Thomas B. Bouetou

On two-loop soliton solution of the Schäfer-Wayne short-pulse equation using hirota's method and Hodnett-Moloney approach

A two-loop soliton solution to the Schafer-Wayne short-pulse equation (SWSPE) is shown. The key step in finding this solution is to transform the independent variables in the equation. This leads to a transformed equation for which it is straightforward to find an explicit two-soliton solution using Hirotas method. The two-loop soliton solution to the SWSPE is then found in implicit form by means of a transformation back to the original independent variables. Following Hodnett and Moloneys approach, some computations of the energy of the one- and two-soliton solutions are made.

On the Conversion of Exact Rotating Loop-Like Soliton Solution to Generalized Schäfer–Wayne Short Pulse Equation

We derive a new equation that is considered as a generalized Schafer–Wayne short pulse equation (SWSPE). We then elicit a soliton solution to this equation using the (1+1)-dimensional coupled nonlinear dispersionless equations (CNLDEs). As a result, this solution may possess a nonzero angular momentum.

Traveling Wave-Guide Channels of a New Coupled Integrable Dispersionless System

In the wake of the recent investigation of new coupled integrable dispersionless equations by means of the Darboux transformation (Zhaqilao, et al., Chin. Phys. B 18 (2009) 1780), we carry out the initial value analysis of the previous system using the fourth-order Runge-Kuttas computational scheme. As a result, while depicting its phase portraits accordingly, we show that the above dispersionless system actually supports two kinds of solutions amongst which the localized traveling wave-guide channels. In addition, paying particular interests to such localized structures, we construct the bilinear transformation of the current system from which scattering amongst the above waves can be deeply studied.

Generating a New Higher-Dimensional Ultra-Short Pulse System: Lie-Algebra Valued Connection and Hidden Structural Symmetries

We carry out the hidden structural symmetries embedded within a system comprising ultra-short pulses which propagate in optical nonlinear media. Based upon the Wahlquist-Estabrook approach, we construct the Lie-algebra valued connections associated to the previous symmetries while deriving their corresponding Lax-pairs, which are particularly useful in soliton theory. In the wake of previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2+1)-dimensional ultra-short pulse equation is unveiled along with its inverse scattering formulation, the application of which are straightforward in nonlinear optics where an additional propagating dimension deserves some attention.

Reply to "Comment on 'On Two-Loop Soliton Solution of the Schafer-Wayne Short-Pulse Equation Using Hirota's Method and Hodnett-Moloney Approach'"

Following a recent comment paid by Zhang and Li to the paper entitled ‘‘On Two-Loop Soliton Solution of the Schafer–Wayne Short-Pulse Equation Using Hirota’s Method and Hodnett–Moloney Approach’’, we show that solving initial value problems may lead to solutions different from that of vanishing boundary value problems. In a recently published work, we have investigated the two-loop soliton solutions to the Schafer–Wayne short pulse (SWSP) equation given by

Fractal structure of ferromagnets: The singularity structure analysis

Following the Weiss-Tabor-Carnevale approach [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 24, 522 (1983)10.1063/1.525721; J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 25, 13 (1984).]10.1063/1.526009 designed for studying the integrability properties of nonlinear partial differential equations, we investigate the singularity structure of a (2+1)-dimensional wave-equation describing the propagation of polariton solitary waves in a ferromagnetic slab. As a result, we show that, out of any damping instability, the system above is integrable. Looking forward to unveiling its complete integrability, we derive its Backlund transformation and Hirotas bilinearization useful in generating a set of soliton solutions. In the wake of such results, using the arbitrary functions to enter into the Laurent series of solutions to the above system, we discuss some typical class of excitations generated from the previous solutions in account of a classification based upon the different expressions of a gene...

Generating a New Higher-Dimensional Coupled Integrable Dispersionless System: Algebraic Structures, Backlund Transformation and Hidden Structural Symmetries

The prolongation structure methodologies of Wahlquist—Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 (1975) 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Backlund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2 + 1)-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.

Comment on: Loop Soliton Solutions of String Interacting with External Field

where q, r, and s are physical observables, x and t representing spaceand time-like independent variables. Taking q 1⁄4 Z, r 1⁄4 X þ iY , and s 1⁄4 X iY with i 1⁄4 1, eq. (4) is transformed to eq. (1) provided 1⁄4 xþ t and 1⁄4 x t. Singularity structure analysis of eq. (4) has recently been investigated. The associated Backlund transformation has been constructed and Hirota’s bilinearization also given through dependent variable transformations. As a result, q, r, and s are expressed as follows r 1⁄4 G=F; s 1⁄4 H=F; q 1⁄4 2@tðlnFÞ: ð5Þ We assume r 1⁄4 s, ð?Þ denoting complex conjugation, and we transform eq. (5) to a new one by looking for some soliton solutions with the following asymptotic behavior jrj ! 0; q ! x; as jxj ! 1: ð6Þ We may then write

Scattering Behavior of Waveguide Channels of a New Coupled Integrable Dispersionless System

Based upon the powerful Hirota method for unearthing soliton solutions to nonlinear partial differential evolution equations, we investigate the scattering properties of a new coupled integrable dispersionless system while surveying the interactions between its self-confined travelling wave solutions. As a result, we ascertain three types of scattering features depending strongly upon a characteristic parameter. Using such findings to depict soliton solutions with nonzero angular momenta, we derive an extended form of the dispersionless system, which is valuable for further physical applications.