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In this work, some types of nonlinear Liouville equation (NLE) and nonlinear Master equations (NME) are studied. We found that the nonlinear terms in the equation can resist state of system damping so that an information solitonic structure appears. Furthermore, the power in the non-linear term is independent of limitation of the solution. This characteristic offers a possibility to construct complicated information solitons from some simple solutions, which allow one to solve complicated NLE or NME. The results obtained in this work may provide an innovated channel for the quantum information transmission over far distance against dissipation and decoherence, and also open a constructive way to resist age decaying of system by designing adjusted field interaction with the system nonlinearly.

It is well known that the nonlinear Schrödinger equation (NSE) provides the solitonic solution which provokes many applications such as optical soliton communications; however it seems to be not clear what is the solution of the corresponding nonlinear Liouville equation (NLE) and the relevant physical meanings? This issue is interesting because the solitonic information communication needs to develop stable and low dissipative channel to transmit or receive singles against dissipation and decoherence. Hence understanding the corresponding NLE with solitonic structure of the density operator may allow us to innovate some methods to control dissipation and decoherence in the information transmission. On the other hand, although quantum information theory in studying transmission and processing of quantum states, entanglement of states for quantum computation, quantum cypotography or quantum teleportation has achieved great progresses [

Actually, if a NSE is defined as

and

then a relevant NLE can be introduced by

where b

which is only true if the “direct” product

Thus an interested solitonic solution of density operator, which can be defined as an information soliton based on the information meaning of density operator, can be constructed by

where

where the amplitude

Then any NLE with higher order of power can be solved by using above formulation, e.g. if a NLE is ex- pressed by

where corresponding NSE is

then, by using

which allows Equation (10) to become

Now let

whose solution likes Equation (6), i.e.

Therefore a general information soliton for Equation (8) can be constructed by

Furthermore, if one assuming

then using the same approach above, one can obtain

which corresponds to a general NLE with a nonlinear term

where defining

Finally a general information solitonic solution can be achieved by

where notice again that

which is just a sort of entropy operator.

The above mentioned NLE motivates us to consider logically to introduce a general nonlinear Liouville equation (GNLE) as

where

where

equilibrium or stable state of the system),

then a NLE can be achieved by

where

However, if

or

This type of GNLE can be used to design a channel of quantum information against dissipation and decoherence. More concretely speaking, for open system plus complicated environment, if

where

For instance, in the amplitude damping model, a master equation with nonlinear term

where

Then using the coherent and entangled state as a basis developed by Fan Hongyi [

which approximately produces

where

so transformations

For solving this nonlinear Equation (30), by left acting

(34)

This yields

so that

Thus a formal solution of this equation is given by

where

By using a thermo-coherent and entangled state

where the relations

Then, in terms of the integral formula based on the Technique of Integration Within an Ordered Product of Operators (IWOP) [

one obtains

The integral term in Equation (41) is calculated as

where the relevant parameters are introduced by

So, this allows one to get

Then by means of Equation (36), Equation (44) gives an asymptotic solution for

Furthermore, if the nonlinear term

where notice

the operator A commutes with a sort of particle number operator

where defining

which gives an equation group described by

and the relevant sum gives,

Then based on the above formalism, a series of asymptotic solutions can be represented as

which enables one to attain

furthermore,

Therefore, when time

In the same way, from Equation (52), one can also obtain

so that one obtains

then an asymptotic solution of Equation (27) is also given by

The above asymptotic configurations can also be defined as a sort of information soliton in the sense: (1) they are invariant structure locally when time elapses enough long, and (2) these structures exist as form of density operator with meaning of QID. The study of the long time evolution of these structures may shed more light on the soliton dynamic of information density as the asymptotic configuration appear through kind of nonlinear self-interaction of the information density reduced from environment. In fact, a type of asymptotic configuration for the (non-)Markovian system with linear interaction between system and environment are found, and a wide class of nonlinear corrections of evolution equations are also found leading to superluminal effects [

which gives naturally a general Liouville equation for the open system constructed by

where

where

Generally, the above results provoke a constructive mechanism to realize a sort of self-organization from the non-equlibrium process for the open system. This may be useful, from physical justification, to construct the stable information transmission among remote space rockets or prolong life span against age decaying of systems. For example, if the original organization is expressed unfortunately by a master equation of the ampli- tude damping model,

then in terms of well known characteristic of master equation, the final asymptotic solution tends to decaying zero state,

So for prolonging life of the organization one can use a driving field F which satisfies

to enable original master equation to become

where

where notice again

This shows that the status of the system remain invariance by means of the interaction of the external field, which allows the original system to prolong life span without decaying.

More generally along this line, a master equation of the amplitude damping model described various decaying processes, after considering a nonlinear term

where

so, one obtains

Thus a formal solution of this equation is given by

where

By using a coherent and entangled state

then one gets an integral form of the normal product as

where the integral part is given by

Then an asymptotic solution of Equation (68) can be achieved

which gives an integral representation as

This allows one to find that the power of

where if giving

hence one obtains

All of these are processed in the open system through a sort of nonlinear interaction expressed as a functional of

nonlinearity to eliminate the dissipation by producing a sort of self-organization, i.e. information soliton.

The nonlinear kinetic equations included NLE and NME based on the micro-representation of the second law of thermodynamics are studied. The nonlinear terms in the equation can resist state of system damping so that an information solitonic structure appears. While the power in the nonlinear term is independent of the limitation of the solution, which permits one to construct more complicated structures of information soliton in the solution of complicated equations. The information soliton can be understood as some invariant structure of information density locally as a self-organization created from nonlinearity. So, these results can provide an innovated channel for the quantum information transmission over far distance against decoherence or damping, and also offer a constructive way to prolong life span of the original system by designing adjusted field interacting with the system nonlinearly.

Bi Qiao, (2015) Nonlinear Liouville Equation and Information Soliton. Journal of Modern Physics,06,2058-2069. doi: 10.4236/jmp.2015.614212