Complex solitons with real energies

Andreas Fring; Julia Cen

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Complex solitons with real energies

Julia Cen, Andreas Fring

Exactly Solvable and Integrable Systems · Submitted 17 Feb 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are PT-symmetric or not.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.65 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.42 · compares with prior results 0.90 · mentions code or data availability 0.02.

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Using Hirota's direct method and Baecklund transformations we construct explicit complex one and two-solutions to the complex Korteweg-de Vries equation, the complex modified Korteweg-de Vries equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be PT-symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex PT-symmetric solutions to the Korteweg-de Vries equation may also be constructed alternatively from real solutions to the modified Korteweg-de Vries by means of Miura transformations. The multi-soliton solutions obtained from Hirota's method break the PT-symmetric, whereas those obtained from Baecklund transformations are PT-invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are PT-symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space-time and the fact that any of our N-soliton solutions may be decomposed into N separate PT-symmetrizable one-soliton solutions.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Using Hirota's direct method and Baecklund transformations we construct explicit complex one and two-solutions to the complex Korteweg-de Vries equation, the complex modified Korteweg-de Vries equation and the complex sine-Gordon equation.	contribution 0.93	0.96	0.10	asserted 1.00
s1	The one-soliton solutions of trigonometric and elliptic type turn out to be PT-symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena.	result 0.99	0.89	0.10	asserted 1.00
s2	We show that alternatively complex PT-symmetric solutions to the Korteweg-de Vries equation may also be constructed alternatively from real solutions to the modified Korteweg-de Vries by means of Miura transformations.	method 0.45	0.90	0.11	asserted 0.66
s3	The multi-soliton solutions obtained from Hirota's method break the PT-symmetric, whereas those obtained from Baecklund transformations are PT-invariant under certain conditions.	result 1.00	0.89	0.13	asserted 1.00
s4	Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are PT-symmetric or not.	result 1.00	0.95	0.55	asserted 1.00
s5	The reason is that the symmetry can be restored by suitable shifts in space-time and the fact that any of our N-soliton solutions may be decomposed into N separate PT-symmetrizable one-soliton solutions.	interpretation 0.99	0.83	0.24	asserted 0.97

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Original abstract

Using Hirota's direct method and Baecklund transformations we construct explicit complex one and two-solutions to the complex Korteweg-de Vries equation, the complex modified Korteweg-de Vries equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be PT-symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex PT-symmetric solutions to the Korteweg-de Vries equation may also be constructed alternatively from real solutions to the modified Korteweg-de Vries by means of Miura transformations. The multi-soliton solutions obtained from Hirota's method break the PT-symmetric, whereas those obtained from Baecklund transformations are PT-invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are PT-symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space-time and the fact that any of our N-soliton solutions may be decomposed into N separate PT-symmetrizable one-soliton solutions.

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4 arXiv papers cite this in 13 sentences

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Builds on it 1 paper

We present here a detailed analysis of the delay times for complex soliton solutions to the Korteweg de-Vries equation (KdV) previously reported in [ref] .

Time-delay and reality conditions for complex solitons · Introduction · 0.81

Cites it as support 1 paper

As demonstrated in [ref] , this can be achieved with suitable real shifts in time or space, but in addition one also requires the model to be integrable.

Regularized degenerate multi-solitons · Conclusions · 0.96

Cites it as background 3 papers

It is well-known that the combination of four Bäcklund transformations combined in a Bianchi-Lamb [ref] commutative fashion gives rise to a “nonlinear superposition principle ”, e.g. [ref] .

Regularized degenerate multi-solitons · Degenerate complex multi-soliton solutions from superposition · 0.83
We refer here to [ref] where different aspects of the system ([ref]) were discussed in the context of the complex KdV equation.

Perfectly invisible $\mathcal{PT}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry · Summary, discussion and outllok · 0.95
Such systems attract attention and find applications in various areas of physics, including optics [ref] , condensed matter physics [ref] , quantum field theory [ref] , gravity and cosmology [ref] , nonlinear waves [ref] , and theory of integrable models of finite [ref] and infinite number of degrees of freedom [ref] .

Conformal bridge transformation, $\mathcal{PT}$- and super- symmetry · Introduction · 0.91

What it cites

Cites as background 1 paper

$𝒫𝒯$-symmetrically deformed nonlinear wave equations have been found to possess various interesting properties [ref] .

PT-symmetry in quasi-integrable models · Introduction · 0.99

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