Non-Paraxial Nonlinear Schr ¨ odinger Equation: Lie Symmetry and Travelling/Solitary Waves K. Sakkaravarthi Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirapalli–620 024, TN. Email: [email protected] Website: www.ksakkaravarthi.ml Abstract We consider a non-paraxial nonlinear Schr ¨ odinger (NNLS) equation describing beam propagation in a planar waveguide with Kerr-like nonlinearity 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 Painlev´ e analysis. 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 evolution equations (NLEEs) and exploring their underlying dynamics remain the central focus of research for the past few decades. Finding exact solutions of these NLEEs (either ODEs or PDEs) is one of the most important tasks and their further investigation plays a crucial role in understanding 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 a given system of NLEE(s) is nothing but an infinitesimal transformation of all of its (their) independent and dependent variables, which leaves the corresponding equations invariant. With such invariance propetry, a given system of equations can be reduced to a set of ODEs/PDEs with lesser number of independent coordinates [2]. 2. Model: NNLS Equation • Ultra-broad beam propagation in a Kerr-like nonlinear medium under the non-paraxial approximation [3]: iq t + kq tt + q xx + γ |q | 2 q = 0, (1) Here q (t , x ): complex envelope mode, t : longitudinal coordinate, x : transverse coordinate, k : non-paraxial coefficient, γ : nonlinearity coefficient. • 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 )): u t + kv tt + v xx + γ v (u 2 + v 2 )= 0, (2a) v t - ku tt - u xx - γ u (u 2 + v 2 )= 0. (2b) • Infinitesimal symmetry transformations: u → u + η 1 (x , t , u , v ), v → v + η 2 (x , t , u , v ), t → t + τ (x , t , u , v ), x → x + ξ (x , t , u , v ). • Infinitesimal coefficients satisfying the symmetry/ invariance conditions: τ = c 1 + 2kc 4 x , ξ = c 2 - 2c 4 t , η 1 =(c 4 x - c 3 )v , η 2 = -(c 4 x - c 3 )u , (3) where c 1 , c 2 , c 3 and c 4 are arbitrary real constants. • Generalized vector fields of Eqn. (2): X = τ∂ t + ξ∂ x + η 1 ∂ u + η 2 ∂ v . • Lie point symmetry generators: X 1 = ∂ t , X 2 = ∂ x , X 3 = -v ∂ u + u ∂ v , X 4 = 2kx ∂ t - 2t ∂ x + xv ∂ u - xu ∂ v . (4) The generators X 1 , X 2 , and X 3 can be associated with translation in time, translation in space, and phase transformations, respectively. • Symmetry reductions are obtained by solving the characteristic equation: dt τ = dx ξ = du 1 η 1 = du 2 η 2 . (5) 4. Sub-groups and Invariants For each case, we analyze the resulting set of coupled ODEs and obtain exact analytical/numerical (travelling/solitary wave) solutions. 4a. Case (i) X 1 + aX 2 : Integrability & Travelling/Periodic waves • On substituting the group-invariant solutions in (2), we get the following system of nonlinear second-order ODEs: (1 + ka 2 )A 00 + aB 0 + γ A(A 2 + B 2 )= 0, (6a) (1 + ka 2 )B 00 - aA 0 + γ B (A 2 + B 2 )= 0. (6b) Here ‘prime’ represents differentiation with respect to the variable y . • Painlev ´ e analysis: Completely integrable for arbitrary a, k & γ . • Two special cases of (6) is also possible for k = -1/a 2 & a = 0. • Modified Prelle-Singer method: First integrals I 1 = 2(A 02 + B 02 )+ γ (A 2 +B 2 ) 2 1+ka 2 & I 2 =(A 0 B - AB 0 )+ a(A 2 +B 2 ) 2(1+ka 2 ) . • 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 with γ = 2 and a = -1. A: solid-red line & B : dashed-blue line. 4b. Case (iii) X 1 + aX 2 : Integrability & Solitary waves • Her we get the following system of nonlinear second-order ODEs: (1 + ka 2 )A 00 - a(1 + 2kb )B 0 - b (1 + kb )A + γ A(A 2 + B 2 )= 0, (7a) (1 + ka 2 )B 00 + a(1 + 2kb )A 0 - b (1 + kb )B + γ B (A 2 + B 2 )= 0. (7b) Painlev ´ e analysis: Completely integrable for arbitrary a, b , k & γ . • Admits a special bright solitary wave solution for k = -1 2b & γ> 0: A = a 1 sech(k 1 y + k 2 ), B = a 2 sech(k 1 y + k 2 ), (8) where k 2 1 = b (1+kb ) 1+ka 2 and a 2 2 = 2b (1+kb ) γ - a 2 1 . • 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, k 2 = -7. 5. Conclusions • Considered a non-integrable non-paraxial nonlinear Schr¨ odinger equation and obtained its equivalent integrable reductions by using the Lie symmetry analysis. • Proved their Painlev´ e 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 National Post-Doctoral Fellowship (PDF/2016/000547). References [1] M Lakshmanan, S Rajasekar. Nonlinear Dynamics: Integrability, Chaos and Patterns, Springer-Verlag, New York, 2003. [2] NH Ibragimov. CRC Handbook of Lie 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.