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Conservation Laws of Deformed N-Coupled Nonlinear Schrödinger Equations and Deformed N-Coupled Hirota Equations

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Abstract

In this paper, we consider two deformed equations namely, deformed N-coupled nonlinear Schrödinger (N-coupled NLS) equations and deformed N-coupled Hirota equations and show that both of them admit an infinitely many conservation laws, which ensures their complete integrability. The conservation laws of the above equations have been constructed by using their Lax representations through Ricatti equations.

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Acknowledgements

The authors are thankful to the anonymous referees for their constructive suggestions. The first author (S.S) would like to thank the Management and Principal of C. Abdul Hakeem College (Autonomous), Melvisharam, for their support and encouragement. The second author (R.S) is supported by Council of Scientific industrial Research(CSIR), New Delhi under Emeritus Scientist Scheme.

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Correspondence to S. Suresh Kumar.

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Appendix

Appendix

For the scalar deformed NLS equation (5), we have constructed another infinite sequence of conserved quantities \((\rho _j,F_j)\). For this purpose, we assume \(\varPhi _1(x,t) = T(x,t) \varPhi _2(x,t)\) and carry out similar analysis outlined in “Conserved quantities of scalar deformed NLS equation” subsection, the scalar deformed NLS equation (5) admits the following conserved quantities \((\rho _j,F_j)\). The conserved densities and associated fluxes are respectively given by,

$$\begin{aligned} \rho _1= & {} u_1= \dfrac{-iqq^*}{2},\\ \rho _2= & {} u_2=\dfrac{q^* q_x}{4},\\ \rho _3= & {} u_3=\dfrac{iq^*(q^*q^{2}+q_{xx})}{8}, \\ \rho _4= & {} u_4=-\dfrac{q q^*(q q^*_x+4q^*q_x)+q^* q_{xxx}}{16}, \\&\vdots \\ \rho _{j}= & {} u_j ,j\ge 1 \end{aligned}$$

and

$$\begin{aligned} F_1= & {} -2\ u_2+\dfrac{iu_1q^*_x}{q^*}+\dfrac{ih}{2},\\ F_2= & {} - 2\ u_3+\dfrac{iu_2q^*_x}{q^*}+\dfrac{u_1g^*}{2 q^*},\\ F_3= & {} -2\ u_4+\dfrac{iu_3q^*_x}{q^*}+\dfrac{u_2g^*}{2 q^*},\\ F_4= & {} -2\ u_5+\dfrac{iu_4q^*_x}{q^*}+\dfrac{u_3g^*}{2 q^*},\\&\vdots \\ F_j= & {} -2\ u_{j+1}+\dfrac{iu_jq^*_x}{q^*}+\dfrac{u_{j-1}g^*}{2 q^*},j=2,3,4,\ldots ,\qquad \end{aligned}$$

where

$$\begin{aligned} u_1= & {} \dfrac{-iqq^*}{2},\\ u_2= & {} \dfrac{iu_{1x}}{2}-\dfrac{iu_1q^*_{x}}{2q^*},\\ u_3= & {} \dfrac{iu_{2x}}{2}-\dfrac{iu_2q^*_{x}}{2q^*}-\dfrac{iu_1^2}{2}, \\ u_4= & {} \dfrac{iu_{3x}}{2}-\dfrac{iu_3q^*_{x}}{2q^*}-iu_1u_2, \\ u_5= & {} \dfrac{iu_{4x}}{2}-\dfrac{iu_4q^*_{x}}{2q^*}-\dfrac{iu_2^2}{2}-iu_1u_3, \\ u_6= & {} \dfrac{iu_{5x}}{2}-\dfrac{iu_5q^*_{x}}{2q^*}-iu_1u_4-iu_2u_3, \\&\vdots \\ u_j= & {} \dfrac{iu_{(j-1)x}}{2}-\dfrac{iu_{(j-1)}q^*_{x}}{2q^*}-\dfrac{i}{2}\sum \limits _{k=1}^{j-2}u_ku_{j-k-1},j\ge 3. \end{aligned}$$

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Suresh Kumar, S., Sahadevan, R. Conservation Laws of Deformed N-Coupled Nonlinear Schrödinger Equations and Deformed N-Coupled Hirota Equations. Int. J. Appl. Comput. Math 6, 19 (2020) doi:10.1007/s40819-019-0766-0

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Keywords

  • Lax pair
  • Conservation laws
  • Deformed N-coupled nonlinear Schrödinger equations
  • Deformed N-coupled Hirota equations
  • Integrability