Time-delay and reality conditions for complex solitons

Abstract

We compute lateral displacements and time-delays for a scattering processes of complex multi-soliton solutions of the Korteweg de-Vries equation.The resulting expressions are employed to explain the precise distinction between solutions obtained from different techniques, Hirota's direct method and a superposition principle based on Baecklund transformations. Moreover they explain the internal structures of degenerate compound multi-solitons previously constructed. Their individual one-soliton constituents are time-delayed when scattered amongst each other. We present generic formulae for these time-dependent displacements. By recalling Gardner's transformation method for conserved charges, we argue that the structure of the asymptotic behaviour resulting from the integrability of the model together with its PT-symmetry ensure the reality of all of these charges, including in particular the mass, the momentum and the energy.

18 pages, 4 figures

References in corpus (5)

• Making Sense of Non-Hermitian Hamiltonians
• PT-symmetric Deformations of the Korteweg-de Vries Equation
• Regularized degenerate multi-solitons
• PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation
• Integrable models from PT-symmetric deformations

Cited by in corpus (10)

• Integrable nonlocal Hirota equations
• Asymptotic and scattering behaviour for degenerate multi-solitons in the Hirota equation
• Degenerate multi-solitons in the sine-Gordon equation
• Non-Hermitian gauge field theories and BPS limits
• Complex BPS solitons with real energies from duality
• Degenerate soliton solutions and their interactions in coupled Hirota equation with trivial and nontrivial background
• Predicting positon solutions of a family of Nonlinear Schrödinger equations through Deep Learning algorithm
• Multicomplex solitons
• Moduli spaces for PT-regularized solitons
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry