To the non-local theory of cold nuclear fusion

In this paper, we revisit the cold fusion (CF) phenomenon using the generalized Bolzmann kinetics theory which can represent the non-local physics of this CF phenomenon. This approach can identify the conditions when the CF can take place as the soliton creation under the influence of the intensive sound waves. The vast mathematical modelling leads to affirmation that all parts of soliton move with the same velocity and with the small internal change of the pressure. The zone of the high density is shaped on the soliton's front. It means that the regime of the ‘acoustic CF’ could be realized from the position of the non-local hydrodynamics.


Introduction
In 1989, two electro-chemists, Martin Fleischmann and Stanley Pons [1] announced about nuclear fusion reactions between deuterium nuclei in a table-top experiment, under ordinary conditions of temperature and pressure, by using electrochemistry. The experimental evidence consisted of the production of large amounts of heat, which could not be attributed to chemical reactions.
The reactions were termed 'cold fusion' (CF), by comparison with the high temperature of thermonuclear fusion. The typical chain of nuclear reactions can be written as follows: Obviously, the following criteria need to be met in order to establish conventional thermonuclear deuterium fusion unquestionably: (1) the experiment has been repeatable by other investigators; (2) there has to be a significant neutron emission statistically well above background level; . From this point of view, there is no reason to estimate fusion efficiency by solving the Rayleigh-Plesset-Keller (RPK) differential equation for bubble collapse. By the way, the RPK equation must be solved numerically together with the equation of state for the bubble gas. No surprise that the resulting solution is quite sensitive to the choice of the equation of state during the last stage of collapse. But the equation of state for ideal gas cannot be used for this stage which is the most interesting stage from the standpoint of nuclear fusion.
The adversaries of the cavitation-induced fusion affirm that theoretical physics does not lead to the acoustic regimes providing CF and all the effect is within the experimental error.
The main objective is to investigate the dynamics of matter under the influence of the sound wave by the methods of non-local physics.

Investigation of the soliton movement under action of the sound wave
Let be the plane sound wave interact with matter. In this case, we can say about the sound pressure P which can be observed for example in the stiff tube. This pressure P was calculated by Rayleigh [16][17][18] and can be written as where ρ m is the density of a surrounding medium without perturbations, v is amplitude of the sound particle velocity in the wave antinodes, E k is the time-and space-average of the kinetic energy density of the sound wave, γ = c p /c V (the ratio of the specific heat at constant pressure to the specific heat at constant volume). Non-local hydrodynamic equations have the form [15,19,20]: where v 0 is the hydrodynamic velocity in the coordinate system at rest, ρ the density, p the pressure, ↔ I unit tensore, F the force acting on the unit of volume and τ the non-local parameter. Several significant remarks follow.
(1) Equations (3.2)-(3.4) should be considered as local approximation of non-local equations (NLE) written in the hydrodynamic form. NLE include quantum hydrodynamics of Schrödinger-Madelung as a deep particular case [15] and can be applied in the frame of the unified theory from the atom scale to the Universe evolution.  (4) The choice of the non-local parameter needs special consideration [19][20][21][22]. The system of equations (3.2)-(3.4) convert in the system of quantum hydrodynamic equations by the suitable choice of the non-local parameter τ . The relation between τ and kinetic energy [21,22] is used in quantum hydrodynamics: where u is the particle velocity and H is the coefficient of proportionality which reflects the state of the physical system. In the simplest case, H is equal to the Plank constanth and the corresponding relation (3.5) correlates with the Heisenberg inequality. From first glance, the approximation (3.5) is distinguished radically from the kinetic relation known from the theory of the rarefied gases (3.6): which is used for the calculation of the non-local parameter in the macroscopic hydrodynamic case (υ is the kinematic viscosity). But it is not a case. In quantum approximation, the value υ qu =h/m has the dimension (cm 2 s −1 ) and can be called as quantum viscosity, for the electron species υ qu =h/m e = 1.1577 cm 2 s −1 . If we take into account that the value p/ρ ∼ V 2 , then the interrelation of (3.5) and (3.6) becomes obvious.
Let us consider now the one-dimensional non-stationary matter movement under action of the wavefront. In this case, equations (3.2)-(3.5) take the form: Then introduce the coordinate system moving along the positive χ -direction of the one-dimensional space with the velocity C = u 0 which is equal to phase velocity of the investigated quantum object (3.10) Taking into account the de Broglie relation, we write that the group velocity u g should be equal 2u 0 . Really, let us write down the energy of the relativistic particle , (3.12) where c is the light velocity, v g is the group velocity and m 0 is the mass of the rest for particle under study. Rewrite (3.11) as follows: is the particle impulse. In the non-relativistic approximation, we have from (3.13) Using the principle of wave-particle parallelism in the de Broglie interpretation, we have for the energy of a particle E =hω =hkv φ , (3.16) where ω is the angular frequency, v φ = ω/κ is the phase velocity, κ = 2π/λ is the wavenumber and λ is the wavelength. Correspondingly, the particle impulse p is p =hk, (3.17) and using (3.17), we find Then in the non-relativistic approach Then in the coordinate system moving with the phase velocity, indestructible soliton has the velocity which is equal to the phase velocity.
We extend the usual definition of the soliton object, which should satisfy two important conditions.
(1) In a moving coordinate system, this object is located in the same restricted area for all time moments including the movement under the influence of external forces. (2) In the coordinate system moving with the phase velocity, indestructible soliton has the velocity which is equal to the phase velocity for all parts of a moving object.
Therefore, we use the moving coordinate system ξ = x − ut. In this system, all dependent hydrodynamic values are functions of (ξ ,t). We investigate the possibility of the soliton creation. For this type of solution, the explicit time dependence does not exist in the considered coordinate system. As a result, equations (3.7)-(3.9) can be written as (continuity equation) (energy equation)  The force scale F 0 is used in (3.27) (3.28) The motion equation and the energy equatioñ are subjected to the same transformations.
In the Rayleigh theory, the valueF is the constant dimensionless parameter. The solution of equations (3.27), (3.29) and (3.30) can be simplified after transformation of the mentioned equations into the one parametric system using the special choice of the lengthscale ξ 0 . Namely then H = 1, and the force scale is The introduced scales have the transparent physical sense. Really, let us introduce the quantum Reynolds number and transform this number using the introduced scales: Then we are dealing with the matter flow for Re = 1. We need to solve the following system of equations:

Numerical simulation
Equations (3.34)-(3.36) constitute the one parametric Cauchy problem as the system of the three ordinary differential equations of the second order. Technical computing software MAPLE allows the realization of the vast mathematical modelling using the variation of the six Cauchy conditions and theF parameter. Let us show the results of some calculations using the MAPLE notations:

Discussion
ParameterF defines the force of the sound action on matter; varying over eight orders of this parameter F leads to the radical reconstruction of the flow. Namely, The structure of the creating solitons has the following very remarkable features.
(1) As it can be expected in the soliton theory, all soliton parts move with the same velocity-the conditionũ = 1 fulfils with high accuracy. The soliton is placed in the bounded region of space. It is important to underline that we deal with the Cauchy problem. It means that the mentioned effect is a product of the matter of self-organization.

Conclusion
In spite of all experimental problems and difficulties, the cavitation-induced fusion or generally speaking, acoustic cold fusion (ACF) has serious experimental confirmation. There is the obvious contradiction between the mentioned experimental results and conclusions of the classical local hydrodynamics. As we see, local hydrodynamics is not applicable to the description of the ACF in principal. The realized mathematical modelling leads to the gross density change on the soliton front without significant pressure changing. It is the desired effect which the CF adherents try to prove.