In the analysis of nonlinear waves in plasma, especially for the search for periodic waves, shock waves, and solitons, mechanical analogy methods are widely applicable. The most famous of them is the Sagdeev pseudopotential method. However, sometimes mathematical difficulties arise when deriving formulas for pseudopotentials. The author proposes three mathematical tricks to get around these difficulties and obtain exact formulas for pseudopotentials in cases where the direct, Sagdeev method is considered inapplicable: a trick based on the Lambert Wfunction, a trick based on the inverse function integration, and a trick based on reducing the theory equations to the Bernoulli differential equation (the Bernoulli pseudopotential method). This article, which is methodological by nature, provides detailed examples of the application of each of these tricks when deriving formulas for pseudopotentials.
I. INTRODUCTION
In the analysis of nonlinear waves in plasma, especially for determining the conditions for the existence of periodic waves, solitons, and shock waves, the methods of mechanical analogy are widely applicable. The most famous of them is the Sagdeev pseudopotential method. This method bears the name of Sagdeev, who, together with his colleagues, applied it to analyze nonlinear ionacoustic waves in collisionless plasmas with cold ions and inertialess isothermal electrons distributed according to Boltzmann.^{1–3} As a result, he showed the possibility of periodic and solitary nonlinear ionsound waves, and determined the limiting velocity of solitary waves (critical Mach number).
It is fair to point to the earlier research by Akhiezer and Polovin,^{4,5} who applied their version of the pseudopotential method to analyze nonlinear electron waves in plasma.
The technique and numerous examples of the Sagdeev pseudopotential method application are described in numerous original works—the total number of which has exceeded several thousand—as well as in books.^{6–11}
The simplest way is to imagine a nonlinear oscillator movement as the movement of a certain hockey puck (or pseudoparticle) sliding without friction along the relief defined by the $ U S g x$ profile (Fig. 1). This is the meaning of the mechanical analogy method.
This analysis is usually carried out as follows: First, graphs of the pseudoforce and pseudopotential profiles, as well as the oscillator phase portrait are plotted. Then, stable equilibrium points (local minima) are determined in the potential well, in which the motion corresponds to a periodic wave. Then, it is necessary to determine the presence of pseudopotential unstable equilibrium points (local maxima) and the shape of separatrices, the motion along which corresponds to solitary waves (solitons) or shock waves. In this case, if the separatrix rests on one unstable equilibrium point, the pseudooscillator motion along it corresponds to a soliton, if it rests on two points, to a shock wave. The positions of the equilibrium points and the range of oscillations are found from the algebraic equations obtained by mathematical analysis of the $ U S g x$ function and its derivative $ \u2212 F S g x$.
The mechanical analogy method, to which the Sagdeev pseudopotential method belongs, turned out to be very fruitful as with its help, it was possible to analyze and solve many problems not only within the framework of theories of nonlinear waves of various types in plasma (for example, Refs. 12–31 and very many others), but also within the framework of theories of nearelectrode sheaths in plasmas,^{32–39} analysis of the dynamics of particle fluxes in a magnetic field,^{40} the arc column theory in the framework of Elenbaas–Heller's equation,^{41} etc.
However, unfortunately, it is not always possible to reduce the original system of equations to the nonlinear oscillator equation (1). With the standard derivation of the exact formula for the pseudopotential in the way suggested by Sagdeev and colleagues,^{1–3} mathematical difficulties, which prevent the calculation from being carried out to the end, often arise. What should be done?
In such cases, an approximate method, based on the expansion of unknowns in a series in a small parameter, is often used. For waves, this method is called the reductive perturbation method. As a result, the equations are reduced to wellknown evolutionary equations such as Korteweg–de Vries, Gardner, Schrödinger, or Kadomtsev–Petviashvili. Some examples of such works are available in Refs. 42–51.
However, these evolutionary equations are valid only for small amplitude waves; they cannot be used to obtain, for example, solutions in the form of a soliton with the maximum possible amplitude (i.e., the socalled extremal soliton^{52}) and it is impossible to obtain solutions in the form of supersolitons^{53–66} or in the form of supernonlinear waves,^{56,67–70} whose amplitude, in principle, cannot be small.
The author proposes three original mathematical tricks that make it possible to obtain exact formulas for pseudopotentials in cases where the indicated difficulties arise and the Sagdeev method is considered inapplicable:

a trick based on the Lambert Wfunction;

a trick based on the inverse function integration; and

a trick based on reducing the theory equations to the Bernoulli ordinary differential equation (the Bernoulli pseudopotential method).
These tricks were tested by the author and his students in numerous works on the theory of nonlinear waves and structures in plasma.
In this article, which is methodological and educational by its nature, examples of the use of each of these tricks for the derivation of pseudopotential formulas are analyzed in detail.
The structure of this article is the following: Sec. II presents a more detailed derivation of the Sagdeev pseudopotential than in the original works^{1–3} for nonlinear ionacoustic waves in cold plasma. This conclusion draws attention to some subtle nuances that are useful for starting researchers and which are usually omitted, being considered trivial. These nuances for more complex cases will no longer be so trivial. Section III shows how to use a trick based on the Lambert Wfunction in deriving the pseudopotential for nonlinear ionacoustic waves in warm isothermal plasma. Section IV presents a method for deriving the Sagdeev pseudopotential for nonlinear ionacoustic waves in warm polytropic plasma, based on the inverse function integration. Section V demonstrates a method for deriving the pseudopotential for nonlinear ionacoustic waves in warm polytropic plasma, based on the Bernoulli differential equation analysis, which results in the Bernoulli pseudopotential. It should be pointed out that the first trick is suitable only for nonlinear waves in isothermal plasma, and the other two tricks are universal. With their help, it is possible to find exact formulas for pseudopotentials for all those cases that were previously analyzed approximately in Refs. 42–51. At the end of the article, there are two appendixes: Appendix A provides some important properties of the Lambert Wfunction, while Appendix B gives a rule for integrating the inverse function, which is usually not given in university mathematics courses and is unavailable in textbooks.
II. IONSOUND WAVES IN COLD ION PLASMA
A. Linear theory
Let us consider the onedimensional wave motion of nonrelativistic collisionless plasma containing a quasineutral mixture of electron and ion gases with $ n 0 e$ and $ n 0 i$ concentrations, respectively, and first consider a cold ion gas at zero temperature. Let us denote $e<0$ as the electron charge and $\u2212Ze$ as the positive charge of the ion. Then, the unperturbed plasma quasineutrality relation is given by $Z e n 0 i\u2212e n 0 e=0$. Since the ion mass $m$ is usually large in comparison with the electron mass, we will neglect the electron mass, i.e., assuming electron gas to be inertialess. We will proceed from the following general system of onedimensional nonrelativistic equations describing the ion gas dynamics in plasma:
 Continuity equations:$ \u2202 n i \u2202 t+ \u2202 v i n i \u2202 x=0.$
 Motion equations:$ \u2202 v i \u2202 t+ v i \u2202 v i \u2202 x= Z e m \u2202 \phi \u2202 x.$
 The Poisson equations:$ \u2202 2 \phi \u2202 x 2=\u22124\pi e n e \u2212 Z e n i.$
Here, $ v i$ is the ions' velocity in the wave; $ n e$ and $ n i$ are the concentration of electrons and ions; and φ is the electrostatic potential.
In this form, the electron concentration can be substituted into the Poisson equation (7).
Along the way, we should point out that if gas is in a field of a different physical nature, then in (10), instead of the electrostatic potential energy $\phi $, it is necessary to substitute the corresponding potential energy of the force field. For example, for a gravitational field, the potential energy of mass $m$ particles has the form $m \phi G$, in this case the formula describing the distribution of particles is called the barometric formula.^{71}
We have discussed in detail the distribution derivation (10) because, below, sometimes a different equation of state, leading to a different barometric formula, will be used.
Such a notation of the variables means that the harmonic disturbance propagates along the $0x$ axis with phase velocity $V= \omega / \kappa $.
Relations (18) and (19) are the dispersion equations of ionacoustic waves in plasma with isothermal inertialess electrons and cold ions, and the waves themselves are electrostatic waves of longitudinal compression–rarefaction of ionic gas and inertialess following of electrons after ions.
The dependence graph $\omega \kappa $ according to (19) is shown in Fig. 2. The phase velocity of waves is determined from the graph by the ray inclination angle tangent, going from the origin at a point on the curve. It is seen that for dependence (19) the phase velocity of the harmonic wave of the ionacoustic wave can take values from interval $V= 0 \u2026 V s$, in which the boundary value $ V s= lim \kappa \u2192 0 d \omega / d \kappa = \lambda D e \omega i = Z k T 0 e / m$ is called an ionic sound linear velocity and is a very important plasma characteristic.
It is worth noting that this velocity is simultaneously determined by the properties of both, plasma components, electron and ion. Outside the specified interval of velocities, including in the shaded angular sector, harmonic waves cannot exist.
The graph in Fig. 2 has two characteristic sections:^{72} a longwavelength section corresponding to linear ionacoustic waves propagating practically without dispersion, when the phase and group velocities are equal to $ V s$, and a shortwave section corresponding to ion plasma oscillations at frequency $\omega = \omega i$ with a group velocity that is practically equal to zero.
Let us explain why a detailed linear analysis of waves is necessary before solving a nonlinear problem.
First, the main questions to be solved in a nonlinear problem are questions about the possibility of the existence of stationary solitons and about their possible characteristics—velocity and amplitude. If stationary solitons are possible, then they correspond to straight rays emerging from zero on the $\omega \kappa $ graph, whose slope should be equal to the soliton velocity. The rays are, of course, not dispersion curves for solitons; they show only the soliton velocity constancy and serve to determine the points of synchronism with a periodic wave on the plane $ \omega , \kappa $. It is clear that if the velocity of any soliton for the considered example of plasma would satisfy the condition $0<V< V s$, i.e., their rays do not fall into the shaded area in Fig. 2, then they would cross the dispersion curve somewhere. This intersection point would correspond to the soliton synchronism with the harmonic wave in which their interaction will take place. As a result, the soliton amplitude would eventually turn out to be nonstationary. Thus, the search for a stationary soliton solution of the nonlinear problem must be somewhere in the shaded area, which excludes the intersection of its ray with the dispersion curve. Looking ahead, let us point out that this area is limited by two beams $ V s<V< V max$ (see below for more details).
Second, it is convenient to express the soliton velocity in a dimensionless normalized form—in the Mach number form. The exact expression found for the linear velocity of ionacoustic waves $ V s= Z k T 0 e / m$ makes this normalization true.
It should be said that linear analysis must necessarily precede the main nonlinear problem solution in accordance with the guidelines.^{73} Otherwise, mistakes are inevitable. A huge list of papers containing such errors can be cited, for example, see Refs. 19 and 74–79, some of which came out after the publication in Ref. 73. In these works, the ion sound velocity, to which the nonlinear equations were normalized, was chosen heuristically, and the dispersion equation analysis was not carried out. This led to the “discovery” of solitons in those areas where they should not be [for example, in Refs. 74 and 77 subsonic solitons were found for $M<1$, i.e., in areas on the plane $ \omega , \kappa $ occupied by periodic waves, in Ref. 78 the speed of sound was found, the Mach number for nonlinear electron plasma waves was calculated, the dispersion curve of which, as it is known, does not pass through the plane origin $ \omega , \kappa $, and these waves themselves do not belong to the acoustic type waves].
B. Derivation of the pseudopotential formula and its analysis (Sagdeev's solution)
Let us assume that a stationary ionacoustic wave propagates in the positive direction of $0x$ axis with phase velocity $V$. The equations describing the nonlinear wave remain the same (5)–(7). They are also supplemented by formula (8).
This means that we are moving from the laboratory reference frame to a new frame related to the wave. In this case, the obtained solution will have the form of a stationary wave, whose profile will be determined by only one variable $\xi $. Plotting the profile in the form of a graph with the abscissa axis $\xi $ directed to the right leads to the fact that a wave traveling to the right in the laboratory frame corresponds to positive $V$.
Let us draw attention in (25) to the fact that the ion velocity in the wave reference frame is negative, i.e., directed to the left. This remark is important if it is necessary, for example, to calculate ion fluxes in an ionacoustic wave.^{81}
This is a very important dependence, which contains the energy conservation law in the motion of ions and the law of conservation of the number of ions in the wave flow. Let us dwell on it in more detail and analyze it. The easiest way to do this is by plotting a graph for (28)—see Fig. 3.
The curve in this graph has two branches. One of them—negative—can always be discarded, because ionic concentration is never negative. However, it can be discarded for another reason, the negative branch does not satisfy the condition of the unperturbed plasma quasineutrality, i.e., does not pass through the $ n i n 0 i \phi = 0=1$ point. In Secs. III–V, we will see that in other plasma models, the dependence $ n i \phi $ always has two branches. In this case, it may be that both of its branches are nonnegative, but only one of these two branches always satisfies the unperturbed plasma quasineutrality condition. Therefore, the quasineutrality condition is a stronger criterion for branch separation!
The remaining branch looks like an increasing function. This corresponds to the fact that in an ionic gas compression phase, an excess of ions is formed in the wave, the potential becomes positive, and, conversely, in the ionic gas rarefaction phase, where there is a shortage of ions, the potential becomes negative.
And yet, the curve is bounded on the left by $ \u2212 2 Z e \phi / m V 2=1$, which determines the maximum value of the electrostatic potential in the wave. Beyond this value, in the maximum compression phase, the ions will stop in the wave reference frame and are reflected from the wave potential barrier (at this boundary, the potential is $\phi >0$, since the electron charge is $e<0$).
Figures 4(a)–4(c) shows three possible versions of $ U S \phi $ pseudopotential plots for different wave velocity $V$ values. All curves on them are bounded on the right by point A, at which the radical expression in (28) vanishes. At higher values of the potential, the root becomes imaginary, which corresponds to the reflection of ions from the wave potential barrier and the appearance of ambiguity in the ion density profile, i.e., breaking the wave back.
The exact solution to this transcendental equation can be expressed in terms of the negative branch of the Lambert Wfunction $ W \u2212 1 x$. Since in the future this new function will often be encountered in this work, we give its mathematical properties in Appendix A. Here, we only point out that the Lambert Wfunctions are the function inverse to $y=x$ exp $x$ function.
Oscillations of the pseudooscillator of maximum swing, starting from the local maximum position, determine the separatrix in the phase portrait, and could correspond to a solitary wave. However, such a solution does not satisfy the condition φ → 0 as $\xi \u2192\xb1\u221e$ and should be discarded. Therefore, subsonic solitary ionacoustic waves (solitons) do not exist.
For a supersonic wave, there are two forms of pseudopotential curves: both with a local maximum at zero and with a local minimum for some $\phi >0$. The only difference between the curves is that point A can be either above [Fig. 4(b)] or below [Fig. 4(c)] the $0\phi $ axis.
Oscillations of a pseudoparticle in a potential well near a local minimum do not satisfy the condition $ \u222b 0 \Lambda \phi \xi d \xi = 0$ for a periodic wave with $\Lambda $ period, which is the unperturbed plasma quasineutrality consequence. Consequently, supersonic periodic ionacoustic waves do not exist.
Thus, the soliton speed lies in the range $ V s<V< V max$, and at the maximum speed its amplitude is maximum and is equal to $ \phi max= \u2212 m V max 2 / 2 Z e\u2248\u2212 1.256 \u2009 431 \u2009 208 \u2026\xd7 k T 0 e / e$. It should be noted that the amplitude of the ionacoustic soliton is positive, since the electron charge $e<0.$ The indicated range of soliton velocities corresponds to the filled sector in Fig. 2.
Thus, as a result of the analysis of the Sagdeev pseudopotential $ U s \phi $, it was found that the subsonic periodic and supersonic solitary waves exhaust the set of nonlinear ionacoustic waves. Their profiles are shown in Figs. 5 and 6, respectively. In this case, other types of waves did not appear in the problem analysis.
III. FIRST TRICK BASED ON THE USE OF LAMBERT WFUNCTION: IONSOUND WAVES IN PLASMA WITH WARM ISOTHERMAL IONS
A. Linear theory
The dependence graph (39) is shown in Fig. 7. It already has three sections: a longwave section corresponding to linear ionacoustic waves propagating practically without dispersion with velocity $ V s= \omega i \lambda D e 2 + V T 2$; a mediumwave section corresponding to ionic plasma oscillations with group velocity, which is significantly less than $ V s$; and a shortwave section, the wave velocity in which asymptotically tends to the thermal velocity of the ions $ V T.$ The last section is of no practical importance, since the wave there rapidly decays according to Landau's decay mechanism. It should be noted that the thermal velocity somewhat increases the ionic sound linear velocity.
The graph lies entirely in the area bounded by rays $\omega = V s\kappa $ and $\omega = V T\kappa $. It is worth noting that the upper ray, corresponding to the ionic sound speed, is determined by the equations of state of the electron and ionic gases, and the lower ray, corresponding to the thermal speed of ions, is determined by the equation of the ionic gas state. When considering models with other equations of state, for example, with an equation of state describing a polytropic process (see below), or with an equation of state for a degenerate Fermi gas, only the slope of the corresponding boundary rays changes.
In the case under consideration, the stationary soliton solution should be sought outside the area containing the dispersion curve, i.e., in the shaded area Fig. 7: supersonic area $V> V s$. Further, we will also see that there are no solitons in the prethermal area $V< V T$.
B. Derivation of the formula for the pseudopotential and its analysis
Let us now consider the features of nonlinear ionacoustic waves in collisionless plasma with hot electrons and hot ions, assuming that the temperatures of the electron and ion gases are constant and not equal to each other. This means that the plasma parameters are such that partial thermodynamic equilibrium is established in the wave, leaving $ T 0 i$ and $ T 0 e$ constant. Before proceeding with the solution of the corresponding equations, let us briefly explain how this equilibrium can be maintained in collisionless plasma.
The collisionlessness of plasma implies the absence of headon interparticle collisions, while longrange Coulomb's collisions always exist. It is Coulomb's collisions, when they are uncorrelated with the wave phase, that provide heat transfer between different areas of gases in the sound wave. It is important that the “heat conduction wave” length is greater than the sound wavelength $ \lambda T \u226b \Lambda $. In the case of the opposite sign of inequality, an adiabatic process occurs. Estimates of transport coefficients, including the thermal conductivity coefficient for collisionless plasma, are given in Ref. 83, which make it possible to express the relationship between wavelengths in terms of plasma parameters.
When choosing a root, just as in case (28), we use the condition of plasma quasineutrality, according to which $ n i= n 0 i$ should be in unperturbed plasma at $\phi =0$. It turns out that for different values of the Lambert Wfunction argument, either one of the roots or the other can satisfy the quasineutrality condition [see dependency plots (46) in Figs. 8(a) and 8(b)].
According to Appendix A, point A at which the real branches of the Lambert Wfunction are conjugated has coordinates $[\u2212exp\u2009 \u2212 1,\u22121]$. It is point A that corresponds to the conjugation point of the roots given by (46). Then, the condition of differentiation in the choice of one or another true root from (46) follows from the condition of finding point A at the plasma quasineutrality point, i.e., $ n i= n 0 i$ at $\phi =0$, i.e., at $\phi =0$, the Wfunction argument must be equal to $\u2212exp\u2009 \u2212 1$, and this is possible when $ m V 2 k T 0 i=1$. The resulting expression is the very border of the true root choice, which can be formulated as follows: for $ m V 2 k T 0 i>1$ (which is equivalen $t\u2009to\u2009 V V T>1$), the true root given by (46) is described by the negative branch of the Lambert Wfunction $ W \u2212 1$, and the root described by the principal branch of the Wfunction $ W 0$ is false and must be discarded from consideration [Fig. 8(a)]. Conversely, for $ m V 2 k T 0 i<1$ (or $ V V T<1$), the true root given by (46) is described by the principal branch of the Wfunction $ W 0$, and the root described by the negative branch of the Wfunction $ W \u2212 1$ is false and should be discarded from consideration [Fig. 8(b)].
It should be said that the first integral of the equations of isothermal gases' motion always has the form of a transcendental quadraticlogarithmic equation of like (45), which has a solution expressed in terms of Lambert's Wfunction (for example, Refs. 84–89). This solution is always twovalued and it is necessary to choose the right root. In plasma, this can be done using the plasma quasineutrality condition. This is the 1st math trick.
For the first time, the selection of roots expressed in terms of the Lambert Wfunction using the quasineutrality condition was carried out in Ref. 90 in the problem of nonlinear waves in symmetric pair plasma (such as an electron–positron plasma). In this work, the possibility of the existence of solitons in symmetric pair plasma if the temperatures of plasma components are different was proved.
This trick with the choice of roots was also widely used in other problems: when calculating the maximum possible Mach numbers of ionacoustic solitons in isothermal plasma,^{52} when considering nonlinear backward dustacoustic waves in dusty plasma,^{91} when separating ambiplasma into matter and antimatter by a chain of solitons in the problem of baryon asymmetry of the universe,^{92} and other research.^{93,94} In research in which supersolitons were discovered^{54} and ion fluxes in supernonlinear ionacoustic waves^{81} were calculated, this trick was also used implicitly, without specifying it in the papers.
The pseudopotential plots are shown in Fig. 9. As can be seen in the dispersion curve (Fig. 7), they should have four qualitatively different shapes depending on the velocity $V$.
The pseudopotential at $V< V T$ presented in Fig. 9(a) is such that no stationary waveforms are possible here. This is easy to understand from the following physical considerations: any slow wave disturbance will be blurred by the thermal motion of ions.
At $ V T<V< V s$, periodic ionsound waves are possible in plasma, which correspond to pseudoparticle oscillations in the potential well [Fig. 9(b)].
At $ V s<V< V max$, when the right end of the pseudopotential curve is above the abscissa axis (point A), ionsound solitons can exist in plasma. This situation is shown in Fig. 9(c). It should be noted that here we denote the end point with letter A (just like in Fig. 8), because point A in Fig. 9 is the image of branch point A in Fig. 8. Physically, point A corresponds to the reflection of ions by a potential barrier in the wave.
At $V> V max,$ neither periodic nor solitary waves are possible [Fig. 9(d)].
The maximum soliton velocity can be determined by equating the pseudopotential value at point A to zero. This gives rise to a transcendental equation for $ V max$, which can only be solved numerically. This was done in Ref. 52, and the dependence of the isothermal ionacoustic soliton maximum velocity depending on $\tau = T 0 i / T 0 e$ parameter was found.
Let us summarize the consideration of this section's results with the following conclusion: a slight complication of the plasma model with the transition from cold ions to hot isothermal ions led the problem of deriving the pseudopotential by direct Sagdeev's method to the analytical solvability limit, when only with the help of a special function, the Lambert Wfunction, it was possible to solve the transcendental equation (45). In even more complex models, as we will see below, this possibility does not exist. Therefore, it is necessary to invent new mathematical tricks.
IV. SECOND TRICK BASED ON THE USE OF INVERSE FUNCTION INTEGRATION: IONSOUND WAVES IN PLASMA WITH WARM POLYTROPIC IONS
A. Linear theory
Recently, the nonlinear theory of ionacoustic waves in ordinary plasma has undergone new developments in papers Refs. 52 and 95–98 in which the acoustic compression–rarefaction wave is considered within the framework of local polytropic processes with an arbitrary polytropic exponent for each plasma component. Such a gasdynamic approach is the most general, containing an adiabat and isotherm, as extreme special cases. In the case of adiabatic compression–rarefaction of the components in a wave, this approach “freezes” the process of heat redistribution in the wave, and in the isothermal case, it makes this process inertialess. In addition, it is known that the dispersion equations derived from the linearized gasdynamic equations in the adiabatic case, as a rule, coincide with the equations obtained within the kinetic description, when the wave process is considered as a small deviation of the distribution function from the equilibrium function.
The dependency plot (52) shown in Fig. 10 does not differ qualitatively from the isothermal case plotted in Fig. 7. The main quantitative difference lies in the inclination angles of the bounding rays: $\omega = \beta i V T\kappa $ from below and $\omega = V s\kappa $ from above, where the speed of sound is $ \u2009 V s= \beta e \omega i \lambda D e 2 + \beta i V T 2$. Stationary soliton solutions should also be sought in this case in the shaded area in Fig. 10.
B. Derivation of the formula for the pseudopotential and its analysis
The polytropic process of plasma components' compression–rarefaction in an ionacoustic wave is the most common gasdynamic process. Let us consider it in more detail.
It is easy to establish, for example, graphically, that the implicit function $ n i \phi $, given by expression (58), as well as (28) and (46), is twovalued, and its plot has two branches, conjugating at branch point A [Figs. 11(a) and 11(b)]. One of the branches is also nonphysical and must be discarded, while the branch left here must also satisfy the unperturbed plasma quasineutrality condition, i.e., must pass through the $ n i \phi = 0= n 0 i$ point.
The analysis showed that for different values, the left branch can be lower or upper. So, for $V< \beta i V T$, the left branch is the upper one [Fig. 11(a)], and for $V> \beta i V T$ the left branch is the lower one [Fig. 11(b)]. Here and below, the thermal velocity of the ions $ \u2009 V T 2= k T 0 i / m$ is also denoted.
Further, if we follow Sagdeev's method, it is necessary to express $ n i$ from Eq. (58), then substitute it into the Poisson equation (55), and integrate to obtain the pseudopotential formula. However, it is not possible to obtain an explicit expression for $ n i$ for an arbitrary $ \beta i$. Therefore, the path based on the application of Sagdeev's method is still closed. We say “still” because perhaps, sometime in the future, a new special function that allows effective resolving of (58) with respect to $ n i$, thus opening Sagdeev's path, will be invented.
How to integrate the Poisson equation, on the right side of which there is the function $ n i \phi $, the form of which is unknown, but the formula for its inverse function $\phi n i$ given by (58) is known?
The differentiation rule of inverse functions is known from university courses in higher mathematics (see Secs. 1.4 and 3.7^{100} and 4.2–2 and 4.5–4^{101}); however, unfortunately, the integration of the inverse function is not included in these courses. Nevertheless, this rule exists,^{102} but it is less well known. This rule is given in Appendix B.
Thus, an exact analytical parametric formula is obtained for the Sagdeev pseudopotential for polytropic wave processes. Figure 12 shows its plots at different velocities of wave $V$, similar to those in Fig. 9 showing pseudopotentials for isothermal waves.
Deriving formulas for the pseudopotential using the inverse function integration method is the 2nd math trick. It was first used for the analytical solution of the problem of solitons in symmetric pairion plasma with different adiabatic exponents in Ref. 103.
In spite of the fact that this trick is versatile, as it makes it possible to find exact formulas for Sagdeev's pseudopotential for almost any equation of state, it is not often used. Indeed, the universality of the trick is based on the fact that the motion equation integral like (57) can always be solved with respect to $\phi $. Nevertheless, there is only one more research,^{52} in which the trick was used to analytically find the extreme ionsound soliton with the largest Mach number $ M max= 3 + 2 3\u22482.5424\u2026$ at $ \beta e=3$.
V. THIRD TRICK BASED ON THE USE OF BERNOULLI ORDINARY DIFFERENTIAL EQUATION: IONSOUND WAVES IN PLASMA WITH WARM POLYTROPIC IONS
The 2nd universal mathematical trick that helps in deriving the Sagdeev pseudopotential in complex cases and described in Sec. IV B is not the only one. We have developed another one.
We carry out one more calculation considering nonlinear ionacoustic waves in plasma with polytropic equations of state of the components. Let us return to formula (58), which gives an exact expression for $\phi n i$.
Substituting (71) and (75) into (74), one can obtain an integral expression, which cannot be calculated analytically for arbitrary polytropic exponents $ \beta e , i$. However, in cases where specific values are assigned to indicators, for example, $ \beta e= \beta i=2$ or $ \beta e= \beta i=3$, the integral in (74) is calculated analytically. The calculation results are very cumbersome (the formulas take several pages) and are not presented here.
The graphs of the Bernoulli pseudopotential at various values of the wave velocity $V$ and at $ \beta e=3$ and $ \beta i= 5 / 3$, which were calculated numerically, are given in Fig. 13. Unequal values $ \beta e$ and $ \beta i$ correspond to adiabatic ion sound in a plasma with magnetized electrons and unmagnetized ions. Thus, obtaining the Bernoulli pseudopotential reduced to calculating the integral in (74).
For the first time, the reduction of a nonlinear wave problem to the solution and analysis of the Bernoulli equation was carried out in Ref. 106 when analyzing space charge waves in a neutralized electron beam, and the pseudopotential $ U B n i$, which arises there, was proposed to be called the Bernoulli pseudopotential in Ref. 107 in which the nonlinear electron waves in plasma was investigated.
In contrast to the Sagdeev pseudopotential, where the role of pseudocoordinate is played by potential φ, in (74) the role of pseudocoordinate is played by the concentration of ions $ n i$. Therefore, it is convenient to choose constant $C$ so that $ U B n i = n 0 i=0$.
Subsequently, the reduction of the equations to the Bernoulli pseudopotential and its analysis took shape as a new independent method.^{108} This method, which is a 3rd math trick, is versatile and always workable, unlike Sagdeev's method. With its help, a large number of plasma problems were considered, e.g., adiabatic ionacoustic waves in plasma,^{98} adiabatic ionacoustic waves in dusty plasma taking into account the variation in charge of dust grains in a wave,^{109} adiabatic waves in electron–positron–ion plasma,^{110} and ionacoustic waves in plasma with quantumdegenerate electrons in the Thomas–Fermi approximation.^{72} Supernonlinear ionacoustic waves were found in dusty electron–positron–ion plasma also.^{68}
A variant of the method was also used to analyze electron and ionacoustic waves in plasma with components having incomplete quantum degeneracy,^{111,112} and the role of pseudocoordinate in $ U B \mu $ in these works was played by the chemical potential $\mu $ of the corresponding plasma component. In these cases, of course, the physical unit of measurement for the Bernoulli pseudopotential turned out to be different here.
We should also note the work presented in Ref. 113 in which the authors, remaining in the laboratory reference frame, derived the pseudopotential $U$( $ v i)$ in a different way, in which the role of pseudocoordinate is played by the ion velocity $ v i$. Thus, we come to an important conclusion that, in principle, any variable can play the role of pseudocoordinate.
Using the Bernoulli pseudopotential, nonlinear oscillations of coalescing magnetic flux filaments in cosmic plasma^{114} were analyzed, solitons in gravitating quantum plasma were investigated,^{115} and localized plasmon perturbations in graphene were considered.^{116} The Bernoulli pseudopotential method is also briefly discussed in Refs. 117–125.
VI. CONCLUSION
In the analysis of nonlinear waves in plasma, especially for the search of periodic waves, shock waves, and solitons, mechanical analogy methods are widely applicable. The most famous of them is the Sagdeev pseudopotential method. However, mathematical difficulties sometimes arise when deriving formulas for pseudopotentials. One of these difficulties is the impossibility of resolving and expressing function $\phi n$ in the form of an explicit function $n \phi $ [see Eqs. (27), (45), and (58)] for substitution into the Poisson equation.
The author proposed three mathematical tricks to get around this difficulty and obtain exact formulas for pseudopotentials in cases where Sagdeev's method was considered inapplicable:

the trick based on the use of Lambert Wfunction, which makes it possible to solve (45) in cases of isothermal wave processes;

the trick based on the inverse function integration, which does not require (58) or any other function and makes it possible to always obtain an exact expression of the pseudopotential in parametric form; and

the trick based on reducing the theory equations to Bernoulli's differential equation (the Bernoulli pseudopotential method)—here, nether condition (58) nor any other function is required, the trick allows you to always obtain an exact expression for the Bernoulli pseudopotential with a different argument.
The tricks were proposed over 10 years ago. They have undergone numerous approbation in various plasmawave problems of the author and his students. However, their use by other researchers is very rare.
This article, which is methodological and educational by nature, provides detailed examples of the use of each of these tricks when deriving exact formulas for pseudopotentials. The use of these tricks will make it possible to reduce the number of models for which it is necessary to apply approximate methods with the subsequent reduction of the system equations to the known model evolutionary equations.
ACKNOWLEDGMENTS
The article partially, explicitly, or indirectly used materials received by the author with his students and published earlier.^{52,67,72,73,81,82,90,92,98,106,108} The author expresses gratitude to all his students.
AUTHOR DECLARATIONS
Conflict of Interest
The author has no conflicts of interest on the topic of this research.
DATA AVAILABILITY
The data that support the findings of this study are available from the author upon reasonable request.
APPENDIX A: LAMBERT'S WFUNCTION AND ITS PROPERTIES
In the mid1990s, five mathematicians, viz., Corless, Gonnet, Hare, Jeffrey, and Knuth, invented a new function,^{126} naming it the Lambert Wfunction after the famous mathematician, mechanic, and optician Johann Heinrich Lambert, known by works on photometry, having established the brightness distribution law in the radiation of an absolutely black body, later named after him. In optics, there is even a special unit of brightness—Lambert.
In other words, the Lambert Wfunction is a function inverse to $y=x\u2009exp\u2009x$, which makes it easy to establish its simplest properties and plot its graph (Fig. 14). The Lambert Wfunction is neither even nor odd. It is defined in the semibounded interval $[\u2212 1 / e ; \u221e )$, where it takes values from −∞ to ∞, and for negative $x$ the function is twovalued. Branch point A with coordinates $(\u2212 1 / e ; \u2212 1 )$ divides the graph of the function into two branches, the upper $ W 0 x$ and the lower $ W \u2212 1 x$ so that both branches in point A have a vertical tangent. The upper branch $ W 0 x$, often called the principal branch, passes through the origin and no longer has any peculiarities. The lower branch $ \u2009 W \u2212 1 x$, called the negative branch, has an inflection point B with coordinates $(\u2212 2 / e 2;\u22122)$ and a vertical asymptote at $x=0$. Other integer values of the index $\nu \u22600,\u22121$ for the $ \u2009 W \nu x$ function refer to complexvalued branches, which are infinite. Further, in those cases where the formulas are valid for all branches, we will write the Wfunction without an index.
Two books^{127,128} and reviews^{129–134} on the properties and applications of the Wfunction in mathematical problems of physics have been published. Reviews and applications of the Lambert Wfunction in a wide range of problems in plasma theory are presented in Refs. 82, 135, and 136.
The Wfunction is a very convenient tool for solving many transcendental equations arising in various problems of the physics of plasma and plasmalike media.^{137–147}
APPENDIX B: INVERSE FUNCTION INTEGRATION
Formula (B1) is illustrated graphically in Fig. 15(b) as the difference between the areas of the figures, as well as in Refs. 148 and 149 and discussed in Ref. 150.