The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential q ( x , t ) can be represented by the solution of this Riemann-Hilbert problem.

Fermionic covariant prolongation structure theory for a (2+1)-dimensional super nonlinear evolution equation

In the framework of the fermionic covariant prolongation structure theory (PST), we investigate the integrability of a new [Formula: see text]-dimensional super nonlinear evolution equation (NEE). The prolongation structure of the super integrable system is presented. Moreover, we derive the Bäcklund transformation of the super integrable system.

The fermionic covariant prolongation structure of the super generalized Hirota equation

The integrability of a super generalized Hirota equation (GHE) is investigated by means of the fermionic covariant prolongation structure theory. We construct the [Formula: see text] prolongation structure for the super GHE and derive the corresponding Lax representation and the Bäcklund transformation. In addition, a solution of the super integrable equation is presented.

Prolongation Structure of a Generalised Inhomogeneous Gardner Equation in Plasmas and Fluids

In this article, the prolongation structure technique is applied to a generalised inhomogeneous Gardner equation, which can be used to describe certain physical situations, such as the stratified shear flows in ocean and atmosphere, ion acoustic waves in plasmas with a negative ion, interfacial solitary waves over slowly varying topographies, and wave motion in a non-linear elastic structural element with large deflection. The Lax pairs, which are derived via the prolongation structure, are more general than the Lax pairs published before. Under the Painlevé conditions, the linear-damping coefficient equals to zero, the quadratic non-linear coefficient is proportional to the dispersive coefficient c(t), the cubic non-linear coefficient is proportional to c(t), leaving no constraints on c(t) and the dissipative coefficient d(t). We establish the prolongation structure through constructing the exterior differential system. We introduce two methods to obtain the Lax pairs: (a) based on the prolongation structure, the Lax pairs are obtained, and (b) via the Lie algebra, we can derive the Pfaffian forms and Lax pairs when certain parameters are chosen. We set d(t) as a constant to discuss the influence of c(t) on the Pfaffian forms and Lax pairs, and to discuss the influence of d(t) on the Pfaffian forms and Lax pairs, we set c(t) as another constant. Then, we get different prolongation structure, Pfaffian forms and Lax pairs.

GEOMETRIC APPROACHES TO PRODUCE PROLONGATIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

The prolongation structure of a two-by-two problem is formulated very generally in terms of exterior differential forms on a standard representation of Pauli matrices. The differential system is general without making reference to any specific equation. An integrability condition is provided which gives by construction the equation to be investigated and whose components involve the structure constants of an SU(2) Lie algebra. Along side this, a related, different kind of prolongation, a type of Wahlquist–Estabrook prolongation, over a closed differential ideal is discussed and some applications are given.

ON TWO-DIMENSIONAL MANIFOLDS WITH CONSTANT GAUSSIAN CURVATURE AND THEIR ASSOCIATED EQUATIONS

The components for the frame field of a two-dimensional manifold with constant Gaussian curvature are determined for arbitrary nonzero curvature. The components of the frame fields are found from the structure equations and lead to specific nonlinear equations which pertain to surfaces with specific values of the Gaussian curvature. For negative curvature, the equation is of sine-Gordon type, and for positive curvature it is of sinh-Gordon type. The integrability and Bäcklund properties of these equations are then investigated by studying a differential ideal of two-forms which leads to the equations. As a consequence of studying the prolongation structure of each equation, a Lax pair and Bäcklund transformation are obtained.