C-integrable nonlinear PDEs. II

Calogero, F.; Xiaoda, Ji

doi:10.1063/1.529112

Cited by 33 publications

(28 citation statements)

References 1 publication

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“…The substitution of (26) into (23) (taking into account (24)) gives the formulas (16)- (18). Theorem 3 is proved.…”

Section: (26)mentioning

confidence: 84%

“…3 that the development of the Gaussian formalism for nonlinear PDE is closely related to the problem of identifying the so-called G-integrable systems [17,18]. These G-integrable systems (equations) may be linearized by means of a certain change of variables.…”

Section: A Discussion Of the Proposed Geometrical Conception Of Diffementioning

confidence: 98%

“…According to any analytical/unction f(z) = v(x,t) + iw(x,t) of complez variables, one can always construct three type of solutions to the Liouville equation by formulas (16)- (18).…”

Section: [(K)2 (Kh Imentioning

confidence: 99%

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Non-euclidean geometry: The Gauss formula and an interpretation of partial differential equations

Poznyak¹,

Попов²

1996

J Math Sci

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“…The substitution of (26) into (23) (taking into account (24)) gives the formulas (16)- (18). Theorem 3 is proved.…”

Section: (26)mentioning

confidence: 84%

Section: A Discussion Of the Proposed Geometrical Conception Of Diffementioning

confidence: 98%

See 1 more Smart Citation

Non-euclidean geometry: The Gauss formula and an interpretation of partial differential equations

Poznyak¹,

Попов²

1996

J Math Sci

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“…For any given Gϭ(G [1] , G [2] ) T B 2 , the equation ͓V, L͔ϭL * (K•G)L Ϫ1 ϪL * (J•G) has the following operator solution: where J, L and L Ϫ1 are defined by ͑6.58͒, ͑6.59͒ and ͑6.64͒, respectively.…”

Section: ͑655͒mentioning

confidence: 99%

Category of nonlinear evolution equations, algebraic structure, and r-matrix

Zhang

Cao

Strampp

2003

Journal of Mathematical Physics

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Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations A system of nonlinear algebraic equations connected with the multisoliton solution of the Benjamin-OnoequationIn this paper we deal with the category of nonlinear evolution equations ͑NLEEs͒ associated with the spectral problem and provide an approach for constructing their algebraic structure and r-matrix. First we introduce the category of NLEEs, which is composed of various positive order and negative order hierarchies of NLEEs both integrable and nonintegrable. The whole category of NLEEs possesses a generalized Lax representation. Next, we present two different Lie algebraic structures of the Lax operator: one of them is universal in the category, i.e., independent of the hierarchy, while the other one is nonuniversal in the hierarchy, i.e., dependent on the underlying hierarchy. Moreover, we find that two kinds of adjoint maps are r-matrices under the algebraic structures. In particular, the Virasoro algebraic structures without a central extension of isospectral and nonisospectral Lax operators can be viewed as reductions of our algebraic structure. Finally, we give several concrete examples to illustrate our methods. Particularly, the Burgers' category is linearized when the generator, which generates the category, is chosen to be independent of the potential function. Furthermore, an isospectral negative order hierarchy in the Burgers' category is solved with its general solution. Additionally, in the KdV category we find an interesting fact: the Harry-Dym hierarchy is contained in this category as well as the well-known Harry-Dym equation is included in a positive order KdV hierarchy.

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“…19 Any c-integrable nonlinear system may be reduced into a linear one under a transformation ͑12͒. That is why we start from a linear system with the constant coefficients:…”

Section: C Integrabilitymentioning

confidence: 99%

Integrability and integrodifferential substitutions

Meshkov

1997

Journal of Mathematical Physics

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A semi-discrete scheme for solving nonlinear hyperbolic-type partial integro-differential equations using radial basis functions A comparative analysis of Painlevé, Lax pair, and similarity transformation methods in obtaining the integrability conditions of nonlinear Schrödinger equations Chen, Lee, and Liu presented in 1979 an algorithm for establishing integrability of two-dimensional partial differential systems. It is proved here that this algorithm is invariant under the point transformations, differential substitutions, and some integrodifferential substitutions. It is also proved that canonical conserved densities of linearizable systems arising in the frameworks of the method are almost all trivial. The integrability of the non-Newtonian liquid equations is investigated and it is proved that there exist two integrable systems only. A preliminary classification of the third-order integrable evolution systems is presented. © 1997 American Institute of Physics.Here the matrices U and V are local functions of u ␣ and a complex parameter ; L, A, and B are the linear scalar differential operators depending on ‫ץ‬ x only ͑but not ‫ץ‬ t ͒. Coefficients of the L, A, and B are local functions of u ␣ and ͑see Zakharov et al., 16 for instance͒. Equation ͑3͒ is called the zero curvature representation of the system ͑1͒. Equation ͑4͒ is called the triadic representation of the system ͑1͒ or (L,A,B) representation. Let us consider the triadic representation for definiteness. It is obvious that an operator equation Hϭ0 is equivalent to its adjoint H ϩ ϭ0, therefore the equation ͑4͒ impliesEquations ͑4͒ and ͑5͒ provide the compatibility for the following linear systems: 6429 A. G. Meshkov: Integrodifferential substitutions If the system (1) is formally integrable, then the system (8) has a solution in the form of the formal Laurent expansions, 6430 A. G. Meshkov: Integrodifferential substitutions

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C-integrable nonlinear PDEs. II

Abstract: A technique to perform a convenient Change of (independent) variables in a PDE is reported, and it is used to generate C-integrable nonlinear PDEs, i.e., nonlinear PDEs solvable by an appropriate Change of variables. Several examples of such PDEs are exhibited.

Cited by 33 publications

References 1 publication

Non-euclidean geometry: The Gauss formula and an interpretation of partial differential equations

Non-euclidean geometry: The Gauss formula and an interpretation of partial differential equations

Category of nonlinear evolution equations, algebraic structure, and r-matrix

Integrability and integrodifferential substitutions

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