Nonlinear Sciences

The Hirota equation is an integrable higher order nonlinear Schrödinger type equation which describes the propagation of ultrashort light pulses in optical fibers. We present a standard Darboux transformation for the Hirota equation and then construct its quasideterminant solutions. The multisoliton and breather solutions of the Hirota equation are given explicitly.

This letter reports some new identities for multisoliton potentials that are based on the explicit representation provided by the Darboux matrix. These identities can be used to compute the complex gradient of the energy content of the tail of the profile with respect to the discrete eigenvalues and the norming constants. The associated derivatives are well defined in the framework of the so-called Wirtinger calculus which can aid a complex variable based optimization procedure in order to generate multisolitonic signals with desired effective temporal and spectral width.

The B ^ (1) n -hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra B ^ (1) n , the Drinfeld-Sokolov B ^ (1) n -KdV hierarchy is obtained by pushing down the B ^ (1) n -flows along certain gauge orbit to a cross section of the gauge action. In this paper, we (1) use loop group factorization to construct Darboux transforms (DTs) for the B ^ (1) n -hierarchy, (2) give a Permutability formula and scaling transform for these DTs, (3) use DTs of the B ^ (1) n -hierarchy to construct DTs for the B ^ (1) n -KdV and the isotropic curve flows of B-type, (4) give algorithm to construct soliton solutions and write down explicit soliton solutions for the third B ^ (1 1 -KdV, B ^ (1) 2 -KdV flows and isotropic curve flows on R 2,1 and R 3,2 of B-type.

We study a discrete Darboux transformation and construct the multi-soliton solutions in terms of ratio of determinants for integrable discrete sine-Gordon equation. We also calculate explicit expressions of single, double, triple, quad soliton solutions as well as single and double breather solutions of discrete sine-Gordon equation. Dynamical features of discrete kinks and breathers have also been illustrated.

A one-fold Darboux transformation between solutions of the semi-discrete massive Thirring model is derived using the Lax pair and dressing methods. This transformation is used to find the exact expressions for soliton solutions on zero and nonzero backgrounds. It is shown that the discrete solitons have the same properties as solitons of the continuous massive Thirring model.

In this paper, based on the Hirota and Maxwell-Bloch (H-MB) system and its application in the theory of the femtosecond pulse propagation through an erbium doped fiber, we define two kinds of nonlocal Hirota and Maxwell-Bloch (NH-MB) systems, namely, PT -symmetric NH-MB system and reverse space-time NH-MB system. Then we construct the Darboux transformations of these NH-MB systems. Meanwhile, we derive the explicit solutions by the Darboux transformations.

We prove a Darboux-Jouanolou type theorem on the algebraic integrability of polynomial differential r -forms over arbitrary fields ( r?? ). Also we investigate the Darboux's method for producing integrating factors. A general algebraic version of Poincaré's Lemma on fields of any characteristic is formulated and proved.

This article is devoted to discovering Lie symmetry algebra of a (3+1)-dimensional Davey-Stewartson system which appears in the field of plasma physics. It is found that the algebra is an infinite dimensional one and of Kac-Moody type. Making use of these symmetries, some reduced equations to lower dimensions are also presented.

Motivated by the proof of Rump of a conjecture of Gateva-Ivanova on the decomposability of square-free solutions to the Yang-Baxter equation, we present several other decomposability theorems based on the cycle structure of a certain permutation associated with the solution.

Within the (2 + 1)-dimensional Korteweg-de Vries equation framework, new bilinear Backlund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function, a family of deformed soliton and deformed breather solutions are presented with the improved Hirotas bilinear method. Choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of obtained solitons. Additionally, two dimensional [2D] rogue waves (localized in both space and time) on the soliton plane are presented, we refer to it as deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane, and its evolution process is analyzed in detail. The deformed 2D rogue wave solutions are constructed successfully, which are closely related to the arbitrary function. This new idea is also applicable to other nonlinear systems.