scholarly journals Lax Representation and Darboux Solutions of the Classical Painlevé Second Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Irfan Mahmood ◽  
Muhammad Waseem
Keyword(s):  
Linear System ◽  
Darboux Transformation ◽  
Integrable Systems ◽  
Lax Pair ◽  
Lax Representation ◽  
Gauge Transformations ◽  
Additive Form

In this article, we present Darboux solutions of the classical Painlevé second equation. We reexpress the classical Painlevé second Lax pair in new setting introducing gauge transformations to yield its Darboux expression in additive form. The new linear system of that equation carries similar structure as other integrable systems possess in the AKNS scheme. Finally, we generalize the Darboux transformation of the classical Painlevé second equation to the N -th form in terms of Wranskian.

scholarly journals Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

2020 ◽  
Vol 16 (4) ◽  
pp. 637-650
Author(s):  
P. Guha ◽  
◽  
S. Garai ◽  
A.G. Choudhury ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Lax Pair ◽  
First Integrals ◽  
Second Order ◽  
Lax Representation ◽  
Lax Pairs ◽  
Second Order Differential Equations ◽  
Autonomous Version ◽  
Autonomous Differential Equations

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

scholarly journals Gauge transformations and symmetries of integrable systems

2007 ◽  
Vol 40 (40) ◽  
pp. 12227-12241 ◽  
Author(s):  
Takeshi Fukuyama ◽  
Kiyoshi Kamimura ◽  
Saša Krešić-Jurić ◽  
Stjepan Meljanac
Keyword(s):  
Integrable Systems ◽  
Gauge Transformations

Lax pair, Darboux transformation and rogue-periodic waves of a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics

2021 ◽  
pp. 2150451 ◽  
Author(s):  
Cheng-Cheng Wei ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Su-Su Chen ◽  
Dan-Yu Yang
Keyword(s):  
Darboux Transformation ◽  
Periodic Wave ◽  
Lax Pair ◽  
Third Order ◽  
Periodic Wave Solutions ◽  
The Third ◽  
Third Order Dispersion ◽  
Wave Solutions ◽  
Minimum Amplitude ◽  
Order Dispersion

For a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics, we derive a Lax pair, a Darboux transformation and two families of the periodic-wave solutions via the Jacobian elliptic functions dn and cn. We construct the linearly-independent and non-periodic solutions of that Lax pair, and substitute those solutions into the Darboux transformation to get the rogue-periodic-wave solutions. When the third-order dispersion or group velocity dispersion (GVD) or inter-modal dispersion (IMD) increases, the maximum amplitude of the rogue-periodic wave remains unchanged. From the rogue-dn-periodic-wave solutions, when the GVD decreases, the minimum amplitude of the rogue-dn-periodic wave decreases. When the third-order dispersion decreases, the minimum amplitude of the rogue-dn-periodic wave rises. Decrease of the IMD causes the period of the rogue-dn-periodic wave to decrease. From the rogue-cn-periodic-wave solutions, when the GVD increases, the minimum amplitude of the rogue-cn-periodic wave decreases. Increase of the third-order dispersion or IMD leads to the decrease of the period.

Darboux transformation for the supersymmetric nonisospectral KdV equation

2021 ◽  
pp. 2150255
Author(s):  
Li Chen ◽  
Shu-Fang Deng
Keyword(s):  
Darboux Transformation ◽  
Kdv Equation ◽  
Lax Pair ◽  
The One

Darboux transformation for the supersymmetric nonisospectral KdV equation is investigated. Based on the Lax pair, we successively construct the one-fold, two-fold and three-fold Darboux tansformations for the supersymmetric nonisospectral KdV equation. Moreover, we present the [Formula: see text]-fold Darboux transformation in the form of superdeterminant.

(1+1)-dimensional m-cKdV, g-cKdV integrable systems, and (2+1)-dimensional m-cKdV hierarchy

2008 ◽  
Vol 86 (12) ◽  
pp. 1367-1380 ◽  
Author(s):  
Y Zhang ◽  
H Tam
Keyword(s):  
Integrable Systems ◽  
Linear Functional ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Integrable Model ◽  
Lax Pair ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Higher Dimensional

A few isospectral problems are introduced by referring to that of the cKdV equation hierarchy, for which two types of integrable systems called the (1 + 1)-dimensional m-cKdV hierarchy and the g-cKdV hierarchy are generated, respectively, whose Hamiltonian structures are also discussed by employing a linear functional and the quadratic-form identity. The corresponding expanding integrable models of the (1 + 1)-dimensional m-cKdV hierarchy and g-cKdV hierarchy are obtained. The Hamiltonian structure of the latter one is given by the variational identity, proposed by Ma Wen-Xiu as well. Finally, we use a Lax pair from the self-dual Yang–Mills equations to deduce a higher dimensional m-cKdV hierarchy of evolution equations and its Hamiltonian structure. Furthermore, its expanding integrable model is produced by the use of a enlarged Lie algebra.PACS Nos.: 02.30, 03.40.K

Localised Nonlinear Waves in the Three-Component Coupled Hirota Equations

2017 ◽  
Vol 72 (11) ◽  
pp. 1053-1070 ◽  
Author(s):  
Tao Xu ◽  
Yong Chen
Keyword(s):  
Darboux Transformation ◽  
Dark Soliton ◽  
Lax Pair ◽  
Higher Order ◽  
Bright Soliton ◽  
Order Effects ◽  
Vector Solution ◽  
Hybrid Solutions ◽  
Three Components ◽  
Hirota Equations

AbstractWe construct the Lax pair and Darboux transformation for the three-component coupled Hirota equations including higher-order effects such as third-order dispersion, self-steepening, and stimulated Raman scattering. A special vector solution of the Lax pair with 4×4 matrices for the three-component Hirota system is elaborately generated, based on this vector solution, various types of mixed higher-order localised waves are derived through the generalised Darboux transformation. Instead of considering various arrangements of the three potential functions q1, q2, and q3, here, the same combination is considered as the same type solution. The first- and second-order localised waves are mainly discussed in six mixed types: (1) the hybrid solutions degenerate to the rational ones and three components are all rogue waves; (2) two components are hybrid solutions between rogue wave (RW) and breather (RW+breather), and one component is interactional solution between RW and dark soliton (RW+dark soliton); (3) two components are RW+dark soliton, and one component is RW+bright soliton; (4) two components are RW+breather, and one component is RW+bright soliton; (5) two components are RW+dark soliton, and one component is RW+bright soliton; (6) three components are all RW+breather. Moreover, these nonlinear localised waves merge with each other by increasing the absolute values of two free parameters α, β. These results further uncover some striking dynamic structures in the multicomponent coupled system.

CONFORMALLY COVARIANT DRESSING TRANSFORMATIONS FOR SUPER SELF-DUALITY EQUATIONS IN D=4

1989 ◽  
Vol 04 (10) ◽  
pp. 971-982
Author(s):  
J. AVAN
Keyword(s):  
Linear System ◽  
Gauge Transformations ◽  
Four Dimensions ◽  
Yang Mills ◽  
Self Duality ◽  
Loop Algebras ◽  
Field Dependent

A set of conformally covariant dressing transformations is constructed for the supersym-metric N=3 self-duality equations in four dimensions, using the associated covariant linear system. They form a closed, 5+6-index algebra, up to field-dependent gauge transformations, containing the previously known loop algebras as a particular subset. This construction generalizes the formerly built set of conformally covariant DT for ordinary self-dual Yang-Mills.

scholarly journals A Five-Component Generalized mKdV Equation and Its Exact Solutions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1145
Author(s):  
Bo Xue ◽  
Huiling Du ◽  
Ruomeng Li
Keyword(s):  
Exact Solutions ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Rational Functions ◽  
Mkdv Equation ◽  
Darboux Transformations ◽  
Lax Pairs ◽  
Gauge Transformations

In this paper, a 3 × 3 spectral problem is proposed and a five-component equation that consists of two different mKdV equations is derived. A Darboux transformation of the five-component equation is presented relating to the gauge transformations between the Lax pairs. As applications of the Darboux transformations, interesting exact solutions, including soliton-like solutions and a solution that consists of rational functions of e x and t, for the five-component equation are obtained.

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