Anjan Kundu

Landau–Lifshitz and higher‐order nonlinear systems gauge generated from nonlinear Schrödinger‐type equations

New Landau–Lifshitz (LL) and higher‐order nonlinear systems gauge generated from nonlinear Schrodinger (NS) type equations are presented. The consequences of gauge equivalence between different dynamical systems are discussed. The gauge connections among various LL and NS equations are found and depicted through a schematic representation.

Quantum and classical integrable sine-Gordon model with defect

Defects which are predominant in a realistic model, usually spoil its integrability or solvability. We on the other hand show the exact integrability of a known sine-Gordon field model with a defect (DSG), at the classical as well as at the quantum level based on the Yang–Baxter equation. We find the associated classical and quantum R-matrices and the underlying q-algebraic structures, analyzing the exact lattice regularized model. We derive algorithmically all higher conserved quantities Cn, n = 1, 2 ,... , of this integrable DSG model, focusing explicitly on the contribution of the defect point to each Cn. The bridging condition across the defect, defined through the Backlund transformation is found to induce creation or annihilation of a soliton by the defect point or its preservation with a phase shift.

Tsunami and nonlinear waves

Propagation.- Waves in shallow water, with emphasis on the tsunami of 2004.- Integrable Nonlinear Wave Equations and Possible Connections to Tsunami Dynamics.- Solitary waves propagating over variable topography.- Water waves generated by a moving bottom.- Tsunami surge in a river: a hydraulic jump in an inhomogeneous channel.- On the modelling of huge water waves called rogue waves.- Numerical Verification of the Hasselmann equation.- Source & Run up.- Runup of nonlinear asymmetric waves on a plane beach.- Tsunami Runup in Lagrangian Description.- Analytical and numerical models for tsunami run-up.- Large waves caused by oceanic impacts of meteorites.- Retracing the tsunami rays.- Modeling and visualization of tsunamis: Mediterranean examples.- Characterization of Potential Tsunamigenic Earthquake Source Zones in the Indian Ocean.- Erratum.

Exact solutions to higher-order nonlinear equations through gauge transformation

Abstract A highly nonlinear hydrodynamic equation proposed by Johnson is solved exactly, by gauge transforming (GT) it to the mixed nonlinear Schrodinger equation. The GT is generalized further, to generate a hierarchy of integrable higher-order nonlinear systems including some known equations, along with their Lax pairs, infinite sets of conserved quantities and exact solutions.

Soliton solutions, Liouville integrability and gauge equivalence of Sasa Satsuma equation

Exact integrability of the Sasa Satsuma equation (SSE) in the Liouville sense is established by showing the existence of an infinite set of conservation laws. The explicit form of the conserved quantities in terms of the fields are obtained by solving the Riccati equation for the associated 3×3 Lax operator. The soliton solutions, in particular, one and two soliton solutions, are constructed by the Hirota’s bilinear method. The one soliton solution is also compared with that found through the inverse scattering method. The gauge equivalence of the SSE with a generalized Landau Lifshitz equation is established with the explicit construction of the new equivalent Lax pair.

Integrable nonautonomous nonlinear Schrödinger equations are equivalent to the standard autonomous equation.

A class of inhomogeneous nonlinear Schrödinger equations (NLS), claiming to be novel integrable systems with rich properties continues appearing in PhysRev and PRL. All such equations are shown to be not new but equivalent to the standard NLS, which trivially explains their integrability features. PACS no: 02.30.Ik , 04.20.Jb , 05.45.Yv , 02.30.Jr Time and again various forms of inhomogeneous nonlinear Schrödinger equations (IHNLS) along with their discrete variants are appearing as central result mostly in the pages of Phys. Rev, and PRL CLU76,RB87,PRA91,Kon93,PRL00,PRL05,PRL07 ̧ , which are either suspected to be integrable due to the finding of particular analytic or stable computer solutions, or assumed to be only Painlevé integrable arxiv08 ̧ , or else claimed to be completely new integrable systems. Apparently the solution of such integrable systems needs generalization of the inverse scattering method (ISM), in which the usual isospectral approach involving only constant spectral parameter λ has to be extended to nonisospectral flow with time-dependent λ(t). Moreover certain features of the soliton solutions of such inhomogeneous NLS, like the changing of the solitonic amplitude, shape and velocity with time were thought to be new and surprising discovery. We show here that all these IHNLS , though completely integrable are not new or independent integrable systems, and in fact are equivalent to the standard homogeneous NLS, linked through simple gauge, scaling and coordinate transformations. The standard NLS is a well known integrable system with known Lax pair, soliton solutions and usual isospectral ISM nls,ALM ̧ . As we see below, a simple time-dependent gauge transformation of the standard isospectral system with constant λ can create the illusion of having complicated nonisospectrality. Similarly, a time-dependent scaling of the standard NLS field Q → q = ρ(t)Q would naturally lead the constant soliton amplitude to a time-dependent one. In the same way a trivial coordinate transformation x → X = ρ(t)x would change the usual constant velocity v of the NLS soliton to a time-variable quantity v(t) = v ρ(t) and the invariant shape of the standard soliton with constant extension Γ = 1 κ to a time-dependent one with variable extension Γ(t) = Γ ρ(t) (see Fig 1a a,b). Therefore all the rich integrability properties of the IHNLS, observed in earlier papers, including more exotic and seemingly surprising features like nonisospectral flow, appearance of shape changing and accelerating soliton etc. can be trivially explained from the timedependent transformations of these IHNLS from the standard NLS and the corresponding explicit result , namely the Lax pair, N-soliton solutions, infinite conserved quantities etc. for the inhomogeneous NLS models can be derived easily from their well known counterparts in the homogeneous NLS case through the same transformations nls ̧ .

Exact accelerating solitons in nonholonomic deformation of the KdV equation with a two-fold integrable hierarchy

The recently proposed nonholonomic deformation of the KdV equation is solved through the inverse scattering method by constructing an AKNS-type Lax pair. Exact N-soliton solutions are found for the basic field and the deforming function showing an unusual accelerated (decelerated) motion. A two-fold integrable hierarchy is revealed, one with the usual higher order dispersion and the other with novel higher nonholonomic deformations.

A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also devise a method to derive explicit general solutions of the nonlinear ODEs. Interestingly, many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models.

Gauge transformations of constrained KP flows: New integrable hierarchies

Integrable systems in 1+1 dimensions arise from the KP hierarchy as symmetry reductions involving square eigenfunctions. Exploiting the residual gauge freedom in these constraints new integrable systems are derived. They include generalizations of the hierarchy of the Kundu–Eckhaus equation and higher‐order extensions of the Yajima–Oikawa and Melnikov hierarchies. Constrained modified KP flows yield further integrable equations such as the hierarchies of the derivative NLS equation, the Gerdjikov–Ivanov equation, and the Chen–Lee–Liu equation.

Classical and quantum integrability of a derivative nonlinear Schrödinger model related to quantum group

An ultralocal structure discovered in a derivative nonlinear Schrodinger model enables us to investigate the complete integrability of the classical system through action‐angle variables and solve exactly the quantum model by quantum inverse scattering transformation. Exploiting the underlying quantum group related Sklyanin‐like algebra an exact lattice version of the model is constructed, and also the deep‐rooted reason for the existence of such an ultralocal model is revealed.