Finite symmetry transformation groups and exact solutions of Lax integrable systems

Sen-Yue, Lou; Ma, Hongcai

doi:10.1016/j.chaos.2005.04.090

Cited by 41 publications

(18 citation statements)

References 25 publications

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“…Recently, for simplicity and finding more general symmetry groups, some types of new simple direct method without the use of any group theory have been established for both Lax-integrable [22] and non-Lax-integrable [23] models.…”

Section: Eementioning

confidence: 99%

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Vortices, circumfluence, symmetry groups, and Darboux transformations of the(2+1)-dimensional Euler equation

et al. 2007

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The Euler equation (EE) is one of the basic equations in many physical fields such as fluids, plasmas, condensed matter, astrophysics, and oceanic and atmospheric dynamics. A symmetry group theorem of the (2+1) -dimensional EE is obtained via a simple direct method which is thus utilized to find exact analytical vortex and circumfluence solutions. A weak Darboux transformation theorem of the (2+1) -dimensional EE can be obtained for an arbitrary spectral parameter from the general symmetry group theorem. Possible applications of the vortex and circumfluence solutions to tropical cyclones, especially Hurricane Katrina 2005, are demonstrated.

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Section: Eementioning

confidence: 99%

“…From Theorem 1, we know that the (2+1)-dimensional Euler equation is weak Lax integrable. So we can apply the new direct method developed in [22] to find some complicated exact solutions from some simple special trivial ones after ruling out the ambiguity mentioned in remark 2.…”

Section: Eementioning

confidence: 99%

Vortices, circumfluence, symmetry groups, and Darboux transformations of the(2+1)-dimensional Euler equation

et al. 2007

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“…Soon after, Lou optimised the above-mentioned approach and presented the improved CK method [6,7]. Most recently, Lou and Ma [8] applied the direct method of symmetry transformation group for Lax integrable system in place of the traditional method of solving symmetry transformation group.…”

Section: Introductionmentioning

confidence: 98%

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

Fei

2016

Zeitschrift Für Naturforschung A

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From the known Lax pair of the Korteweg-de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

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“…A new method was later developed to directly find finite symmetry transformation groups for Lax integrable nonlinear physical systems by considering the integrable system and its Lax pair to be form invariant under a type of symmetry group transformation, and then point symmetry algebras can be readily obtained. It has been realized for many systems such as the Kadomtsev Petviashvili equation, the dispersive long wave equation and the extended self dual Yang Mills equations [22]. More recently, the Gardner method which is traditionally utilized to find conservation laws of integrable equations has been generalized to generate the infinite hierarchy of symmetries of integrable equations, and its relation with the Lenard recursion has been also discussed [28].…”

Section: Introductionmentioning

confidence: 99%

Nonlocal symmetries and conservation laws of the Sinh-Gordon equation

Tang

Liang

2021

JNMP

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Nonlocal symmetries of the (1+1)-dimensional Sinh-Gordon (ShG) equation are obtained by requiring it, together with its Bäcklund transformation (BT), to be form invariant under the infinitesimal transformation. Naturally, the spectrum parameter in the BT enters the nonlocal symmetries, and thus through the parameter expansion procedure, infinitely many nonlocal symmetries of the ShG equation can be generated accordingly. Making advantages of the consistent conditions introduced when solving the nonlocal symmetires, some new nonlocal conservation laws of the ShG equation related to the nonlocal symmetries are obtained straightforwardly. Finally, taking the nonlocal symmetries as symmetry constraint conditions imposing on the BT, some new finite and infinite dimensional nonlinear systems are constructed.

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Finite symmetry transformation groups and exact solutions of Lax integrable systems

Cited by 41 publications

References 25 publications

Vortices, circumfluence, symmetry groups, and Darboux transformations of the(2+1)-dimensional Euler equation

Vortices, circumfluence, symmetry groups, and Darboux transformations of the(2+1)-dimensional Euler equation

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

Nonlocal symmetries and conservation laws of the Sinh-Gordon equation

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