Studies on Coherent Structures and their Dynamics in Nonlinear Systems

Modak, Subhrajit (2019) Studies on Coherent Structures and their Dynamics in Nonlinear Systems. PhD thesis, Indian Institute Of Science Education and Research Kolkata.

PDF (PhD thesis of Subhrajit Modak (10RS007)) 10RS007.pdf - Submitted Version Restricted to Repository staff only Download (2MB)

Abstract

This thesis is devoted to the study of nonlinear equations of mean Ąeld and hydrodynamic origins, namely the nonliner Schrödinger equation and its generalizations, applicable to Bose-Einstein condensates (BECs) and KortewegŰde Vries and its modiĄed version. Exact solutions to all these equations are obtained containing solitons and cnoidal wave solutions. For atomic-molecular condensates, solutions are obtained in quasi-one dimensional scenario, relevant for cigar shaped BEC. They are found to describe chemical oscillations in the chemistry of ultracold regime, which is beginning to be explored in present times. Exact localized soliton solutions are also obtained, which coherently convert atomic condensate to the molecular one in a controlled manner. The area of reaction front dynamics in the ultracold regime is also an area of strong theoretical and expeimental interest. The stability of the mean Ąeld condensate solutions have been explored, which leads to Schr¨odinger type eigenvalue equations, under linear stability analysis. The possibility of instability naturally leads to complex eigenvalues of the Schrödinger type equations. Therefore, we have systematically explored the spectral problem of Pöschl-Teller class of potentials which are known to arise in the stability analysis of BEC systems. The real and complex eigenvalues, are shown to appear as a bifurcation phenomena. Interestingly, it is found that parity-time reversal symmetry naturally appears in this system. The thesis is organized as follows: After a brief introduction, in chapter two, we start with a demonstration of the occurrence of oscillatory reactions in the ultra-cold chemistry of atom-molecular Bose-Einstein condensate. Nonlinear oscillations in the mean-Ąeld dynamics is identiĄed for a speciĄc range of elliptic modulus, giving rise to both in- and out-phase The thesis is organized as follows: After a brief introduction, in chapter two, we start with a demonstration of the occurrence of oscillatory reactions in the ultra-cold chemistry of atom-molecular Bose-Einstein condensate. Nonlinear oscillations in the mean-Ąeld dynamics is identiĄed for a speciĄc range of elliptic modulus, giving rise to both in- and out-phase Chapter 3 deals with the P¨oschl-Teller potential (Scarf II) with PT-symmetry. The appearance of merging of levels and spectral bifurcation near an exceptional point is found to connect the PT -broken and unbroken phases. The bound states in the PT -broken phase are manifested as spectral singularities. The intimate connection of PT -symmetry breaking and breaking of supersymmetry is pointed out, as also the phenomenon of hysteresis near exceptional point. Intriguingly, the PT -symmetric Hamiltonians related by SUSY are also found to be isospectrally deformed counterparts for a speciĄc parametric condition with the deformation satisfying the Korteweg-deVries equation. Chapter 4 shows the existence of complex solitary wave and periodic solutions of the Kortweg de-vries (KdV) and modiĄed Kortweg de-Vries (mKdV) equations. The solutions of the KdV (mKdV) equation appear in complex-conjugate pairs and are even (odd) under the simultaneous actions of parity (P) and time-reversal (T ) operations. The corresponding localized solitons are hydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishing intensity. In chapter 5, one of the less understood ambiguities of quantization is emphasized to result from the presence of higher-order time derivatives in Lagrangians, giving rise to multiple-valued Hamiltonians. We explore certain classes of branched Hamiltonians in the context of nonlinear autonomous diferential equation of Liénard type. Elementary nonlinear models that emerge are shown to admit a feasible quantization along these lines. We conclude in the chapter 6 after summarizing the obtained results and providing directions for future study.

Actions (login required)

View Item