Stability Catalyzer for a Relativistic Non-Topological Soliton Solution

TL;DR

This paper introduces a method to stabilize a specific solitary wave solution in a nonlinear Klein-Gordon system by adding a massless term, making it energetically stable and akin to stable particles.

Contribution

It proposes a novel stability catalyzer by adding a massless term to the Lagrangian, ensuring the stability of the solitary wave solution without altering its fundamental dynamics.

Findings

Adding the term guarantees energetic stability of the SSWS.

Large parameter B enhances stability and prevents formation of non-trivial solutions with finite energy.

Stable configurations resemble multiple isolated SSWSs, similar to particles.

Abstract

For a real nonlinear Klein-Gordon Lagrangian density with a special solitary wave solution (SSWS), which is essentially unstable, it is shown how adding a proper additional massless term could guarantee the energetically stability of the SSWS, without changing its dominant dynamical equation and other properties. In other words, it is a stability catalyzer. The additional term contains a parameter $B$, which brings about more stability for the SSWS at larger values. Hence, if one considers $B$ to be an extremely large value, then any other solution which is not very close to the free far apart SSWSs and the trivial vacuum state, require an infinite amount of energy to be created. In other words, the possible non-trivial stable configurations of the fields with the finite total energies are any number of the far apart SSWSs, similar to any number of identical particles.

Equations

$${\mathcal{L}}_o={\partial }^{\mu }\phi {\partial }_{\mu }\phi -U(\phi ),$$

$$\square \phi =\frac{{\partial }^2\phi }{\partial t^2}-\frac{{\partial }^2\phi }{\partial x^2}=-\frac{1}{2}\frac{dU}{d\phi }.$$

$$T^{\mu \nu }=2{\partial }^{\mu }\phi {\partial }^{\nu }\phi -g^{\mu \nu }{\mathcal{L}}_o,$$

$$T^{00}={\epsilon }_o(x,t)={\dot{\phi }}^2+{\phi }^{\prime 2}+U(\phi )(T^{01}=2\dot{\phi }{\phi }^{\prime }),$$

$$U(\phi )={\phi }^4(1-{\phi }^2),$$

$$\frac{d^2{\phi }_o}{d{x}^2}=2{\phi }_o^3-3{\phi }_o^5,$$

$${\phi }_o(x)=\frac{\pm 1}{\sqrt{1+x^2}}.$$

$${\phi }_v(x,t)={\phi }_o(\tilde{x})=\frac{\pm 1}{\sqrt{1+{\tilde{x}}^2}},$$

$$E=m={\int }_{-\infty }^{+\infty }{T}^{00}dx={\int }_{-\infty }^{+\infty }[{\dot{{\phi }_v}}^2+{\phi }_v^{\prime 2}+U({\phi }_v)]dx=\gamma E_o=\gamma m_o,$$

$$P={\int }_{-\infty }^{+\infty }{T}^{01}dx=2{\int }_{-\infty }^{+\infty }\dot{{\phi }_v}{\phi }_v^{\prime }dx=\gamma m_o\text{v}.$$

$$\mathcal{L}={\mathcal{L}}_o+F={\partial }^{\mu }\phi {\partial }_{\mu }\phi -U(\phi )+F,$$

$$\left[\square \phi +\frac{1}{2}\frac{dU}{d\phi }\right]+\frac{1}{2}\left[\frac{\partial }{\partial x^{\mu }}\left(\frac{\partial F}{\partial ({\partial }_{\mu }\phi )}\right)-\left(\frac{\partial F}{\partial \phi }\right)\right]=0.$$

$$\mathcal{K}={\partial }_{\mu }\phi {\partial }^{\mu }\phi +U(\phi )={\dot{\phi }}^2-{\phi }^{\prime 2}+{\phi }^4(1-{\phi }^2),$$

$$F=B{\mathcal{K}}^3,$$

$$\frac{\partial }{\partial x^{\mu }}\left(\frac{\partial F}{\partial ({\partial }_{\mu }\phi )}\right)=B\mathcal{K}\left[6\mathcal{K}{\partial }_{\mu }{\partial }^{\mu }\phi +12{\partial }_{\mu }\mathcal{K}{\partial }^{\mu }\phi \right],$$

$$\frac{\partial F}{\partial \phi }=\left[3B{\mathcal{K}}^2\frac{\partial \mathcal{K}}{\partial \phi }\right],$$

$$T^{00}=\frac{\partial \mathcal{L}}{\partial \dot{\phi }}\dot{\phi }-\mathcal{L}=\left[{\dot{\phi }}^2+{\phi }^{\prime 2}+U(\phi )\right]+\left[B{\mathcal{K}}^2(6{\dot{\phi }}^2-\mathcal{K})\right]={\epsilon }_o+{\epsilon }_1.$$

$${\epsilon }_1=B{\mathcal{K}}^2(5{\dot{\phi }}^2+{\phi }^{\prime 2}+{\phi }^4({\phi }^2-1)),$$

$$F=B\sum _{i=1}^3{\mathcal{K}}_i^3,$$

$${\mathcal{K}}_1={\phi }^4{\mathbb{S}}_1,$$

$${\mathcal{K}}_2={\phi }^4{\mathbb{S}}_1+{\mathbb{S}}_2,$$

$${\mathcal{K}}_3={\phi }^4{\mathbb{S}}_1+{\mathbb{S}}_2+2{\phi }^2{\mathbb{S}}_3,$$

$${\mathbb{S}}_1={\partial }_{\mu }\theta {\partial }^{\mu }\theta -1,$$

$${\mathbb{S}}_2={\partial }_{\mu }\phi {\partial }^{\mu }\phi +{\phi }^4(1-{\phi }^2),$$

$${\mathbb{S}}_3={\partial }_{\mu }\phi {\partial }^{\mu }\theta .$$

$$k^{\mu }\equiv (k^0,k^1)=(\omega ,k)=(\omega ,\omega v),$$

$$\square \phi +\frac{1}{2}\frac{dU}{d\phi }+\frac{1}{2}\left[\frac{\partial }{\partial x^{\mu }}\left(\frac{\partial F}{\partial ({\partial }_{\mu }\phi )}\right)-\left(\frac{\partial F}{\partial \phi }\right)\right]=0,$$

$$\frac{\partial }{\partial x^{\mu }}\left(\frac{\partial F}{\partial ({\partial }_{\mu }\theta )}\right)=0.$$

$${\phi }_s(x)={\phi }_o(x)=\frac{\pm 1}{\sqrt{1+x^2}},{\theta }_s(t)={\omega }_{s}t=\pm t.$$

$${\phi }_v(x,t)={\phi }_s(\tilde{x})=\frac{\pm 1}{\sqrt{1+{\tilde{x}}^2}},{\theta }_v(x,t)={\theta }_s(\tilde{t})=k_{\mu }{x}^{\mu }=\omega t-kx.$$

$$\epsilon (x,t)=T^{00}=\frac{\partial \mathcal{L}}{\partial \dot{\phi }}\dot{\phi }+\frac{\partial \mathcal{L}}{\partial \dot{\theta }}\dot{\theta }-\mathcal{L}={\epsilon }_o+{\epsilon }_1+{\epsilon }_2+{\epsilon }_3=$$

$$\left[{\dot{\phi }}^2+{\phi }^{\prime 2}+U(\phi )\right]+B\sum _{i=1}^3\left[3C_i{\mathcal{K}}_i^2-{\mathcal{K}}_i^3\right],$$

$$C_i=\frac{\partial {\mathcal{K}}_i}{\partial \dot{\theta }}\dot{\theta }+\frac{\partial {\mathcal{K}}_i}{\partial \dot{\phi }}\dot{\phi }=\{\begin{array}{ll}2{\phi }^4{\dot{\theta }}^2& i=1\\ 2({\dot{\phi }}^2+{\phi }^4{\dot{\theta }}^2)& i=2\\ 2(\dot{\phi }+{\phi }^2\dot{\theta }{)}^2& i=3.\end{array}$$

$${\epsilon }_1=B{\mathcal{K}}_1^2[5{\phi }^4{\dot{\theta }}^2+{\phi }^4{\theta }^{\prime 2}+{\phi }^4]\ge 0,$$

Full text

Keywords : non-topological soliton; solitary wave solution; nonlinear Klein-Gordon equation; energetical stability; stability catalyzer.

1 Introduction

For decades, the classical relativistic field equations with stable solitary wave solutions (solions111According to some well-known references such as [1], a solitary wave solution is a soliton when it reappears without any distortion after collisions. The stability is essentially a necessary condition for a solitary wave solution to be a soliton. However, in this paper, we only adopt the stability condition for the definition of a soliton solution.) have drawn the interest of many physicists [1, 2, 3, 4, 5]. In fact, soliton solutions behave like real particles as they have the non-disperse localized energy density functions and satisfy the standard relativistic energy-momentum relations. For example, the real nonlinear Klein-Gordon (RNKG)[6, 7] systems in $1+1$ dimensions with kink (anti-kink) solutions [1, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], Skyrme model of baryons [5, 39, 40, 41, 42] and ’t Hooft Polyakov model which yields magnetic monopole solutions [1, 5, 43, 44, 45, 46, 47] in $3+1$ dimensions are three well-known systems which yield stable solitary wave solutions or solitons. One should note that, all the three systems mentioned above, yield topological solitons and the topological feature is the main reason behind their stability. With topological solutions, there are generally complicated conditions on the boundaries to have a multi particle-like solution. However, with the non-topological solitary wave solutions, each arbitrary multi particle-like solution can be obtained easily just by adding distinct far apart solitary wave solutions together.

There have been many works on the non-topological solitary wave solutions so far [1, 2, 3, 4, 5, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. However, the famous relativistic non-topological solitary wave solutions are Q-balls [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. With non-topological solitary wave solutions, an important criterion for examining the stability is the classical (or Vakhitov-Kolokolov) criterion [61, 62, 63, 64, 79, 80, 81]. The classical stability criterion is based on examining dynamical equations when they are linearized for small fluctuations above the background of the solitary wave solution. If the linearized equation does not lead to any unstable growing mode, the solitary wave solution is called a stable solution classically. There is another stability criterion called the energetical stability criterion. In fact, a special solitary wave solution is energetically stable if any arbitrary variation above its background leads to an increase in the total energy. In other words, an energetically stable solitary wave solution would be stable against any arbitrary deformation [82, 83, 84, 85, 86].

A solitary wave solution which is energetically stable would be a single solution among the other (close) solutions. For example, the kinks (anti-kinks) as the well-known topological solitary wave solutions of the real nonlinear Klein-Gordon (RNKG) systems are inevitably energetically stable [1, 82]. However, a solitary wave solution, which is classically stable, is not necessarily an energetically stable solution; or it is not a single solution with the minimum energy among the other (close) solutions. For example, some of the Q-ball solutions are classically stable [61, 62, 63, 64, 81], but none of the them are energetically stable [85]. In fact, the energetical stability criterion is at a higher level than the classical stability criterion. In other words, if a solitary wave solution is energetically stable, it would undoubtedly be classically stable, as well. Moreover, if a solitary wave solution is not classically stable, it would not be an energetically stable solution, as well.

In this paper, in line with the previous works [83, 84, 85, 86], we introduce a special RNKG model in $1+1$ dimensions with a well-formed non-topological solitary wave solution which is essentially unstable [1, 61]. But we will show how adding a proper term to the original RNKG Lagrangian density, transforms the special solitary wave solution (SSWS) into an energetically stable object. We call this additional term the “* stability catalyzer*”, because it behaves as a massless spook222We chose the word “spook” so that it will not be confused with words like “ghost” and “phantom”, which have meaning in the literature. which surrounds the SSWS and guarantees its energetical stability. In other words, it prevents any change in the internal structure of the SSWS, and leaves the dominant dynamical equations and other properties of the SSWS invariant. It should be noted that, we consider the model in $1+1$ dimension just for the sake of simplicity, as it can be extended to $3+1$ dimensions with some modifications.

There is a parameter $B$ in the stability catalyzer term, which leads to more stability for the SSWS at larger values. In other words, the larger the values the greater will be the increase in the total energy for any arbitrary small variation above the background of the SSWS. Hence, if one considers a system with an extremely large value of parameter $B$, then the other configurations of the fields (which are not very close to the SSWS and the vacuum state) need extremely large energies to be created; meaning that, the possible solutions of the system with the finite energies are only the free far apart SSWSs, as multi particle-like solutions.

The present paper has been organized as follows: In the next section, we set up the basic equations for the RNKG systems with a single scalar field and consider a special RNKG model with a special non-topological non-vibrational solitary wave solution which is essentially unstable. In section 3, we introduce the stability catalyzer term, which serves to introduce an extended KG system333The extended KG systems were introduced in Ref. [83]. with a single non-topological energetically stable SSWS. Section 4 presents an in-depth study of the stability of the SSWS for any arbitrary small deformations according to the energetical stability criterion. The last section is devoted to summary and conclusion.

2 Single field RNKG systems in $1+1$ dimensions

The simplest form of the real nonlinear Klein Gordon (RNKG) systems in $1+1$ dimensions can be introduced by the following Lagrangian density:

[TABLE]

in which $\varphi$ is a single real scalar field and $U(\varphi)$ is called the field potential. Note that, we set the speed of light to one ($c=1$) throughout the paper for the sake of simplicity. Using the principle of least action, the related equation of motion is

[TABLE]

Using the Noether’s theorem [6, 7], one can simply obtain the energy-momentum tensor:

[TABLE]

in which $g^{\mu\nu}$ is the Minkowski metric. The $T^{00}$ ($T^{10}=T^{01}$) component of this tensor is the same energy (momentum) density function, which for the Lagrangian density (1), is simplified to

[TABLE]

in which the dot and the prime are symbols for time and space derivatives respectively. The integration of $T^{00}$ ($T^{10}$) over the whole space yields the same total energy $E$ (momentum $P$) of the system and always remains constant.

In general, there is not a stable non-topological non-vibrational solitary wave solution for the RNKG systems in $1+1$ dimensions [1, 61]. For example, if one considers a special field potential as follows:

[TABLE]

then the equation of motion (2), for a static (non-moving and non-vibrational) solution $\varphi=\varphi_{o}(x)$, is simplified to

[TABLE]

which has a non-topological solution as follows:

[TABLE]

Applying the Lorentz transformations, the moving version of this solution (7) can be obtained as well:

[TABLE]

where $\tilde{x}=\gamma(x-vt)$, $\gamma=1/\sqrt{1-v^{2}}$ and $v$ is the velocity. However, the field potential (5) for $|\varphi|>\frac{\sqrt{6}}{3}$ is decreasing and for $|\varphi|>1$ takes negative values. Therefore, the special solitary wave solution (8) is essentially unstable and without violating the conservation energy law, the effect of any small perturbation, causes the profile of the localized solution (7) to blow up [1](see Figs. 1 and 2). In [61], the instability of the non-topological non-vibrational solitary wave solutions of the RNKG systems in $1+1$ dimensions are generally referred to the existence of the growing modes.

In general, since the theory is relativistic, the same well-known relativistic relations between the moving and non-moving versions of any arbitrary solution, which has a localized energy density function, would be obtained, meaning that:

[TABLE]

where $E_{o}$ ($m_{o}$) is the same rest energy (mass) of the solution. For the special solution (7), according to Eq. (9), the related rest energy is $E_{o}=\pi/4$. Moreover, the width of any arbitrary moving solution is always smaller than its non-moving version, exactly according to the Lorentz contraction law.

3 The stability catalyzer term for a SSWS

Here, we attempt to find a proper additional term for the original Lagrangian density (1) that could guarantee the energetical stability of the SSWS (7). However, similar to a catalyzer, we expect that it has no role in the dominant dynamical equation and the other properties of the SSWS (7). In other words, we first expect that the standard Eq. (2) to remain the dominant dynamical equation only for the SSWS (7), and second, the rest energy of the SSWS (7) to be a minimum among the energies of the other (close) solutions. The other complementary discussions are the same as those sufficiently presented in [85].

However, we assume a new extended KG Lagrangian density as follows:

[TABLE]

where $F$ is the same unknown additional (stability catalyzer) term, which should be properly identified. We expect the new extended Lagrangian density (11) to be reduced to the same original version (1) just for the SSWS (7), that is, the new additional term $F$ should be zero only for the SSWS (7). Note that, the new extended system (11) and the original RNKG system (1) are essentially different relativistic field systems with different solutions except for the SSWS (7), which is considered to be a common solution. According to the standard relativistic Lagrangian densities in physics, we expect the unknown additional scalar term $F$ to be a function of the allowed scalars $\varphi$ and $\partial_{\mu}\varphi\partial^{\mu}\varphi$. However, the new equation of motion is

[TABLE]

For the SSWS (7) to still remain a solution of the new equation of motion (12) (or the new equation of motion (12) is reduced to the same original version (2)), since the first part of this new equation, according to the same original Eq. (2), is satisfied automatically, i.e. $\left[\Box\varphi_{o}+\frac{1}{2}\frac{dU}{d\varphi_{o}}\right]=0$, and the functional $F$ is not essentially linear in ${\cal L}_{o}$, so we conclude that the two distinct terms $\frac{\partial}{\partial x^{\mu}}\left(\frac{\partial F}{\partial(\partial_{\mu}\varphi)}\right)$ and $\frac{\partial F}{\partial\varphi}$ must be zero independently for the SSWS (7).

To meet all these requirements, one can conclude that $F$ must be a function of the powers of ${\cal K}$ (${\cal K}^{n}$’s with $n\geq 3$), where ${\cal K}$ is a special scalar functional

[TABLE]

which is defined to be zero when we have a SSWS (7). For example, a simple choice for the functional $F$ is

[TABLE]

where $B$ is a real positive number. For this special choice (14), we obtain

[TABLE]

which are both obviously zero for the SSWS (7). In fact, each term on the right hand side of the above relations contains a power of ${\cal K}$ and hence all are zero for the SSWS (7). Therefore, with this special choice (14), we can be certain that the previous SSWS (7) is again a solution of the new extended system (11), and the new dynamical equation (12) is reduced to the same original one (2), as its dominant dynamical equation.

The energy density functional of the new extended system (11) can be obtained easily

[TABLE]

According to Eq. (13), the second part of the energy density (15) becomes

[TABLE]

which is zero for the SSWS (7) and the vacuum state $\varphi=0$. However, it is not a positive definite function, because function $\varphi^{4}(\varphi^{2}-1)$ in the range $0<|\varphi|<1$ is negative. Hence, we cannot be certain about the energetical stability of the SSWS (7).

In order to introduce a new proper additional term $F$ for which the energetical stability of the SSWS (7) is properly guaranteed, we have to use a new scalar field $\theta$ which can be called the phase field or the catalyzer field. However, the new proper additional term $F$ can be introduced as follows:

[TABLE]

where

[TABLE]

and

[TABLE]

In general, since $\mathbb{S}_{i}$’s are three independent scalars, it is not possible for them to be zero simultaneously except for the non-trivial SSWS (7) with $\theta=\omega_{s}t=\pm t$. In other words, $\mathbb{S}_{1}=0$, $\mathbb{S}_{2}=0$ and $\mathbb{S}_{3}=0$ are three independent coupled nonlinear PDE’s which do not have any non-trivial common solutions except for the SSWS (7) with $\theta=\pm t$ (see the Appendix A). In fact, the same result applies to ${\cal K}_{i}$’s, since ${\cal K}_{i}$’s are three independent linear combinations of the scalars $\mathbb{S}_{i}$’s, they are not zero simultaneously except for the SSWS (7) with $\theta=\pm t$. Note that, for a moving version of the SSWS (8), which moves at the velocity of $v$, the proper phase function $\theta$, for which all $\mathbb{S}_{i}$’s would be zero simultaneously, is $\theta=k_{\mu}x^{\mu}$, i.e. the boosted version of function $\theta=\pm t$, provided

[TABLE]

where $\omega=\gamma\omega_{s}$ and $\omega_{s}=\pm 1$.

However, the dynamical equations of motion of the extended KG system (11) with the new additional term (14) can be obtained easily as follows:

[TABLE]

Again, it is easy to show that all the different first and second derivatives of $F$ (17), which were seen in the Eqs. (25) and (26), for the SSWS (7) with $\theta=\pm t$, would be zero simultaneously. In other words, for the SSWS (7) with $\theta=\pm t$, Eq. (26) is automatically satisfied and Eq. (25) is reduced to the same standard original version (2) as the dominant dynamical equation of the free SSWS (7). Note that the SSWS (7) in the new extended system (11) must be now considered along with a scalar field $\theta=\pm t$. However, from here on in this paper, the non-moving SSWS is as follows:

[TABLE]

Hence, the moving version of the SSWS (27) would be

[TABLE]

where $\tilde{t}=\gamma(t-vx)$ and $\tilde{x}=\gamma(x-vt)$.

The energy density functional of the extended KG system (11) with the new additional term (17) is

[TABLE]

which are divided into four distinct parts and

[TABLE]

After a straightforward calculation, one can obtain:

[TABLE]

All terms in the above relations are now positive definite, therefore all the $\varepsilon_{i}$’s ($i=1,2,3$) are positive definite functions and bounded from below by zero. All $\varepsilon_{i}$’s ($i=1,2,3$) are zero simultaneously just for the trivial vacuum state $\varphi=0$ and the non-trivial SSWS (27), just as we expected from the catalyzer. Now, if parameter $B$ is considered to be a large number, since at least one of the ${\cal K}_{i}$’s is a non-zero function for any other solution, then at least one of the $\varepsilon_{i}$’s ($i=1,2,3$), which all contain parameter $B$, would be a large positive function. It means that for other solutions (except for the ones which are very close to the vacuum $\varphi=0$), the related energies are always larger than the rest energy of the SSWS (27). Unlike $\varepsilon_{i}$’s ($i=1,2,3$), which are three absolute positive functions and are minimum for the SSWS (27), $\varepsilon_{o}$ is not an absolute positive function and is not a minimum for the SSWS (27). In the next section, we will show the role of $\varepsilon_{o}$ in the stability considerations, meaning that if we take an extended system (11) with a large parameter $B$ (approximately $B>10^{2}$), it would be completely ineffective.

4 the energetically stability of the SSWS

In general, a solitary wave solution (such as kink and anti-kink solutions of the RNKG systems) is energetically stable if its rest energy is at a minimum among the energies of the other (close) solutions. In other words, for an energetically stable solitary wave solution, any arbitrary deformation (variation) above the background, leads to an increase in the total energy. In this section, we specifically examine the energetical stability of the SSWS (27). In fact, we are going to consider the variation of the total energy versus any arbitrary small deformation above the background of the SSWS (27), which is at rest. In general, any deformed version of the SSWS (27) can be introduced as follows:

[TABLE]

where $\delta\varphi$ and $\delta\theta$ are considered as arbitrary small functions of space-time. Now, if we insert (34) in $\varepsilon_{o}(x,t)$ and keep the terms up to the second order of small variation $\delta\varphi$, then it yields

[TABLE]

where $\varepsilon_{os}$, $\delta\varepsilon_{o1}$ and $\delta\varepsilon_{o2}$ are defined on the right hand side of the above equation, respectively. $\varepsilon_{os}(x)=\left(\varphi_{s}^{\prime 2}+U(\varphi_{s})\right)$ is the energy density function of the non-moving SSWS (27). $\delta\varepsilon_{o1}$ and $\delta\varepsilon_{o2}$ are functionals of the first and second order of the small variation $\delta\varphi$, respectively. Note that, for a non-moving SSWS (27), $\dot{\varphi_{s}}=0$, $\theta_{s}^{\prime}=0$ and $\dot{\theta_{s}}=\omega_{s}=\pm 1$. It is obvious that $\delta\varepsilon_{o1}$, $\delta\varepsilon_{o2}$, and hence $\delta\varepsilon_{o}$ are not necessarily the positive definite small functionals. Now, one can do a similar procedure for the additional terms $\varepsilon_{i}$’s ($i=1,2,3$). If one inserts a slightly deformed SSWS (34) in $\varepsilon_{i}$ ($i=1,2,3$), it yields

[TABLE]

in which $\varepsilon_{is}=0$, ${\cal K}_{is}=0$ and $C_{is}=\omega_{s}^{2}\varphi_{s}^{4}$ are related to the SSWS (27). According to Eq. (4), since $C_{i}>0$ (30), $\delta\varepsilon_{i}$’s are positive definite, as is generally expected from Eqs. (31), (32) and (33).

According to Eqs. (18)-(23), keeping up the terms to the first order of small variations for the deformed SSWS (34), we have

[TABLE]

Since $\delta{\cal K}_{i}$’s are linear in the first order of small variations $\delta\varphi$, $\delta\theta$ and their derivatives ($\delta\dot{\theta}$, $\delta\varphi^{\prime}$, $\delta\dot{\varphi}$ and $\delta\theta^{\prime}$), thus, according to Eq. (4), $\delta\varepsilon_{i}$’s are positive definite linear functions of the second order of small variations and their derivatives, which are all multiplied by $B$.

For any arbitrary small variations $\delta\varphi$ and $\delta\theta$ above the background of a non-moving SSWS (27), the variation of the total energy can be calculated by the integration of $\delta\varepsilon$ over the whole space:

[TABLE]

To show that the SSWS (27) is energetically stable, we must prove that $\delta E$ is always positive for any arbitrary small deformation. In other words, if any arbitrary deformation needs external energies to occur, then the SSWS (27) is an energetically stable solitary wave solution. Since $\delta\varepsilon_{1}$, $\delta\varepsilon_{2}$ and $\delta\varepsilon_{3}$ are positive definite functions, then their integration over the whole space, i.e. $\delta\mathbb{E}_{1}$, $\delta\mathbb{E}_{2}$ and $\delta\mathbb{E}_{3}$, always leads to positive values. Now, let us to focus on the $\delta\mathbb{E}_{o}$:

[TABLE]

where, $\delta\mathbb{E}_{o1}$ is the contribution of the first order of variations in $\delta\mathbb{E}_{o}$, which we will show that it would be zero in general. For the not-deformed non-moving SSWS (27), according to Eq. (2), as its dominant dynamical equation, we can use $\varphi^{\prime\prime}_{s}=\frac{d^{2}\varphi_{s}}{dx^{2}}$ instead of $\frac{1}{2}\frac{dU(\varphi_{s})}{d\varphi_{s}}$ to obtain:

[TABLE]

Hence, the integration of $\delta\varepsilon_{o1}$ over the whole space leads to

[TABLE]

Note that, $\delta\varphi$ and $\frac{d\varphi_{s}}{dx}$ are zero at $\pm\infty$. Therefore, the following result is generally valid:

[TABLE]

where $\delta\varepsilon_{e}=\delta\varepsilon-\delta\varepsilon_{o1}=\delta\varepsilon_{o2}+\sum_{i=1}^{3}\delta\varepsilon_{i}$, and can be called the effective variation of the energy density function. Now, if one can prove that for all arbitrary small variations, $\delta\varepsilon_{e}$ always remains a positive function, then the integration of that would be always positive as well, and the energetical stability condition is fulfilled. Note that, $\sum_{i=1}^{3}\delta\varepsilon_{i}$ is essentially positive definite, but $\delta\varepsilon_{o2}$ is not necessarily a positive function.

Since $\frac{1}{2}\frac{d^{2}U(\varphi_{s})}{d\varphi_{s}^{2}}=-15\varphi_{s}^{4}+6\varphi_{s}^{2}>0$ for $|\varphi_{s}|<\frac{\sqrt{10}}{5}$, hence undoubtedly, $\delta\varepsilon_{o2}=(\delta\dot{\varphi})^{2}+(\delta\varphi^{\prime})^{2}+\frac{1}{2}\frac{d^{2}U(\varphi_{s})}{d\varphi_{s}^{2}}(\delta\varphi)^{2}$ itself would be positive for the points $x$ that $|\varphi_{s}(x)|$ is less than $\frac{\sqrt{10}}{5}$, and then $\delta\varepsilon_{e}>0$ for such points. But, function $\frac{1}{2}\frac{d^{2}U(\varphi_{s})}{d\varphi_{s}^{2}}$, for the points $x$ that $|\varphi_{s}(x)|>\frac{\sqrt{10}}{5}$, would be negative, and we cannot be certain that $\delta\varepsilon_{o2}$ (and then $\delta\varepsilon_{e}$) is always positive. Nevertheless, if one considers a system with a large value of parameter $B$, we can be certain that $\delta\varepsilon_{e}>0$ for all points. In fact, $|\delta\varepsilon_{o2}|$ is a function of the second order of $\delta\varphi$, $\delta\varphi^{\prime}$ and $\delta\dot{\varphi}$ which does not contain parameter $B$, but $\delta\varepsilon_{i}$’s ($i=1,2,3$) are also functions of the second order of variations $\delta\varphi$, $\delta\theta$ and their derivatives which are multiplied by $B$. Hence, we are certain that always $\sum_{i=1}^{3}\delta\varepsilon_{i}\gg|\delta\varepsilon_{o2}|$ or $\delta\varepsilon_{e}\approx\sum_{i=1}^{3}\delta\varepsilon_{i}>0$, provided that $B$ is a large number (approximately $B>10^{2}$). Accordingly, for the arbitrary variations $\delta\varphi$ and $\delta\theta$, $\delta E$ would be always positive and then we ensure that the SSWS (27) is an energetically stable object, meaning that, its energy would be at a minimum among the other (close) solutions. It should be noted that, the theory is relativistic, hence confirming the energetical stability of the SSWS at rest, is generalized to all moving versions at any arbitrary speed.

To summarize, according to the previous considerations, for any small deformation above the background of the SSWS (27), we finally have

[TABLE]

where, $E$ is the total energy of the small deformed SSWS (34), and $E_{s}=\pi/4$ is the rest energy of the SSWS (27). It is true that for the hypothetical solutions (34), which are close to the SSWS (27), the field variations $\delta\varphi$, $\delta\theta$ and hence $\delta{\cal K}_{i}$’s are small, but the term $[3BC_{is}(\delta{\cal K}_{i})^{2}]$ is not necessarily small, because it contains the large parameter $B$. Hence, $\delta E$ is not necessarily small as well (see Fig. 3). For any close solution (34), with two specific small variations $\delta\varphi$ and $\delta\theta$, there are three specific $\delta{\cal K}_{i}$’s that are used to obtain the total energy (43). Since $\delta E$ is proportional to the integration of $\sum_{i=1}^{3}C_{is}(\delta{\cal K}_{i})^{2}$, thus for any arbitrary close solution (34), there are always continuously closer solutions with smaller $\delta\varphi$ and $\delta\theta$ and hence smaller $\sum_{i=1}^{3}C_{is}(\delta{\cal K}_{i})^{2}$, which leads to smaller $\delta E$. Therefore, none of the close solutions (34), are energetically stable. Note that, the close solutions (34) are those for which the approximations (4) and (4) are valid.

Numerically, we should study the stability of the SSWS (27) for some arbitrary small hypothetical deformations. For example, six arbitrary slightly deformed SSWSs (34) can be introduced as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

in which $\xi$ is a small parameter, which can be an indication of the order of deformations (variations) for any kind of the small deformations (44)-(49). All of the deformed functions (44)-(49) turn to the same free non-deformed SSWS (27) for $\xi=0$. For all arbitrary deformations (44)-(49), Fig. 3(a-f) show how larger values of parameter $B$ lead to more stability. In other words, the larger is the value of $B$ the greater would be the increase in the total energy versus $|\xi|$. Figure 3(a-c) show clearly why the case $B=0$ leads to an energetically unstable SSWS. In other words, for the case $B=0$, in Figure. 3(a-c), the rest energy of the SSWS, i.e. $E_{s}=E(\xi=0)=\frac{\pi}{4}$, is not a minimum. Note that, the case $B=0$ is the same original RNKG system (1) with the same SSWS (7).

Some hypothetical deformations for the SSWS (27) can be considered as those that appear in Figs. 1 and 2. According to Fig. 1, the profile of the initial SSWS (7) at $t=0$, does not remarkably change at the time interval $0<t<3$. In fact, it changes so slightly that is not visible in Fig. 1. If one numerically calculates the total energy of the deformed SSWS at different times $0<t<5$, Fig. (4) is obtained. It reaffirms that for arbitrary small deformations above the background of the SSWS (27), for example in the range $0<t<3$, the larger values of $B$ lead to more stability, that is, the larger the values the greater will be the increase in the total energy for any arbitrary small variation above the background of the SSWS (27). In general, the total energy always increases (and increases more for larger values of $B$) versus the amount of any arbitrary variation above the background of the SSWS (27). Although the parameter $B$ can be taken a large value, it would not affect the dominant dynamical equation (2) and the observable of the SSWS (27).

At the initial times (the times that are close to $t=0$), a multi lump (particle-like) solution with different velocities can be easily constructed just by adding distinct far apart SSWSs (28) together. For example, for $N$ distinct SSWSs (28) at different velocities $v_{j}$ and different initial positions $a_{j}$, provided that $a_{j+1}-a_{j}\gg 1$, the multi particle-like solution at initial times is as follows:

[TABLE]

where $\gamma_{j}=1/\sqrt{1-v_{j}^{2}}$. Since the phase field $\theta$ for each SSWS (28) depends on its velocity, hence it must change from one to another. That is to say, if there are two SSWSs (i.e. $N=2$) with one of them being at rest ($v_{1}=0$) and the other moving ($v_{2}\neq 0$), then the phase field must change from $\theta=\omega_{s}t$ at the position of the first SSWS to $\theta=k_{\mu}x^{\mu}=\omega_{2}t-k_{2}x$ at the position of the second SSWS. In the regions between two SSWSs, the scalar field $\varphi$ is almost zero and hence $\varepsilon$ is almost zero everywhere. Thus, there is not any rigorous restriction on $\theta$ to be in the standard forms $\theta=\omega_{s}t$ and $\theta=k_{\mu}x^{\mu}$ as the special solutions of the PDE $\mathbb{S}_{1}=0$. In other words, where the scalar field $\varphi$ is almost zero, the phase field $\theta$ is completely free and evolve without any rigorous restriction, i.e. it can change slowly from $\theta=\omega_{s}t$ to $\theta=k_{\mu}x^{\mu}$ in the spaces for which $\varphi\thickapprox 0$. In fact, for the case $\varphi\thickapprox 0$, it is not necessary to satisfy $\mathbb{S}_{i}=0$ or ${\cal K}_{i}=0$ ($i=1,2,3$) simultaneously, because for such situations, all $\varepsilon_{i}$’s ($i=1,2,3$) would automatically be almost zero simultaneously without any restrictive condition.

The general dynamical equations (25) and (26), as two coupled nonlinear PDEs, have infinite solutions. Similar to the SSWS (27), some of these solutions may be stable or there may not have any other stable solution at all. However, if one considers a system with a large parameter $B$, as we have already shown, there is not any energetically stable solution among the close solutions (34) at all. But how is it possible to know if the system has other stable solutions or not? In the present situation, it is by no means our goal to answer this question that can be mathematically important. In general, if one demonstrates that the other possible stable solutions have very large energies, then physically, it is not an important issue, because they require very large external energies to be created and it is outside the scope of our research. In fact, except for the solutions which are very close444In order for a better understanding of the matter, we can provide a qualitative definition of a conventional criterion. We could designate “very close solutions” for those “close solutions” (34) whose energy, for example for the case $B=10^{20}$, is less than $10^{3}$, or the ones for which the magnitude of the variations $\delta\varphi$ and $\delta\theta$ is approximately less than $10^{-9}$. to the free far apart SSWSs (50) and trivial vacuum state $\varphi=0$, the other (possible stable) solutions can not be physically simply created. To put it differently, for any (possible stable) solution, it is not possible for all ${\cal K}_{i}$’s to be zero simultaneously, thus at least one of the $\varepsilon_{i}$’s, which contain the large parameter $B$, is a non-zero large function and then the energy is much larger than the rest energy of a SSWS (27). Accordingly, the other (possible stable) solutions need large external energies to be created, which is very unlikely to occur physically. For example, if $B=10^{20}$, for a hypothetical small deformation (45) with $\xi=\pm 10^{-5}$, the total energy is in the order of $10^{11}$, that is, the required external energy must be of the order $10^{11}$ to create a small deformed SSWS (45) with $\xi=\pm 10^{-5}$!

Furthermore, for other (possible stable) solutions, the additional term $F$ is no longer zero and is a large functional. Thus, in the coupled PDEs (25) and (26), the terms $\Box\varphi$ and $\frac{1}{2}\frac{dU}{d\varphi}$ are very small compared with the terms which contain $F$. In other words, the general dynamical equations (25) and (26), for the other solutions, which are not very close to the free far apart SSWSs (50) and the vacuum state $\varphi=0$, are reduced to

[TABLE]

as their dominant dynamical equations. In terms of functional ${\cal K}_{i}$’s, since $F=B\sum_{i=1}^{3}{\cal K}_{i}^{3}$, equivalently Eqs. (51) and (52), turn into

[TABLE]

The important point is that the parameter $B$ is again ineffective in these equations. In other words, any solution of the coupled PDEs (53) and (54) would be approximately a solution of the general dynamical equations (25) and (26) as well, provided that $B$ is considered to be a large number.

For the far apart free SSWSs (50), the dominant dynamical equation is the same simple equation (2), and the role of the catalyzer term (17) is approximately zero. However, when they get close to each other and their profiles change slightly, the role of the stability catalyzer term $F$ becomes important and strongly opposes a closer approach and causes more change in their profiles. In this situation, the dominant dynamical equations are the same general forms (25) and (26). Accordingly, as expected in the collision between the SSWSs (50), they reappear after collision processes without any significant distortion. For example, for two SSWSs (50) which are initialized to collide with each other at the same speed $v=1-10^{-11}$, that is very close to the speed of light, the kinetic energy of each one is $K=E-E_{s}\thickapprox 2.2\times 10^{5}$. Now, if $B=10^{20}$, for a hypothetical deformation like (45), such kinetic energy can only cause a variation in the order of $\xi\approx 10^{-8}$ for each SSWS, that is, they reappear after the collision without any significant distortion. In general, for two free SSWSs (50), while they are far apart, the total energy is finite, but when they come close together to interact, their profiles can not have notable deformations, because their initial kinetic energy must be a huge value to generate a remarkable deformation for each SSWS, which is not physically simple to be provided.

Based on all that has been said so far, it is obvious that if one considers a system with an extremely large value of $B$ (for example $B=10^{20}$ or even more), the other configurations of the fields $\varphi$ and $\theta$, which are not very close to the free far apart SSWSs (50) and the vacuum state $\varphi=0$, require extreme energy to be created. From a physical point of view, this issue can be interesting, as it classically explains how a system leads to many identical particles with specific characteristics. In other words, if one considers this system as a real physical system, it is not possible to provide an extremely large external energy at a special place for creating the other (possible stable) configurations of the fields. Thus, the only non-trivial stable configurations of the fields with the finite energies would be any number of the far apart SSWSs (50) as a multi particle-like solution. Similar to the quantum field theory, the free far apart SSWSs (50) can be classically called the quanta of the system. Again, it should be noted that, it does not matter to us whether the system has other possible stable solutions. What is important to us here is that the only non-trivial stable solutions with finite energies are any number of the free far apart SSWSs (50), as many identical particles with the specific characteristics, which can be interesting for physicists.

5 Summary and conclusion

In this paper, we introduced an extended Klein-Gordon system as an example (11), which analytically yields an energetically stable solitary wave solution (27). In other words, it leads to a soliton solution. The new Lagrangian density (11) is composed of two distinct parts, first, the original part which is a known standard RNKG system (1), and second, an additional part, which can be called the stability catalyzer term (17). The original standard RNKG Lagrangian density (1) is introduced for a single scalar field $\varphi$. But, to introduce a proper stability catalyzer term, it is necessary to use a different scalar field $\theta$ (phase field) along with the original scalar field $\varphi$, meaning that the stability catalyzer term is a functional of $\varphi$ and $\theta$ simultaneously. The role of the stability catalyzer term seems as a massless spook which surrounds the SSWS (27) and guarantees the stability of the SSWS (27). The stability catalyzer term has no role in the dominant dynamical equation and the other properties of the SSWS (27), meaning that the general dynamical equations (25) are reduced to the same known standard RNKG version (2) just for the SSWS (27). However, it guarantees the energetical stability of the SSWS (27). Therefore, any arbitrary small deformation above the background of the SSWS (27) leads to an increase in the total energy. In other words, the rest energy of the SSWS is at a minimum among the other solutions of the new extended KG system (11) except for the ones which are very close to the vacuum state $\varphi=0$.

There is a parameter $B$ in the stability catalyzer term, the larger values of which, leads to more stability; meaning that, the larger the values the greater will be the increase in the total energy for any arbitrary small variation above the background of the SSWS (27). Hence, considering a system with an extremely large value of $B$, leads to a classical system only with multi particle-like solutions. In fact, the other solutions of the system, which are not very close to the free far apart SSWSs (50) and the vacuum state $\varphi=0$, require infinite amounts of energy to be created, and are not physically possible to occur. Thus, physically the possible stable solutions of the system with the finite energies, are either any number of the far apart SSWSs (as any number of identical free particles) or the trivial vacuum state. In other words, the SSWS (27) can be considered as the quantum of this classical system.

Acknowledgement

The author wishes to express his appreciation to the Persian Gulf University Research Council for their constant support.

Appendix A

Here, we are going to show that the following three PDE’s

[TABLE]

do not have any non-trivial common solutions except for the SSWS (27). Equation (57) generates $\dot{\theta}$ in terms of $\theta^{\prime}$, $\varphi^{\prime}$ and $\dot{\varphi}$ as follows:

[TABLE]

If we insert this into Eq. (55), we can obtain $\theta^{\prime}$ in terms of $\varphi^{\prime}$ and $\dot{\varphi}$ as follows:

[TABLE]

Using Eqs. (58) and (59), $\dot{\theta}$ can be obtained as well:

[TABLE]

The obvious mathematical expectation $(\dot{\theta})^{\prime}=\dfrac{d}{dx}\dfrac{d\theta}{dt}=\dfrac{d}{dt}\dfrac{d\theta}{dx}=\dot{(\theta^{\prime})}$ leads to the following result:

[TABLE]

which can be simply written in a covariant form:

[TABLE]

Therefore, to find the common solutions of the three independent nonlinear PDE’s (55), (56) and (57), equivalently we can search for the common solutions of the two different PDE’s (56) and (62). In general, it is easy to show that each non-vibrational function $\varphi_{v}(x,t)=\varphi_{o}(\gamma(x-vt))$, would be a solution of the PDE (62) or (61). Moreover, for any non-vibrational solitary wave solution, Eqs. (59) and (60) lead to $\theta^{\prime}=\pm\gamma v=\omega_{s}\gamma v=\omega v$ and $\dot{\theta}=\pm\gamma=\gamma\omega_{s}=\omega$ as we expected. On the other hand, we know that the SSWS (28) is the single non-vibrational solution of the PDE (56). Hence, for PDE’s (56) and (62), the single common non-vibrational solitary wave solution is the SSWS (28), as we expected. Accordingly, for the scalar field $\varphi$, there are two completely different PDE’s (56) and (62). Therefore, it does not seem that other common vibrational solutions exist along with the single non-vibrational SSWS (28).