scholarly journals Deformed Sine-Gordon Models, Solitons and Anomalous Charges

2021 ◽  
Author(s):  
Harold Blas ◽  
Hector F. Callisaya ◽  
João P.R. Campos ◽  
Bibiano M. Cerna ◽  
Carlos Reyes
Keyword(s):  
Conservation Laws ◽  
Infinite Number ◽  
Soliton Solutions ◽  
Linear Formulation ◽  
Reflection Symmetry ◽  
Local Conservation ◽  
Time Symmetry ◽  
Non Local ◽  
Sine Gordon Model ◽  
Special Space

We study certain deformations of the integrable sine-Gordon model (DSG). It is found analytically and numerically several towers of infinite number of anomalous charges for soliton solutions possessing a special space–time symmetry. Moreover, it is uncovered exact conserved charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kink-antikink (even parity) scatterings with equal and opposite velocities. Moreover, we provide a linear formulation of the modified SG model and a related tower of infinite number of exact non-local conservation laws. We back up our results with extensive numerical simulations for kink-kink, kink-antikink and breather configurations of the Bazeia et al. potential V q w = 64 q 2 tan 2 w 2 1 − sin w 2 q 2 , q ∈ R , which contains the usual SG potential V 2 w = 2 1 − cos 2 w .

scholarly journals Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 2. Pseudoenergy-based theory

1995 ◽  
Vol 290 ◽  
pp. 105-129 ◽  
Author(s):  
Paul J. Kushner ◽  
Theodore G. Shepherd
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Structure ◽  
Central Body ◽  
Stability Criteria ◽  
Local Conservation ◽  
Hamiltonian Description ◽  
Time Symmetry ◽  
Stability Theorems ◽  
Basic States ◽  
Boundary Stability

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy.A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number.The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states.The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

N-fold Darboux transformation of a six-field integrable lattice system

2021 ◽  
pp. 2150248
Author(s):  
Yanan Qin
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Lattice System ◽  
Local Conservation ◽  
Integrable Lattice ◽  
Coupled Equation ◽  
Math Phys

In this paper, we studied a semidiscrete coupled equation, which is integrable in the sense of admitting Lax representations. Proposed first by Vakhnenko in 2006, local conservation laws and one-fold Darboux transformation were presented with different forms, respectively, in O. O. Vakhnenko, J. Phys. Soc. Jpn. 84, 014003 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015). On the basis of these results, we principally construct [Formula: see text]-fold Darboux transformation by means of researching gauge transformation of its Lax pair, and work out its explicit multisolutions. Given a set of seed solutions and appropriate parameters, we can calculate two-soliton solutions and plot their figures when [Formula: see text].

scholarly journals A kinetic decomposition for singular limits of non-local conservation laws

2009 ◽  
Vol 247 (12) ◽  
pp. 3338-3356 ◽  
Author(s):  
Frederike Kissling ◽  
Philippe G. LeFloch ◽  
Christian Rohde
Keyword(s):  
Conservation Laws ◽  
Local Conservation ◽  
Singular Limits ◽  
Kinetic Decomposition ◽  
Non Local
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