Research Interests
Nonlinear Dynamics
Nonlinear Phenomena; Differential Equations; Integrable & Non-integrable Systems; Symmetries;
Nonlinear Waves; Vector Solitons; Breathers; Rogue waves; Higher Dimensional Models;
Optical Systems; Bose-Einstein Condensates; Hydrodynamics; Biological Systems
Nonlinear Dynamics
Nonlinear Phenomena; Differential Equations; Integrable & Non-integrable Systems; Symmetries;
Nonlinear Waves; Vector Solitons; Breathers; Rogue waves; Higher Dimensional Models;
Optical Systems; Bose-Einstein Condensates; Hydrodynamics; Biological Systems
Research Grants/Fellowships Received
- Duration: June 2021 - (ongoing)
Funding Agency: Asia-Pacific Center for Theoretical Physics (APCTP), Pohang, Korea
Scheme: Young Scientist Training program
Project Title: Dynamics of Localized Nonlinear Coherent Structures and their Applications
Implementing Institute: Asia-Pacific Center for Theoretical Physics (APCTP), Pohang
- Duration: August 2016–July 2018
Funding Agency: DST - Science & Engineering Research Board, Govt. of India
Scheme: National Post-Doctoral Fellowship
Project Title: Symmetry and Geometry Associated with Certain Nonlinear Evolution Equations
Implementing Institute: Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli, India - Duration: April 2013–March 2016
Funding Agency: Council of Scientific & Industrial Research (CSIR), Govt. of India
Scheme: Direct Senior Research Fellowship
Implementing Institute: Bishop Heber College (Autonomous), Tiruchirappalli, India - Duration: August 2014
Funding Agency: DAE - National Board for Higher Mathematics (NBHM), Govt. of India
Scheme: International Travel Grant
Purpose: To visit Instituto de Ciances Matemáticas (ICMAT), Madrid, Spain - Duration: July 2014
Funding Agency: DST - Scientific and Engineering Research Board (SERB), Govt. of India
Scheme: International Travel Grant
Purpose: To visit Churchill College of University of Cambridge, UK - Duration: April 2011 – March 2013
Funding Agency: DST - Scientific and Engineering Research Board (SERB), Govt. of India
Scheme: Project Assistantship from the major research project (Multicomponent solitons in nonlinear optics) of Dr. T. Kanna
Implementing Institute: Bishop Heber College (Autonomous), Tiruchirappalli, India
Summary of Research Contributions
My research is focussed on Nonlinear Dynamics with special emphasis on exploring the dynamics of different coherent nonlinear wave structures.
Nonlinear dynamics is one of the frontier research topics as it explains various physical systems in almost all branches of science, engineering, and technology. There exist different types of waves in diverse context of nonlinear models starting from simple travelling waves to periodic, solitary and several localized wave structures. Solitons (self-reinforcing waves that maintain their properties) are important nonlinear entities in the recent years that arise as the solutions of integrable nonlinear partial differential equations governing various dynamical systems and find multifaceted applications due to their ability to propagate over extraordinary distances without any loss of energy and remarkable stability under collisions. Consequently, they give rise to several interesting features in the associated nonlinear dynamical systems. Especially, the multicomponent solitons show distinct propagation characteristics and posses fascinating energy-sharing collisions which are not possible in scalar (single component) solitons. Based on the above, my study was focussed on the dynamics of multicomponent solitons in several nonlinear systems with different types of nonlinearities arising in the context of nonlinear optics, Bose-Einstein condensates and hydrodynamics.
1. Solitons in Coherently Coupled NLS systems
We have successfully implemented a non-standard Hirota’s bilinearization procedure to an integrable multicomponent coherently coupled nonlinear Schrödinger (M-CCNLS) system describing the simultaneous propagation of multiple coherently coupled orthogonally polarized beams in Kerr type nonlinear media and constructed soliton solutions which are further classified as coherently coupled solitons and incoherently coupled solitons based on the presence and absence of four-wave mixing nonlinearity, respectively, admitting various profiles like single-hump, double-hump and flat-top structures. We found from a detailed asymptotic analysis that, though the total energy of the system remains conserved, energy in individual components are not conserved which leads to an interesting observation of energy-switching collision of solitons. Also, we have explored the energy-sharing and elastic collisions as special cases. These can be realized in soliton collision based optical computing for intriguing applications.
My research is focussed on Nonlinear Dynamics with special emphasis on exploring the dynamics of different coherent nonlinear wave structures.
Nonlinear dynamics is one of the frontier research topics as it explains various physical systems in almost all branches of science, engineering, and technology. There exist different types of waves in diverse context of nonlinear models starting from simple travelling waves to periodic, solitary and several localized wave structures. Solitons (self-reinforcing waves that maintain their properties) are important nonlinear entities in the recent years that arise as the solutions of integrable nonlinear partial differential equations governing various dynamical systems and find multifaceted applications due to their ability to propagate over extraordinary distances without any loss of energy and remarkable stability under collisions. Consequently, they give rise to several interesting features in the associated nonlinear dynamical systems. Especially, the multicomponent solitons show distinct propagation characteristics and posses fascinating energy-sharing collisions which are not possible in scalar (single component) solitons. Based on the above, my study was focussed on the dynamics of multicomponent solitons in several nonlinear systems with different types of nonlinearities arising in the context of nonlinear optics, Bose-Einstein condensates and hydrodynamics.
1. Solitons in Coherently Coupled NLS systems
We have successfully implemented a non-standard Hirota’s bilinearization procedure to an integrable multicomponent coherently coupled nonlinear Schrödinger (M-CCNLS) system describing the simultaneous propagation of multiple coherently coupled orthogonally polarized beams in Kerr type nonlinear media and constructed soliton solutions which are further classified as coherently coupled solitons and incoherently coupled solitons based on the presence and absence of four-wave mixing nonlinearity, respectively, admitting various profiles like single-hump, double-hump and flat-top structures. We found from a detailed asymptotic analysis that, though the total energy of the system remains conserved, energy in individual components are not conserved which leads to an interesting observation of energy-switching collision of solitons. Also, we have explored the energy-sharing and elastic collisions as special cases. These can be realized in soliton collision based optical computing for intriguing applications.
The exact bright one- and two-soliton solutions of a two-component coherently coupled nonlinear Schrödinger (mixed type 2-CCNLS) system with opposite signs of self-phase modulation, cross-phase modulation and four-wave-mixing nonlinearities are obtained. We found that the system admits only the coherently coupled solitons, with double-hump and flat-top profiles in addition to asymmetric single-hump structure, and undergo elastic collision always. Dynamics of bright two-soliton bound states having single-hump and double-hump bound profiles are identified and their characteristic features are investigated. Interestingly, we found the existence of regular two-soliton bound state even for the choice for which the mixed type 2-CCNLS system admits singular one-soliton solution. Also, we have revealed the possibility of controlling the breathing oscillations of soliton bound state.
2. Solitons in Long wave Short wave Systems
We have derived a generalized version of multi- component Yajima-Oikawa (M-YO) system, that results due to the resonance interaction between several short-waves and a long-wave, from multiple coupled nonlinear Schrödinger equation by asymptotic reduction and proved its integrability for arbitrary real nonlinearity coefficients. By constructing explicit bright N-soliton solution in the form of Gram determinant we have studied their propagation and collisions. Particularly, we have identified two types of energy-sharing collisions in the short-wave components corresponding to same and mixed signs of nonlinearities, which have not been reported earlier. Also, obtained the standard elastic collisions for special choice of nonlinearities along with the interactions of asymptotic bright, dark, anti-dark and W-shaped solitons. Additionally, we have explored the three-soliton collision and confirmed their pair-wise nature of collisions. Apart from these, we have constructed dark soliton solutions of M-YO system and revealed the feature of nonlinearity coefficients in their propagation and elastic collisions.
We have derived a generalized version of multi- component Yajima-Oikawa (M-YO) system, that results due to the resonance interaction between several short-waves and a long-wave, from multiple coupled nonlinear Schrödinger equation by asymptotic reduction and proved its integrability for arbitrary real nonlinearity coefficients. By constructing explicit bright N-soliton solution in the form of Gram determinant we have studied their propagation and collisions. Particularly, we have identified two types of energy-sharing collisions in the short-wave components corresponding to same and mixed signs of nonlinearities, which have not been reported earlier. Also, obtained the standard elastic collisions for special choice of nonlinearities along with the interactions of asymptotic bright, dark, anti-dark and W-shaped solitons. Additionally, we have explored the three-soliton collision and confirmed their pair-wise nature of collisions. Apart from these, we have constructed dark soliton solutions of M-YO system and revealed the feature of nonlinearity coefficients in their propagation and elastic collisions.
In continuation, we have generalized a (2 + 1)D version of M-YO system (called M-LSRI system) and studied its integrability as well as constructed the exact bright N-soliton solution. We have explored two types of energy-sharing collisions of the higher dimensional system for same-sign and mixed-signs of nonlinearities that can also exhibit head-on and overtaking collisions simultaneously in two different dimensions. Additionally, we have constructed the higher dimensional dark soliton solutions and analyzed their propagation as well as collision dynamics that are always elastic.
3. Soliton-like Coherent Structures
Interestingly, we have demonstrated the formation of resonant solitons admitting localized coherent structures akin to breathers and rogue-waves co-existing with solitons as a long-living intermediate state under special parametric choices for which the phase-shift after collision becomes infinity during collisions. This opens a new way for the generation and evolution of breathers/rogue waves that needs further research. We have carried out a critical analysis and unraveled the dynamics of resonant solitons/breathers resulting for infinite phase-shift. Especially, we have obtained an inclined breather, time localized (space- periodic) Akhmediev breather, space-localized (time-periodic) Ma breather and space-time localized rogue wave for different choices of soliton parameters. Also, we have shown the presence of resonant solitons with multiple oscillating side- bands. Investigations on the dynamics of bright- soliton bound-state of LSRI system shows that the periodic beating and breathing oscillations can be tailored by the parameters resulting from the higher-dimensional and multicomponent nature of the system. Also, it reveals the possibility for a transition from soliton bound-state to colliding solitons (elastic or energy sharing collisions) and its collision with a standard soliton.
Interestingly, we have demonstrated the formation of resonant solitons admitting localized coherent structures akin to breathers and rogue-waves co-existing with solitons as a long-living intermediate state under special parametric choices for which the phase-shift after collision becomes infinity during collisions. This opens a new way for the generation and evolution of breathers/rogue waves that needs further research. We have carried out a critical analysis and unraveled the dynamics of resonant solitons/breathers resulting for infinite phase-shift. Especially, we have obtained an inclined breather, time localized (space- periodic) Akhmediev breather, space-localized (time-periodic) Ma breather and space-time localized rogue wave for different choices of soliton parameters. Also, we have shown the presence of resonant solitons with multiple oscillating side- bands. Investigations on the dynamics of bright- soliton bound-state of LSRI system shows that the periodic beating and breathing oscillations can be tailored by the parameters resulting from the higher-dimensional and multicomponent nature of the system. Also, it reveals the possibility for a transition from soliton bound-state to colliding solitons (elastic or energy sharing collisions) and its collision with a standard soliton.
4. Solitons in Spinor BECs
We have explored the integrability nature of the three-component Gross-Pitaevskii (3-GP) equa- tions describing the dynamics of spin-1 Bose- Einstein condensates and found that the 3-GP system passes Painlevé test only for two choices of coupling constants. We have obtained exact bright one- and two-soliton solutions by using the non-standard Hirota’s bilinearization method and classified them as polar and ferromagnetic solitons exhibiting single-hump, double-hump and flat-top profiles. Different types of two-soliton interactions among these solitons are also investi- gated by performing the asymptotic analysis and revealed the spin-switching, inelastic and elastic interactions. Apart from these, we have also investigated such soliton propagation and interactions in a non-autonomous 3-GP model with a kink-type nonlinearity and harmonic potential.
We have explored the integrability nature of the three-component Gross-Pitaevskii (3-GP) equa- tions describing the dynamics of spin-1 Bose- Einstein condensates and found that the 3-GP system passes Painlevé test only for two choices of coupling constants. We have obtained exact bright one- and two-soliton solutions by using the non-standard Hirota’s bilinearization method and classified them as polar and ferromagnetic solitons exhibiting single-hump, double-hump and flat-top profiles. Different types of two-soliton interactions among these solitons are also investi- gated by performing the asymptotic analysis and revealed the spin-switching, inelastic and elastic interactions. Apart from these, we have also investigated such soliton propagation and interactions in a non-autonomous 3-GP model with a kink-type nonlinearity and harmonic potential.
5. Solitons in Networks
It is well known that solitons in integrable systems recover their original profiles after their mutual collisions. But, this is not necessarily be true in the case of optical fibre arrays, governed by a set of integrable coupled nonlinear Schrödinger (CNLS) equations. We have considered the Manakov and mixed-type ‘two-component’ CNLS systems which possess two important characteristics: (1) The polarizations of the two-component solitons are changed through their mutual collisions (Manakov system) and (2) the energy/intensity switching occurs through the head-on collision (mixed system). By placing the above solitons on the primary star graph (PSG), we have seen that soliton collisions give rise to interesting phase changes in PSG: (a) The transition in PSG from its depolarized state to polarized one; (b) A state with selectively amplified bond is generated on PSG from its homogeneous state. These results will be applicable to network protocols using optical fibre arrays.
It is well known that solitons in integrable systems recover their original profiles after their mutual collisions. But, this is not necessarily be true in the case of optical fibre arrays, governed by a set of integrable coupled nonlinear Schrödinger (CNLS) equations. We have considered the Manakov and mixed-type ‘two-component’ CNLS systems which possess two important characteristics: (1) The polarizations of the two-component solitons are changed through their mutual collisions (Manakov system) and (2) the energy/intensity switching occurs through the head-on collision (mixed system). By placing the above solitons on the primary star graph (PSG), we have seen that soliton collisions give rise to interesting phase changes in PSG: (a) The transition in PSG from its depolarized state to polarized one; (b) A state with selectively amplified bond is generated on PSG from its homogeneous state. These results will be applicable to network protocols using optical fibre arrays.
6. Symmetries of Nonlinear Systems
We have investigated the symmetry reductions of the nonlinear Helmholtz (NLH) equation arising in the context of nonlinear optics by using the Lie symmetry analysis. Especially, with an infinitesimal transformation, we have obtained the symmetry generators (vector fields), identified a set of invariants from the optimal system of one- dimensional sub-algebra and symmetry reductions in the form of coupled ODEs. We have studied the integrability property of the reduced ODEs by using the Painlevé analysis and constructed explicit first integrals by using the modified Prelle-Singer method. In the study of NLH system, we have also obtained nonlinear periodic and solitary wave solutions for some ODEs resulting for specific symmetry reductions. We have explored some interesting nonlinear wave structures resulting for different choices of arbitrary parameters. The obtained nonlinear periodic wave solutions can find applications in the context of fiber Bragg gratings, bimodal fibers, etc. Further, we have compared the known Lie symmetry analysis of the nonlinear Schrödinger equation with the results of the present NLH equation. Our analysis shows the possible symmetry reductions of a scalar nonlinear partial differential equation. This study can be extended to investigate several coupled nonlinear systems of physical interest.
We have investigated the symmetry reductions of the nonlinear Helmholtz (NLH) equation arising in the context of nonlinear optics by using the Lie symmetry analysis. Especially, with an infinitesimal transformation, we have obtained the symmetry generators (vector fields), identified a set of invariants from the optimal system of one- dimensional sub-algebra and symmetry reductions in the form of coupled ODEs. We have studied the integrability property of the reduced ODEs by using the Painlevé analysis and constructed explicit first integrals by using the modified Prelle-Singer method. In the study of NLH system, we have also obtained nonlinear periodic and solitary wave solutions for some ODEs resulting for specific symmetry reductions. We have explored some interesting nonlinear wave structures resulting for different choices of arbitrary parameters. The obtained nonlinear periodic wave solutions can find applications in the context of fiber Bragg gratings, bimodal fibers, etc. Further, we have compared the known Lie symmetry analysis of the nonlinear Schrödinger equation with the results of the present NLH equation. Our analysis shows the possible symmetry reductions of a scalar nonlinear partial differential equation. This study can be extended to investigate several coupled nonlinear systems of physical interest.
7. Rogue waves & Breathers in Water Wave Systems
We have investigated certain nonlinear wave equations Boussinesq, Benjamin-Ono, and KMN model, arising in the context of water wave systems and studied the existence and dynamics of localized wave structures such as solitons, breathers, rogue waves by constructing explicit analytic solutions through bilinear forms and polynomial functions. Our analyses revealed the control mechanism of these waves by manipulating the parameters suitably. Further, we have explored the formation of different rational/solitary waves along with the bound states, fusion, fission and bending properties. The results presented in this work will be encouraging to the studies on the localized waves in other higher dimensional systems.
We have investigated certain nonlinear wave equations Boussinesq, Benjamin-Ono, and KMN model, arising in the context of water wave systems and studied the existence and dynamics of localized wave structures such as solitons, breathers, rogue waves by constructing explicit analytic solutions through bilinear forms and polynomial functions. Our analyses revealed the control mechanism of these waves by manipulating the parameters suitably. Further, we have explored the formation of different rational/solitary waves along with the bound states, fusion, fission and bending properties. The results presented in this work will be encouraging to the studies on the localized waves in other higher dimensional systems.
8. Tunable Solitons, Breathers & Rogue waves
We have studied a system of coherently coupled nonlinear Schrödinger equations with modulated self-phase modulation, cross-phase modulation, and four-wave mixing nonlinearities and varying refractive index in anisotropic graded index nonlinear medium. By identifying a generalized similarity transformation, we obtain a general localized wave solutions and investigate their dynamics with a proper set of modulated nonlinearities. Our study reveals different manifestations of localized waves such as stable solitons, Akhmediev breathers, Ma breathers, and rogue waves of bright, bright-dark, and dark-dark type and explores their manipulation mechanism with suitably engineered nonlinearity parameters.
Further, with the help of a non-standard Hirota’s bilinearization method and exact soliton solutions, we have explored the impact of varying nonlinearities and refractive index in the propagation and collisions of optical bright solitons by reverse engineering. Interestingly, we have shown the emergence of several modulated solitonic phenomena such as periodic oscillation, amplification, compression, tunneling/cross-over, excitons, as well as their combined effect in the single-soliton propagation and two-soliton collisions with appropriate forms of nonlinearity. Notably, we have identified a tool to transform the nature of soliton collisions with certain type of inhomogeneous nonlinearities.
Presently, working on the theoretical exploration of dynamics and control of various nonlinear waves in one- and higher-dimensional nonlinear systems.
We have studied a system of coherently coupled nonlinear Schrödinger equations with modulated self-phase modulation, cross-phase modulation, and four-wave mixing nonlinearities and varying refractive index in anisotropic graded index nonlinear medium. By identifying a generalized similarity transformation, we obtain a general localized wave solutions and investigate their dynamics with a proper set of modulated nonlinearities. Our study reveals different manifestations of localized waves such as stable solitons, Akhmediev breathers, Ma breathers, and rogue waves of bright, bright-dark, and dark-dark type and explores their manipulation mechanism with suitably engineered nonlinearity parameters.
Further, with the help of a non-standard Hirota’s bilinearization method and exact soliton solutions, we have explored the impact of varying nonlinearities and refractive index in the propagation and collisions of optical bright solitons by reverse engineering. Interestingly, we have shown the emergence of several modulated solitonic phenomena such as periodic oscillation, amplification, compression, tunneling/cross-over, excitons, as well as their combined effect in the single-soliton propagation and two-soliton collisions with appropriate forms of nonlinearity. Notably, we have identified a tool to transform the nature of soliton collisions with certain type of inhomogeneous nonlinearities.
Presently, working on the theoretical exploration of dynamics and control of various nonlinear waves in one- and higher-dimensional nonlinear systems.