-pulse equation