Soliton equations and Backlund transformations: Explicit Solutions and Algebraic Properties

Sandra Carillo (Roma Sapienza)

Monday 22nd June 11:00-12:00
Maths 311B

Abstract

Soliton equations play a central role from an applied perspective because they admit solutions whose shape remains unchanged during propagation and whose amplitude is conserved over time. Such equations frequently appear in applied mathematics, with applications spanning fluid dynamics, nonlinear optics, acoustics, theoretical physics and, more recently, biophysics. We will consider nonlinear evolution equations of soliton type. Specifically, 3rd order ones are considered. A chain of links connecting a variety of such equations is constructed. The Backlund transformations can be generalized from the commutative to the non-commutative setting. Accordingly they also
connect 3rd non-commutative equations. As a consequence, solutions of the matrix modified Korteweg-de Vries (mKdV) equation are considered. Both soliton as well as breather-type solutions admitted by the d × d matrix mKdV equation are constructed. In addition, algebraic properties, such as the admitted Hamiltonian structure are proved via Backlund transformations.

The obtained solutions can be termed soliton solutions, since they manifest the characteristic behavior of solitons. In particular, two-soliton solutions of the d × d matrix mKdV equation are constructed. This work builds on a general explicit formula for N-soliton solutions in the infinite-dimensional (operator) case, while the explicit finite-dimensional (matrix) case is treated in detail. Additionally, invariance properties of third-order KdV-type equations in non-commutative (matrix) contexts have been investigated more recently. Solutions of matrix mKdV equation are presented which can be termed soliton solutions since they exhibit the typical behaviour of solitons. Specifically, two-soliton solutions of the d × d-matrix modified Korteweg de-Vries equation are obtained. An explicit formula in the matrix case is studied, while invariance properties of third order KdV-type equations are investigated. Extensions to 5th order nonlinear evolution equations are presented.

Add to your calendar

Download event information as iCalendar file (only this event)