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Analytic methods to generate integrable mappings

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Abstract

Systematic analytic methods of deriving integrable mappings from integrable nonlinear ordinary differential, differential-difference and lattice equations are presented. More specifically, we explain how to derive integrable mappings through four different techniques namely, (i) discretization technique, (ii) Lax pair approach, (iii) periodic reduction of integrable nonlinear partial difference equations and (iv) construction of sufficient number of integrals of motion. The applicability of methods have been illustrated through Ricatti equation, a scalar second-order nonlinear ordinary differential equation with cubic nonlinearity, 2- and 3-coupled second-order nonlinear ordinary differential equations with cubic nonlinearity, lattice equations of Korteweg–de Vries, modified Korteweg–deVries and sine-Gordon types.

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Acknowledgement

This work forms part of the start-up grant given by UGC and DST PURSE grant given by University of Madras.

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Authors and Affiliations

  1. Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, 600 005, India

    C UMA MAHESWARI & R SAHADEVAN

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  1. C UMA MAHESWARI
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  2. R SAHADEVAN
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Correspondence to R SAHADEVAN.

Appendix A

Appendix A

The entries of M 1(n,λ), M 2(n,λ), M 3(n,λ), M 4(n,λ), L 4(n) for mappings (29)–(34) are listed below. Mapping (29)

$$\begin{array}{@{}rcl@{}} && M_{1}(n,\lambda)=\frac{a_{1}x_{n+1}+a_{4}+(a_{1}x_{n+1}+a_{1})[x_{n}+x_{n+2}]+a_{1}x_{n}x_{n+1}x_{n+2}}{x_{n}x_{n+2}},\\ && M_{2}(n,\lambda)=\frac{1}{x_{n}},\quad L_{4}(n)=-M_{1}(n,\lambda)x_{n}x_{n+1} ,\\ && M_{3}(n,\lambda)=(\lambda +M_{1}(n,\lambda))x_{n},\\ && M_{4}(n,\lambda)\,=\,\frac{a_{1}x_{n}x_{n+2}\,+\,a_{1}x_{n}\,+\,a_{1}x_{n+2}\,+\,a_{1}\,+\,[a_{1}x_{n+2}\,+\,a_{5}x_{n}x_{n+2}\,+\,a_{1}x_{n}\,+\,1]x_{n+1}}{x_{n+1}}. \end{array} $$

Mapping (30)

$$\begin{array}{@{}rcl@{}} M_{1}(n,\lambda)&=&\frac{a_{4}x_{n+1}+a_{2}x_{n+1}^{2}+a_{2}x_{n+1}[x_{n}+x_{n+2}]}{x_{n}x_{n+2}},\\ M_{2}(n,\lambda)&=&\frac{1}{x_{n}},\quad L_{4}(n)=-M_{1}(n,\lambda)x_{n}x_{n+1} ,\\ &&M_{3}(n,\lambda)=(\lambda +M_{1}(n,\lambda))x_{n},\\ M_{4}(n,\lambda)&=&\frac{[a_{5}x_{n}x_{n+2}+a_{2}x_{n+2}+a_{2}x_{n}]x_{n+1}+a_{2}x_{n}x_{n+2}+x_{n+1}^{2}}{x_{n+1}^{2}}. \end{array} $$

Mapping (31)

$$\begin{array}{@{}rcl@{}} M_{1}(n,\lambda)&=&\frac{1}{2}t[a_{4}[x_{n}+x_{n+1}+x_{n+2}]+a_{2}[x_{n}+x_{n+2}]x_{n+1}\\&&+a_{3}x_{n}x_{n+2}+a_{1}x_{n}x_{n+1}x_{n+2}],\\ M_{2}(n,\lambda)&=&\frac{1}{x_{n}},\quad L_{4}(n)=-M_{1}(n,\lambda)x_{n}x_{n+1} ,\\ &&M_{3}(n,\lambda)=(\lambda +M_{1}(n,\lambda))x_{n},\quad M_{4}(n,\lambda)=M_{1}(n,\lambda)+1. \end{array} $$

Mapping (32)

$$\begin{array}{@{}rcl@{}} M_{1}(n,\lambda)&=&\frac{1}{2}\left[a_{4}x_{n}x_{n+1}+a_{4}x_{n+1}x_{n+2}+a_{3}x_{n}x_{n+1}x_{n+2}+a_{1}x_{n}x_{n+1}^{2}x_{n+2}\right]\\ M_{2}(n,\lambda)&=&\frac{1}{x_{n}},\quad L_{4}(n)=-M_{1}(n,\lambda)x_{n}x_{n+1} ,\\ &&M_{3}(n,\lambda)=(\lambda +M_{1}(n,\lambda))x_{n},\quad M_{4}(n,\lambda)= M_{1}(n,\lambda)+1. \end{array} $$

Mapping (33)

$$\begin{array}{@{}rcl@{}} M_{1}(n,\lambda)&=&\frac{1}{2}[a_{1}[(x_{n}+x_{n+1})x_{n+2}^{2}+{x_{n}^{2}}(x_{n+2}+x_{n+1})\\&&+2x_{n}x_{n+1}x_{n+2}-(x_{n+2}+x_{n})x_{n+1}^{2}]\\&&+a_{2}[(x_{n}+x_{n+1})x_{n+2}+(x_{n}-x_{n+1})x_{n+1})\\&&+a_{3}(x_{n}+x_{n+2}] +a_{4}[(x_{n+2}+x_{n}-x_{n+1}]x_{n}x_{n+1}x_{n+2})\\&&+a_{5}[{x_{n}^{2}}+x_{n+2}^{2}+x_{n+1}^{2}+x_{n}x_{n+2}-x_{n+1}x_{n+2}-x_{n}x_{n+1}]],\\ M_{2}(n,\lambda)&=&\frac{1}{x_{n}},\quad L_{4}(n)=-M_{1}(n,\lambda)x_{n}x_{n+1} ,\\ &&M_{3}(n,\lambda)=(\lambda +M_{1}(n,\lambda))x_{n},\quad M_{4}(n,\lambda)=M_{1}(n,\lambda)+1. \end{array} $$

Mapping (34)

$$\begin{array}{@{}rcl@{}} M_{1}(n,\lambda)&=&\frac{1}{2}[a_{1}[(x_{n}+x_{n+1})x_{n+2}^{2}+x_{n+1}^{2}(x_{n+2}+x_{n})\\&&+{x_{n}^{2}}(x_{n+1} +x_{n+2})+2x_{n}x_{n+1}x_{n+2}]\\ && +a_{2}[(x_{n}+x_{n+1})x_{n+2}+x_{n}x_{n+1}]\\&&+a_{3}[x_{n}+x_{n+1}+x_{n+2}] +a_{4}[{x_{n}^{2}}+x_{n+1}^{2}+x_{n+2}^{2}]], \\ M_{2}(n,\lambda)&=&\frac{1}{x_{n}},\quad L_{4}(n)=-M_{1}(n,\lambda)x_{n}x_{n+1} ,\\ &&M_{3}(n,\lambda)=(\lambda +M_{1}(n,\lambda))x_{n} , \quad M_{4}(n,\lambda)=M_{1}(n,\lambda)+1. \end{array} $$

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MAHESWARI, C.U., SAHADEVAN, R. Analytic methods to generate integrable mappings. Pramana - J Phys 85, 807–821 (2015). https://doi.org/10.1007/s12043-015-1102-9

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