Non-Paraxial Nonlinear Schrodinger Equation:Lie Symmetry and Travelling/Solitary Waves

K. SakkaravarthiCentre for Nonlinear Dynamics, Bharathidasan University, Tiruchirapalli–620 024, TN.

Email: ksakkaravarthi@gmail.com Website: www.ksakkaravarthi.ml

Abstract

We consider a non-paraxial nonlinear Schrodinger (NNLS) equation describing beam propagation in a planar waveguide with Kerr-likenonlinearity under non-paraxial approximation and obtain its symmetry reductions in the form of coupled ordinary differential equations(ODEs) by using the Lie symmetry analysis. Then we study the integrability nature of the reduced ODEs by performing the Painleveanalysis. Finally, we obtain their first integrals and construct periodic as well as solitary wave solutions by analytical/numerical method.

1. Introduction

• Investigation of physical systems described by nonlinear evolutionequations (NLEEs) and exploring their underlying dynamics remain thecentral focus of research for the past few decades. Finding exactsolutions of these NLEEs (either ODEs or PDEs) is one of the mostimportant tasks and their further investigation plays a crucial role inunderstanding several nonlinear phenomena [1].•Lie symmetry analysis has been proved to be a powerful tool for studying

nonlinear problems arising in many scientific fields. Lie symmetry of agiven system of NLEE(s) is nothing but an infinitesimal transformation ofall of its (their) independent and dependent variables, which leaves thecorresponding equations invariant. With such invariance propetry, a givensystem of equations can be reduced to a set of ODEs/PDEs with lessernumber of independent coordinates [2].

2. Model: NNLS Equation

•Ultra-broad beam propagation in a Kerr-like nonlinear medium under thenon-paraxial approximation [3]:

iqt + k qtt + qxx + γ |q|2q = 0, (1)Here q(t , x): complex envelope mode, t : longitudinal coordinate, x :transverse coordinate, k : non-paraxial coefficient, γ: nonlinearitycoefficient.•The above NNLS equation (1) is non-integrable.•But admits interesting special solitary wave solutions.•For γ > 0 (γ < 0): (1)⇒ Focusing (Defocusing) type NNLS equation.•For k = 0: (1)⇒ Standard integrable NLS equation.

3. Lie Symmetry Analysis

•Write complex equation (1) into two real equations (q = u(x , t) + iv(x , t)):ut + kvtt + vxx + γ v(u2 + v2) = 0, (2a)vt − kutt − uxx − γ u(u2 + v2) = 0. (2b)

• Infinitesimal symmetry transformations:u → u + ε η1(x , t ,u, v), v → v + ε η2(x , t ,u, v),t → t + ε τ (x , t ,u, v), x → x + ε ξ(x , t ,u, v).• Infinitesimal coefficients satisfying the symmetry/ invariance conditions:τ = c1 + 2kc4x , ξ = c2 − 2c4t , η1 = (c4x − c3)v , η2 = −(c4x − c3)u, (3)

where c1, c2, c3 and c4 are arbitrary real constants.•Generalized vector fields of Eqn. (2): X = τ∂t + ξ∂x + η1∂u + η2∂v .

•Lie point symmetry generators:X1 = ∂t, X2 = ∂x, X3 = −v ∂u + u∂v , X4 = 2kx ∂t − 2t ∂x + xv ∂u − xu ∂v .(4)The generators X1, X2, and X3 can be associated with translation in time,translation in space, and phase transformations, respectively.•Symmetry reductions are obtained by solving the characteristic equation:

dtτ

=dxξ

=du1

η1=

du2

η2. (5)

4. Sub-groups and Invariants

For each case, we analyze the resulting set of coupled ODEs and obtainexact analytical/numerical (travelling/solitary wave) solutions.

4a. Case (i) X1 + aX2: Integrability & Travelling/Periodic waves

•On substituting the group-invariant solutions in (2), we get the followingsystem of nonlinear second-order ODEs:

(1 + ka2)A′′+ aB′ + γA(A2 + B2) = 0, (6a)

(1 + ka2)B′′ − aA′ + γB(A2 + B2) = 0. (6b)

Here ‘prime’ represents differentiation with respect to the variable y .•Painleve analysis: Completely integrable for arbitrary a, k & γ.•Two special cases of (6) is also possible for k = −1/a2 & a = 0.•Modified Prelle-Singer method: First integrals

I1 = 2(A′2 + B′2) + γ(A2+B2)2

1+ka2 & I2 = (A′B − AB′) + a(A2+B2)2(1+ka2)

.•Existence of two functionally independent ‘time-independent’ integrals

itself ensures the complete integrability.

Fig. 1: Periodic wave trains of Eq. (6) for (a) k = 0 and (b) k = 0.5 withγ = 2 and a = −1. A: solid-red line & B: dashed-blue line.

4b. Case (iii) X1 + aX2: Integrability & Solitary waves

•Her we get the following system of nonlinear second-order ODEs:(1 + ka2)A

′′ − a(1 + 2kb)B′ − b(1 + kb)A + γA(A2 + B2) = 0, (7a)(1 + ka2)B

′′+ a(1 + 2kb)A′ − b(1 + kb)B + γB(A2 + B2) = 0. (7b)

Painleve analysis: Completely integrable for arbitrary a, b, k & γ.•Admits a special bright solitary wave solution for k = −1

2b & γ > 0:A = a1 sech(k1y + k2), B = a2 sech(k1y + k2), (8)

where k21 = b(1+kb)

1+ka2 and a22 = 2b(1+kb)

γ − a21.

•a: Only the width and central position alter, without affecting amplitudes.•Symbiotic solitary waves: Appears only in A & vanishes in B.

Fig. 2: Solitary waves for (a) γ = 2, and a = ±1 & (b) γ = 2, and a = 0.(c) Symbiotic solitary wave for γ = 1, and a = 1. Here b = 1, k2 = −7.

5. Conclusions

•Considered a non-integrable non-paraxial nonlinear Schrodingerequation and obtained its equivalent integrable reductions by using theLie symmetry analysis.•Proved their Painleve intrgrability and obtained first integrals.•Constructed periodic/travelling/solitary wave solutions by using an

appropriate analytical/numerical method.•Extensively analyzed the influence of non-paraxial effect.

Acknowledgements

This work was carried out with the support of DST-SERB NationalPost-Doctoral Fellowship (PDF/2016/000547).

References[1] M Lakshmanan, S Rajasekar. Nonlinear Dynamics: Integrability, Chaos andPatterns, Springer-Verlag, New York, 2003. [2] NH Ibragimov. CRC Handbook ofLie group analysis of differential equations, Vol. 1-3. CRC Press, Florida, 1993-1996.[3] B Crosignani, PD Porto, A Yariv. Opt Lett 22:778:1997.

Collaborators: Dr. A.G. Johnpillai, Dr. T. Kanna and Prof. M. Lakshmanan.

******5th International Conference on Complex Dynamical Systems and Applications, 4-6 Dec. 2017, IIT Guwahati.