PT symmetry implies real energy andlinear stability of complex soliton
Andreas Fring, Julia Cen ,Francisco Correa, Takanobu TairaCity, U-London, Los Alamos National Lab, Austral U-Chile, U-Tokyo Hatano lab

Union Canal in Scotland

Based on: Nuclear Phys B 979 (2022) 115783,
J. Phys. A: Math. Theor. 53 (2020) 455701

Table of contents (What I want to tell you today)
Section 1: Introduction
Section 2: PT symmetry implies real energy
Section 3: PT symmetry implies stability
Section 4: Conclusion

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1. Introduction (Do you recognise some of them?)● Kink● Breather● Peakon● Compacton● BPS soliton● Vortex● Monopole● Instanton● Skyrmion
And more …
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1. Introduction (What I define soliton to be in this talk)Time dependent Solution to a non-linear differential equation which
1. Keeps its shape as time changes.
I will refer to classical solution of the2.
Collisionequationsof two solitons
doesasnot
alter
nonlinearof motion“soliton”
theshapes
in thistalk. of two solitons before and after
the collision.

I. Takyi, H. Weigel PRD94 2016

Bouncing and wobbling kinks4/24

1. Introduction

Is called a collective coordinate or moduli space coordinate
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1. Introduction
Example: Sine-Gordon

One soliton:(Position of the kink)
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Table of contents (What I want to tell you today)
Section 1: Introduction
Section 2: PT symmetry implies real energy
Section 3: PT symmetry implies stability
Section 4: Conclusion

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2. Complex Soliton and PT symmetry
E. Wigner. Normal form of antiunitary operators. J. Math. Phys., 1:409–413, 1960.

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2. Complex Soliton and PT symmetry
Example: Complex Sine-Gordon [Bender, Jones, Rivers. Physics Letters B 625.3-4 (2005): 333340.] [Fring, Taira. Journal of Physics A: Mathematical and Theoretical 53.45 (2020): 455701.]

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2. Complex Soliton and PT symmetry
Example: Complex Sine-Gordon [Bender, Jones, Rivers. Physics Letters B 625.3-4 (2005): 333340.] [Fring, Taira. Journal of Physics A: Mathematical and Theoretical 53.45 (2020): 455701.]

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Table of contents (What I want to tell you today)
Section 1: Introduction
Section 2: PT symmetry implies real energy
Section 3: PT symmetry implies stabilitySection 4: Conclusion

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3. Stability analysis: General argument

is stable wrt perturbation withif

Sturm-Liouville equation
Real eigenvalue to S-L problem
leads to stability of the soliton.
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3. Stability analysis: General argument
PT symmetric quantum mechanics

PT symmetricquantummechanics
Sturm-Liouvilleequation

PT symmetry

PT symmetry

Real eigenvalues

Real eigenvalues

Stabilit
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3. Stability analysis: Example

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3. Stability analysis: Example

Strat with this and find Sturm-Liouville equation

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3. Stability analysis: ExampleC removed by redefining x.

Two equations are equivalent upto moduli space of the soliton
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3. Stability analysis: Example
From [Morse feshbach 1953] page 768 (or landau Lipsitz) we have
Bound states (discrete spectrum)
Zero mode

Shape mode

Scattering states (continuous spectrum)
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3. Stability analysis: Example
From [Morse feshbach 1953] page 768 we have
Scattering states (continuous spectrum)

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3. Stability analysis: Example

Scattering states (continuous spectrum)

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3. Stability analysis: Example

Scattering states (continuous spectrum)

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Zero mode

Shape mode

Solution is stable if themodel is PT symmetricin the equivalent class ofmoduli space

Scattering
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Table of contents (What I want to tell you today)
Section 1: Introduction
Section 2: PT symmetry implies real energy
Section 3: PT symmetry implies stability
Section 4: Conclusion

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4. Conclusion
is linearly stable if following equation is satisfied

Technical point: Usefultools are supersymmetricquantum mechanics

PT symmetry

Real Energy

Stability
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4. Conclusion
What about two soliton, three soliton ?
[ICTS talk] Complex Solitons in Integrable Systems with
Real Energies ,by Andreas Fring

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Thank you for listening25Supersymmetric Quantum Mechanics

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1. Introduction (Complex soliton)
Cen, Julia, and Andreas Fring. Journal of Physics A:
Mathematical and Theoretical 49.36 (2016): 365202.

Hirota-Sastuma
equation

Ito equation

Fring, Andreas, and Takanobu Taira. Journal of Physics A:
Mathematical and Theoretical 53.45 (2020): 455701.

PT symmetry
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