This follows from the fact that *W Y*~, *Y* ¼ *W* Φ~ , Φ ¼ 0.

that the equality (61) holds, it is sufficient to prove the identity

we have *C x*ð Þ� , *x*, *λ* 0. But in this case *Cxx*00j*t*¼*x* ¼ ð Þ *V x*ð Þ� *λI C*j

Φ~ ð Þþ *x*, *λ* Φ0

ð Þj *x*, *t*, *λ t*¼*x* � Ω1ð Þ*λ* , i.e.,

It shows that Ω1ð Þ¼ *λ I*, we obtain (61).

*W*~ Φ~ , *Y* ¼ sin *A* � Φð Þ� 0, *λ* cos *A* � Φ0

(69) is performed on *x* identically, we obtain

Similarly we get *W*~ Ψ~, *Y* ¼ �*W Y*~, Ψ . Thus,

Ω1ð Þ¼ *λ* lim*x*!∞ Ψ0

*A*ð Þ¼� *λ W* Ψ~, *Y* , *B*ð Þ¼ *λ W* Φ~ , *Y* ,

Φ~ ð Þþ *x*, *λ* Φ0

ð Þ *x*, *λ* Φ~ ð Þ *x*, *λ* Φ0

ð Þ *x*, *λ Y x*ð Þ , *λ*

To make sure that the jump in the first derivative at *t* ¼ *x* is equal to ð Þ �*I* , i.e.,

which is an analogue of the Cauchy function. This function is the solution of Eq. (1) of the variable *x*, and it is the solution of Eq. (59) of the variable *t*. By (62),

Since operator solutions Φð Þ *x*, *λ* and Ψð Þ *x*, *λ* form a fundamental system of solutions of Eq. (1), the operator solution *Y x*ð Þ , *λ* of Eq. (1) satisfying the initial conditions (57), can be written as *Y x*ð Þ¼ , *λ* Φð Þ *x*, *λ A*ð Þþ *λ* Ψð Þ *x*, *λ B*ð Þ*λ* , where

Similarly, operator solution *Y x* ~ð Þ , *λ* of Eq. (59) can be represented in the form

ð Þ� *x*, *λ* Φ~ 0

*Y*�1

ð Þ *x*, *λ* Φ�1

ð Þ *x*, *λ W Y*~, Φ �1

Φ~ ð Þþ *t*, *λ* Φð Þ *x*, *λ W Y*~, Φ �1

ð Þ *x*, *λ W Y*~, Φ �1

*Y x*ð Þ¼ , *λ* Ψð Þ *x*, *λ W* Φ~ , *Y* � Φð Þ *x*, *λ W* Ψ~, *Y :* (70)

*Y x* ~ð Þ¼ , *λ W*~ Φ~ , *Y* Ψ~ð Þ� *x*, *λ W*~ Ψ~, *Y* Φ~ ð Þ *x*, *λ* , (71)

*Y x* ~ð Þ¼ , *λ W Y*~, Ψ Φ~ ð Þ� *x*, *λ W Y*~, Φ Ψ~ð Þ *x*, *λ :* (73)

ð Þ *x*, *λ* Ψ~ð Þ *x*, *λ :* (74)

Substituting (70) and (73) into the formula (69), using the fact that the equality

ð Þ *x*, *λ* Φ~ ð Þ� *x*, *λ* Φ0

By Theorem 2.1, on the asymptotic behavior of functions Φð Þ *x*, *λ* and Ψð Þ *x*, *λ* at

ð Þ¼� 0, *λ* Ωð Þ¼� 0, *λ W Y*~, Φ *:* (72)

ð Þ� *x*, *λ*

Φ�1

ð Þ *x*, *λ :* (66)

*Y x* ~ ð Þ� , *λ I:* (67)

*Y t*

*t*¼*x* � 0, and,

*Y x* ~ ð Þ� , *λ* Ω1ð Þ*λ :* (69)

~ ð Þ , *λ* , (68)

ð Þ *x*, *λ*

*Recent Developments in the Solution of Nonlinear Differential Equations*

$$
\overline{\chi}\_3(\mathbf{x}, \boldsymbol{\lambda}) = \Phi(\mathbf{x}, \boldsymbol{\lambda}) \,\mathsf{W}\{\boldsymbol{\Psi}, \boldsymbol{Y}\} \left(\mathsf{W}\{\boldsymbol{\Phi}, \boldsymbol{Y}\}\right)^{-1} \int\_{\boldsymbol{x}}^{\boldsymbol{\Phi}} \tilde{\Phi}\left(t, \boldsymbol{\lambda}\right) \tilde{f}(t) \,dt,\tag{82}
$$

$$
\overline{\chi}\_4(\mathbf{x}, \boldsymbol{\lambda}) = \Psi(\mathbf{x}, \boldsymbol{\lambda}) \int\_{\mathbf{x}}^{\mathbf{x}} \tilde{\Phi}\left(t, \boldsymbol{\lambda}\right) \overline{f}(t) \, dt. \tag{83}
$$

*χ*j j 2ð Þ *x*, *λ* ≤*c*2ð Þ*λ* 1

we have *χ*j j 4ð Þ *x*, *λ* ≤*c*1ð Þ*λ γ*∞ð Þ *x*, *λ* Ð ∞

the case *α*þ1

, we obtain

*χ*j j 4ð Þ *x*, *λ* ≤ *c*2ð Þ*λ x*�*α*

*λ* 2 � �*n*

lows *χ*j j 4ð Þ *x*, *λ* ≤*c*1ð Þ*λ γ*0ð Þ *x*, *λ γ*∞ð Þ *x*, *λ* Ð ∞

2*α* ¼ *n*∈ *N*, i.e. *α* ¼ 1

ð ∞

*γ*0ð Þ *t*, *λ γ*0ð Þ *x*, *λ f t*ð Þ � � �

> ð ∞

0 @

*t x*

� �2*a*ð Þ� *α*,*λ α*

*x*

2 ∞ð

> 2 ∞ð

Hence it follows that *χ*j j 4ð Þ *x*, *λ* ≤*c*4ð Þ *α*, *λ x*�*α*�1

0 @

1

Thus, *Rλf* ∈*L*2ð Þ **H**, 0, ð Þ ∞ for any function *f* ∈*L*2ð Þ **H**, 0, ð Þ ∞ . □

Since the resolvent *Rλ* is a meromorphic function of *λ*, the poles of which coincide with the eigenvalues of the operator *L*, the statement of Theorem 3.1 can

*are satisfied for the Eq. (1), then the spectrum of the operator L is real, discrete and coincides with the union of spectra of self-adjoint operators Lk*, *k* ¼ 1, *m, i.e.*

**Theorem 4.2** *If the conditions (3)-(5)) whereα* >1 *or condition (6) where* 0<*α* ≤1

0 @

1

*x*

Replacing variables *t* ¼ *xu*, we get

Since the inequality exp �*xα*þ1 *u*ð Þ *α*þ1�1

with sufficiently large *u*∈ ½ Þ 1, ∞ , we have

*χ*j j 4ð Þ *x*, *λ* ≤*c*3ð Þ*λ x*�*α*�1

In case, where 0< *α*≤1 and *α*þ1

*k*¼1*σ*ð Þ *Lk* .

be refined.

*σ*ð Þ¼ *L* ∪*r*

**57**

*χ*j j 4ð Þ *x*, *λ* ≤*c*3ð Þ*λ x*�*α*þ1

*χ*j j 4ð Þ *x*, *λ* ≤*c*3ð Þ*λ x*�*α*

ffiffiffiffiffiffiffiffiffi *v x*ð Þ p4

By formula (3), we get *χ*j j 2ð Þ *x*, *λ* ≤ *c*3ð Þ*λ*

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

ð *x*

0 @

1 ffiffiffiffiffiffiffiffi *v t*ð Þ p *dt*

case of *v x*ð Þ¼ *x*2*α*, 0 <*α* ≤1, the assertion can be proved similarly.

1 A

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

1 2

ffiffiffiffiffiffi

of the Eq. (1) satisfy the conditions (3)-(5), we have *χ*2ð Þ *x*, *λ* ∈*L*2ð Þ **H**, 0, ð Þ ∞ . In the

For the function *χ*4ð Þ *x*, *λ* we will conduct the proof for the case when *v x*ð Þ¼ *x*2*α*, 0 <*α* ≤1 and the coefficients of the Eq. (1) satisfy the condition (6). As in (85)

> *γ*0ð Þ *t*, *λ γ*0ð Þ *x*, *λ f t*ð Þ � � � �*dt*.

�*dt* ≤*c*2ð Þ*λ x*�*α*

exp

� �

Let us use the asymptotics of the functions *γ*0ð Þ *x*, *λ* and *γ*∞ð Þ *x*, *λ* , for example, in

*x γ*0ð Þ *t*, *λ f t*ð Þ �

*x*

≤ *c*2ð Þ*λ* 1

ffiffiffiffiffiffiffiffiffi *v x*ð Þ p4

2*n*�1 (see (36) and (40)).  Since the operator solution Φð Þ *x*, *λ* decays fairly quickly as *x* ! ∞, then j j Φð Þ *x*, *λ* ∈*L*2ð Þ 0, ∞ . It follows that

$$\left|\overline{\chi}\_{1}(\mathbf{x},\lambda)\right| \leq c(\lambda) \cdot \left|\Phi(\mathbf{x},\lambda)\right| \cdot \left\|\bar{\Phi}(t,\lambda)\right\| \cdot \left|\tilde{f}(t)\right| dt \leq$$

$$\leq c(\lambda) \cdot \left|\Phi(\mathbf{x},\lambda)\right| \cdot \left(\int\_{a}^{\mathbf{x}} \left|\bar{\Phi}(t,\lambda)\right| dt\right)^{\frac{1}{2}} \cdot \left(\int\_{a}^{\mathbf{x}} \left|\tilde{f}(t)\right| dt\right)^{\frac{1}{2}} <$$

$$\leq c(\lambda) \cdot \left|\Phi(\mathbf{x},\lambda)\right| \cdot \left(\int\_{a}^{\infty} \left|\bar{\Phi}(t,\lambda)\right| dt\right)^{\frac{1}{2}} \cdot \left(\int\_{a}^{\infty} \left|\tilde{f}(t)\right| dt\right)^{\frac{1}{2}} \leq c\_{1}(\lambda) \cdot \left|\Phi(\mathbf{x},\lambda)\right|,\tag{84}$$

and therefore *χ*1ð Þ *x*, *λ* ∈*L*2ð Þ **H**, 0, ð Þ ∞ . Similarly we get that *χ*3ð Þ *x*, *λ* ∈*L*2ð Þ **H**, 0, ð Þ ∞ . First we prove the assertion for the function *χ*2ð Þ *x*, *λ* , when

*α*> 1 and the coefficients of the Eq. (1) satisfy the conditions (3)-(5). In this case, we have *χ*j j 2ð Þ *x*, *λ* ≤j j Φð Þ *x*, *λ* Ð *x a* Ψ~ ð Þ *t*, *λ* � � � � *f t*ð Þ � � � �*dt*.

By virtue of the asymptotic formulas for the operator solutions Φð Þ *x*, *λ* and Ψð Þ *x*, *λ* we obtain that

$$\left|\overline{\chi}\_{2}(\varkappa,\lambda)\right| \leq c\_{1}(\lambda)\chi\_{0}(\varkappa,\lambda)\int\_{a}^{\varkappa} \chi\_{\varkappa}(t,\lambda)\left|\overline{f}(t)\right|dt.\tag{85}$$

Let us rewrite this relation in the following form

$$\left|\overline{\chi}\_{2}(\varkappa,\lambda)\right| \leq c\_{1}(\lambda)\chi\_{0}(\varkappa,\lambda)\chi\_{\infty}(\varkappa,\lambda)\int\_{a}^{\varkappa}\frac{\chi\_{\infty}(t,\lambda)}{\chi\_{\infty}(\varkappa,\lambda)}\left|\overline{f}(t)\right|dt.\tag{86}$$

By using the definition of the functions *γ*0ð Þ *x*, *λ* and *γ*∞ð Þ *x*, *λ* (see (9)) and by applying the Cauchy-Bunyakovskii inequality we obtain

$$|\overline{\chi}\_2(\mathbf{x}, \boldsymbol{\lambda})| \le \frac{1}{2} c\_1(\boldsymbol{\lambda}) \frac{1}{\sqrt{\nu(\boldsymbol{\kappa})}} \left( \int\_a^\mathbf{x} \sqrt{\frac{\nu(\boldsymbol{\kappa})}{\nu(t)}} \exp\left(-2 \int\_t^\mathbf{x} \sqrt{\nu(u)} du\right) dt \right)^{\frac{1}{2}} \left( \int\_0^\mathbf{\omega} \left| \tilde{f}(t) \right|^2 dt \right)^{\frac{1}{2}}.\tag{87}$$

Since *t* ≤*x*, we get exp �2 Ð *x t* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *v u*ð Þ*du* p � � ≤1, and then the latter estimate for *χ*2ð Þ *x*, *λ* can be rewritten as follows

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

$$|\overline{\chi}\_2(\mathbf{x}, \boldsymbol{\lambda})| \le c\_2(\boldsymbol{\lambda}) \frac{1}{\sqrt[4]{\nu(\mathbf{x})}} \left( \int\_a^\mathbf{x} \frac{1}{\sqrt{\nu(t)}} dt \right)^{\frac{1}{2}} \le c\_2(\boldsymbol{\lambda}) \frac{1}{\sqrt[4]{\nu(\mathbf{x})}} \left( \int\_a^\mathbf{x} \frac{1}{\sqrt{\nu(t)}} dt \right)^{\frac{1}{2}}.\tag{88}$$

By formula (3), we get *χ*j j 2ð Þ *x*, *λ* ≤ *c*3ð Þ*λ* ffiffiffiffiffiffi *v x*ð Þ p4 , and hence, if *α* >1 and the coefficients of the Eq. (1) satisfy the conditions (3)-(5), we have *χ*2ð Þ *x*, *λ* ∈*L*2ð Þ **H**, 0, ð Þ ∞ . In the case of *v x*ð Þ¼ *x*2*α*, 0 <*α* ≤1, the assertion can be proved similarly.

For the function *χ*4ð Þ *x*, *λ* we will conduct the proof for the case when *v x*ð Þ¼ *x*2*α*, 0 <*α* ≤1 and the coefficients of the Eq. (1) satisfy the condition (6). As in (85) we have *χ*j j 4ð Þ *x*, *λ* ≤*c*1ð Þ*λ γ*∞ð Þ *x*, *λ* Ð ∞ *x γ*0ð Þ *t*, *λ f t*ð Þ � � � �*dt*, which can be rewritten as follows *χ*j j 4ð Þ *x*, *λ* ≤*c*1ð Þ*λ γ*0ð Þ *x*, *λ γ*∞ð Þ *x*, *λ* Ð ∞ *x γ*0ð Þ *t*, *λ γ*0ð Þ *x*, *λ f t*ð Þ � � � �*dt*.

Let us use the asymptotics of the functions *γ*0ð Þ *x*, *λ* and *γ*∞ð Þ *x*, *λ* , for example, in the case *α*þ1 2*α* ¼ *n*∈ *N*, i.e. *α* ¼ 1 2*n*�1 (see (36) and (40)). Setting *a*ð Þ¼ *α*, *λ* 1�3� … �ð Þ 2*n*�3 *n*! � *λ* 2 � �*n* , we obtain

$$\left|\overline{\chi}\_{4}(\mathbf{x},\boldsymbol{\lambda})\right| \leq c\_{2}(\boldsymbol{\lambda})\mathbf{x}^{-a} \left(\frac{\boldsymbol{\gamma}\_{0}(\mathbf{t},\boldsymbol{\lambda})}{\boldsymbol{\gamma}\_{0}(\mathbf{x},\boldsymbol{\lambda})} \, \Big|\, \overline{f}(t)\, \middle|\, dt \leq c\_{2}(\boldsymbol{\lambda})\mathbf{x}^{-a} \left(\int\_{a}^{\mathbf{x}} \left(\frac{\boldsymbol{\gamma}\_{0}(\mathbf{t},\boldsymbol{\lambda})}{\boldsymbol{\gamma}\_{0}(\mathbf{x},\boldsymbol{\lambda})}\right)^{2} dt\right)^{\frac{1}{2}} \left(\int\_{0}^{\mathbf{x}} \left|\, \overline{f}(t)\right|^{2} dt\right)^{\frac{1}{2}}\right),\tag{89}$$

$$|\overline{\chi}\_4(\varkappa,\lambda)| \le c\_3(\lambda)\varkappa^{-a} \left( \bigcap\_{x}^{\infty} \left( \frac{t}{\varkappa} \right)^{2a(a,\lambda)-a} \exp \frac{-2\varkappa^{a+1} \left( \left( \frac{t}{\varkappa} \right)^{a+1} - 1 \right)}{1+a} dt \right)^{\frac{1}{2}}.\tag{90}$$

Replacing variables *t* ¼ *xu*, we get

$$|\overline{\chi}\_4(\mathbf{x}, \boldsymbol{\lambda})| \le c\_3(\boldsymbol{\lambda}) \pi^{-a + \frac{1}{2}} \left( \int\_1^\infty u^{2a(a, \boldsymbol{\lambda}) - a} \exp \frac{-2\pi^{a+1}(u^{a+1} - \mathbf{1})}{1 + a} du \right)^{\frac{1}{2}}.\tag{91}$$

Since the inequality exp �*xα*þ1 *u*ð Þ *α*þ1�1 1þ*α* ≤ *x*�2 holds for all *α* ∈ð � 0, 1 and *u* ∈½ Þ 1, ∞ with sufficiently large *u*∈ ½ Þ 1, ∞ , we have

$$|\overline{\chi}\_4(\varkappa,\lambda)| \le c\_3(\lambda) \pi^{-a-\frac{1}{2}} \left( \int\_1^\infty u^{2a(a,\lambda)-a} \exp\frac{-\varkappa^{a+1}(u^{a+1}-1)}{1+a} du \right)^{\frac{1}{2}}.\tag{92}$$

Hence it follows that *χ*j j 4ð Þ *x*, *λ* ≤*c*4ð Þ *α*, *λ x*�*α*�1 2, therefore *χ*4ð Þ *x*, *λ* ∈*L*2ð Þ **H**, 0, ð Þ ∞ *:* In case, where 0< *α*≤1 and *α*þ1 2*α* ∉ *N* and where *α*> 1, the proof is similar.

Thus, *Rλf* ∈*L*2ð Þ **H**, 0, ð Þ ∞ for any function *f* ∈*L*2ð Þ **H**, 0, ð Þ ∞ . □

Since the resolvent *Rλ* is a meromorphic function of *λ*, the poles of which coincide with the eigenvalues of the operator *L*, the statement of Theorem 3.1 can be refined.

**Theorem 4.2** *If the conditions (3)-(5)) whereα* >1 *or condition (6) where* 0<*α* ≤1 *are satisfied for the Eq. (1), then the spectrum of the operator L is real, discrete and coincides with the union of spectra of self-adjoint operators Lk*, *k* ¼ 1, *m, i.e. σ*ð Þ¼ *L* ∪*r k*¼1*σ*ð Þ *Lk* .

*χ*3ð Þ¼ *x*, *λ* Φð Þ *x*, *λ W* Ψ~, *Y* � � *W* Φ~ , *Y* � � � � �1

*Recent Developments in the Solution of Nonlinear Differential Equations*

*x* ! ∞, then j j Φð Þ *x*, *λ* ∈*L*2ð Þ 0, ∞ . It follows that

≤*c*ð Þ� *λ* j j Φð Þ *x*, *λ* �

ð ∞

0 @

Φ~ ð Þ *t*, *λ* � � �

*a*

<*c*ð Þ� *λ* j j Φð Þ *x*, *λ* �

we have *χ*j j 2ð Þ *x*, *λ* ≤j j Φð Þ *x*, *λ* Ð *x*

Ψð Þ *x*, *λ* we obtain that

*χ*j j 2ð Þ *x*, *λ* ≤

**56**

1 2 *c*1ð Þ*λ*

Since *t* ≤*x*, we get exp �2

*χ*2ð Þ *x*, *λ* can be rewritten as follows

*χ*4ð Þ¼ *x*, *λ* Ψð Þ *x*, *λ*

*χ*j j 1ð Þ *x*, *λ* ≤ *c*ð Þ� *λ* j j Φð Þ *x*, *λ* �

ð *x*

0 @

�*dt*

and therefore *χ*1ð Þ *x*, *λ* ∈*L*2ð Þ **H**, 0, ð Þ ∞ .   First we prove the assertion for the function *χ*2ð Þ *x*, *λ* , when *α*> 1 and the coefficients of the Eq. (1) satisfy the conditions (3)-(5). In this case,

> � *f t*ð Þ � � � �*dt*.

By virtue of the asymptotic formulas for the operator solutions Φð Þ *x*, *λ* and

ð *x*

*a*

ð *x*

*a*

ð *x*

*t*

By using the definition of the functions *γ*0ð Þ *x*, *λ* and *γ*∞ð Þ *x*, *λ* (see (9)) and by

exp �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *v u*ð Þ*du* p � �

0 @

In this section, a resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.

#### **5. Spectral singularities of differential operator with triangular matrix coefficients**

**Remark 5.1** *If the perturbation U x*ð Þ *in Eq. (1) does not satisfy conditions (3)-(5) or condition (6), then the statement of Theorem 4.2 ceases to be true, which is shown by the following example.*

Example 5.1 Consider the equation:

$$l[\overline{\eta}] = -\overline{\eta}'' + \begin{pmatrix} \varkappa^2 & q(\varkappa) \\ 0 & \pi^2 \varkappa^2 \end{pmatrix} \overline{\eta} = \lambda \overline{\eta}, \quad 0 \le \varkappa < \infty, \overline{\eta} = \begin{pmatrix} \varkappa\_1 \\ \varkappa\_2 \end{pmatrix} \tag{93}$$

with the boundary condition

$$
\overline{\wp}(\mathbf{0}) = \mathbf{0}.\tag{94}
$$

�*y*12

satisfying the initial conditions *y*12ð Þ¼ 0, *λ y*120

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

ð Þ¼ 0, *λ* 1, *v*ð Þ¼� 0, *λ* 1, *v*0

*y*12ð Þ¼ *x*, *λ*0

Choose the coefficient *q x*ð Þ¼ *y*22ð Þ *x*, *λ*0 *exμ*

∞ 0 *y*2

4 ≤*t*≤ *x*

*l*½ �¼� *y y*00 þ *p x*ð Þ *q x*ð Þ

*p x*ð Þ,*r x*ð Þ! ∞ monotonically as *x* ! ∞.

Consider the separated system

Let the boundary condition is given at *x* ¼ 0:

cos *A* � *y*0

where *A* is a triangular matrix, *cosA* ¼ cos *α*11 *cos α*12

*l*1 *y*1 � � ¼ �*y*1

*l*2 *y*2

� � ¼ �*y*00

4 for *t*≤ *x* 2

are fulfilled simultaneously, therefore, *y*12ð Þ *x*, *λ*0 > *c*3

4�2*α t* 2 >*e* 1 2 *t* 4 ≥ *e x*4

∞. It follows that *y*12ð Þ *x*, *λ*0 ∉ *L*2ð Þ 0; ∞ and *λ*0 ∉ *σ*ð Þ *L* .

*u*ð Þ¼ 0, *λ* 0, *u*0

solution *y*12ð Þ *x*, *λ*0 is given by

show that the integral Ð

In the case of *x*

≥*e*�*x*2

4 , *x* 2 � �, we have *et*

*x*≥ *N*1 >0,

we have

*e*�ð Þ *x*�*t* 2

**59**

and *t*∈ *x*

00 þ *x*2

ð *x*

0

*u x*ð Þ , *λ* , *v x*ð Þ , *λ* be the solutions of the Eq. (95), satisfying the initial conditions

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

*u x*ð Þ , *λ v t*ð Þ� , *λ v x*ð Þ , *λ u t*ð Þ , *λ* - be the Cauchy function of the Eq. (95). Then the

since the solution *y*22ð Þ *x*, *λ*0 has finitely many zeros, we conclude that, for any

�*αx*2

4 Ð *x* 2 *x* 4 *et* 4 � *e*�2*α t* 2

0 *r x*ð Þ � �*y* ¼ *λy*, 0≤ *x*< ∞, *y* ¼ *y*1

where *p x*ð Þ, *q x*ð Þ,*r x*ð Þ are scalar functions, *p x*ð Þ,*r x*ð Þ are real functions and

�ð Þ *x*�*t* 2

2 and *x*≥ max 4ð Þ *N*1, 2*N*2 , the inequalities (100) and (101)

32 , hence for *x* ! ∞*y*12ð Þ *x*, *λ*0 > *c*3 *x*

Ð *x* 2 *x* 4 *et* 4 � *e*�2*α t* 2

*y*22ð Þ *x*, *λ*0 ≥*c*1*e*

*C x*ð Þ , *t*, *λ*0 ≥ *c*2*e*

, we get *y*12ð Þ *x*, *λ*0 >*c*3*e*�*x*2

There arises the question on the nature of such values *λ*. Consider the equation with a triangular matrix potential:

and the Cauchy function decays no faster than *e*�ð Þ *x*�*t* 2

*y*12 þ *q x*ð Þ*y*22 ¼ *λ*0*y*12, (98)

*q t*ðÞ� *C x*ð Þ� , *t*, *λ*0 *y*22ð Þ *t*, *λ*0 *dt:* (99)

ð Þ¼ 0, *λ* 0, and let *C x*ð Þ¼ , *t*, *λ*

12ð Þ *x*, *λ*0 *dx* diverges and, consequently, *λ*0 ∉ *σ*ð Þ *L* . Indeed,

ð Þ¼ 0, *λ* 0, belongs to *L*2ð Þ 0; ∞ . Let

, where *μ*> 2 (for instance, *μ* ¼ 4), and

, *α*>0, (100)

. Hence, if j j *x* � *t* > *N*2,

*:* (101)

� *e*�ð Þ *x*�*t* 2

*y*2

ð Þ� 0 sin *A* � *y*ð Þ¼ 0 0, (103)

00 þ *p x*ð Þ*y*1 ¼ *λy*1, (104)

2 þ *r x*ð Þ*y*2 ¼ *λy*2*:* (105)

0 cos *α*22 � �.

*dt*. If *x* is sufficiently large

� �, (102)

4 *e*�*x*2 4 þ*x*4 32 !

*dt*. Since

Together with the problem (93), (94), consider the separated system

$$d\_1[\mathcal{y}\_1] = -\mathcal{y}\_1 \prime + \varkappa^2 \mathcal{y}\_1 = \lambda \mathcal{y}\_1,\tag{95}$$

$$\mathbb{I}l\_2\left[\mathbf{y}\_2\right] = -\mathbf{y}\_2'' + \pi^2 \mathbf{x}^2 \mathbf{y}\_2 = \lambda \mathbf{y}\_2 \tag{96}$$

with the boundary conditions.

$$y\_1(\mathbf{0}) = \mathbf{0}, y\_2(\mathbf{0}) = \mathbf{0}.\tag{97}$$

As above, denote by *L*0 the differential operator generated by the differential expression *l*½ � *y* (93) and the boundary condition (94), and by *L*1, *L*2 denote the minimal symmetric operators on *L*2ð Þ 0; ∞ , generated by the differential expressions *l*1 *y*1 , *l*2 *y*2 and the boundary conditions (97). Their self-adjoint extensions *L*~ 1, *L*~ 2 are the closures of the operators *L*1, *L*2, respectively. The operators *L*~ 1, *L*~ 2 are semibounded; let us denote their spectra by *σ*1 ¼ *σ L*~ 1 , *σ*2 ¼ *σ L*~ 2 .

The Eq. (95) (cf. (49)) has the solution *y*1,*n*ð Þ¼ *x Hn*ð Þ� *x exp* � *x*2 2 for *λ* ¼ 2*n* þ 1. Since *H*2*n*þ1ð Þ¼ 0 0, the eigenvalues of the operator *L*~ 1 are *λn* ¼ 4*n* þ 3. The sets *σ*1 and *σ*2 do not intersect.

Denote by *L* the extension of the operator *L*0 generated by the requirement on the functions from the domain of the operator *L* to belong to *L*2ð Þ *H*2, 0; ð Þ ∞ , and by *σ*ð Þ *L* its spectrum.

Denote by *Y x*ð Þ¼ , *λ y*11ð Þ *x*, *λ y*12ð Þ *x*, *λ* 0 *y*22ð Þ *x*, *λ* the matrix solution of the Eq. (93), satisfying the initial conditions *Y*ð Þ¼ 0, *λ* 0, *Y*0 ð Þ¼ 0, *λ I*.

If some *λ*0 ∈*σ L*~1 , and *y x*ð Þ , *λ*0 - is the corresponding eigenfunction of the operator *L*~1, then the vector function *y x*ð Þ¼ , *λ*0 *y x*ð Þ , *λ*0 0 is the eigenfunction of the operator *L*, corresponding to the eigenvalue *λ*0, i.e. *λ*0 ∈*σ*ð Þ *L* . Moreover, *λ*0 ∈*σ L*~2 is the eigenvalue of the operator *L* if and only if the solution *y*12ð Þ *x*, *λ*0 of the equation

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

$$-\mathcal{y}\_{12}{}^{\prime} + \mathfrak{x}^2 \mathcal{y}\_{12} + q(\mathfrak{x})\mathfrak{y}\_{22} = \lambda\_0 \mathfrak{y}\_{12},\tag{98}$$

satisfying the initial conditions *y*12ð Þ¼ 0, *λ y*120 ð Þ¼ 0, *λ* 0, belongs to *L*2ð Þ 0; ∞ . Let *u x*ð Þ , *λ* , *v x*ð Þ , *λ* be the solutions of the Eq. (95), satisfying the initial conditions *u*ð Þ¼ 0, *λ* 0, *u*0 ð Þ¼ 0, *λ* 1, *v*ð Þ¼� 0, *λ* 1, *v*0 ð Þ¼ 0, *λ* 0, and let *C x*ð Þ¼ , *t*, *λ u x*ð Þ , *λ v t*ð Þ� , *λ v x*ð Þ , *λ u t*ð Þ , *λ* - be the Cauchy function of the Eq. (95). Then the solution *y*12ð Þ *x*, *λ*0 is given by

$$\mathcal{Y}\_{12}(\mathbf{x}, \boldsymbol{\lambda}\_0) = \bigcap\_{0}^{\mathbf{x}} q(t) \cdot \mathbf{C}(\mathbf{x}, t, \boldsymbol{\lambda}\_0) \cdot \mathcal{Y}\_{22}(t, \boldsymbol{\lambda}\_0) dt. \tag{99}$$

Choose the coefficient *q x*ð Þ¼ *y*22ð Þ *x*, *λ*0 *exμ* , where *μ*> 2 (for instance, *μ* ¼ 4), and show that the integral Ð ∞ 0 *y*2 12ð Þ *x*, *λ*0 *dx* diverges and, consequently, *λ*0 ∉ *σ*ð Þ *L* . Indeed, since the solution *y*22ð Þ *x*, *λ*0 has finitely many zeros, we conclude that, for any *x*≥ *N*1 >0,

$$(y\_{22}(\varkappa, \lambda\_0) \ge c\_1 e^{-a\overline{\varkappa}^2}, a > 0,\tag{100}$$

and the Cauchy function decays no faster than *e*�ð Þ *x*�*t* 2 . Hence, if j j *x* � *t* > *N*2, we have

$$\mathbf{C}(\mathbf{x}, t, \lambda\_0) \ge c\_2 e^{-\left(\mathbf{x} - t\right)^2}. \tag{101}$$

In the case of *x* 4 ≤*t*≤ *x* 2 and *x*≥ max 4ð Þ *N*1, 2*N*2 , the inequalities (100) and (101) are fulfilled simultaneously, therefore, *y*12ð Þ *x*, *λ*0 > *c*3 Ð *x* 2 *x* 4 *et* 4 � *e*�2*α t* 2 � *e*�ð Þ *x*�*t* 2 *dt*. Since *e*�ð Þ *x*�*t* 2 ≥*e*�*x*2 4 for *t*≤ *x* 2 , we get *y*12ð Þ *x*, *λ*0 >*c*3*e*�*x*2 4 Ð *x* 2 *x* 4 *et* 4 � *e*�2*α t* 2 *dt*. If *x* is sufficiently large and *t*∈ *x* 4 , *x* 2 � �, we have *et* 4�2*α t* 2 >*e* 1 2 *t* 4 ≥ *e x*4 32 , hence for *x* ! ∞*y*12ð Þ *x*, *λ*0 > *c*3 *x* 4 *e*�*x*2 4 þ*x*4 32 ! ∞. It follows that *y*12ð Þ *x*, *λ*0 ∉ *L*2ð Þ 0; ∞ and *λ*0 ∉ *σ*ð Þ *L* .

There arises the question on the nature of such values *λ*.

Consider the equation with a triangular matrix potential:

$$l[\overline{\eta}] = -\overline{\eta}'' + \begin{pmatrix} p(\mathbf{x}) & q(\mathbf{x}) \\ \mathbf{0} & r(\mathbf{x}) \end{pmatrix} \overline{\eta} = \lambda \overline{\eta}, \quad \mathbf{0} \le \mathbf{x} < \infty, \overline{\eta} = \begin{pmatrix} y\_1 \\ y\_2 \end{pmatrix},\tag{102}$$

where *p x*ð Þ, *q x*ð Þ,*r x*ð Þ are scalar functions, *p x*ð Þ,*r x*ð Þ are real functions and *p x*ð Þ,*r x*ð Þ! ∞ monotonically as *x* ! ∞.

Let the boundary condition is given at *x* ¼ 0:

$$
\cos A \cdot \overline{\mathfrak{y}}'(0) - \sin A \cdot \overline{\mathfrak{y}}(0) = 0,\tag{103}
$$

where *A* is a triangular matrix, *cosA* ¼ cos *α*11 *cos α*12 0 cos *α*22 � �. Consider the separated system

$$d\_1[\mathcal{y}\_1] = -\mathcal{y}\_1 \prime + p(\mathbf{x})\mathcal{y}\_1 = \lambda \mathcal{y}\_1,\tag{104}$$

$$\mathbb{I}\left[\mathbf{y}\_2\right] = -\mathbf{y}\_2'' + r(\mathbf{x})\mathbf{y}\_2 = \lambda \mathbf{y}\_2. \tag{105}$$

In this section, a resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient

**5. Spectral singularities of differential operator with triangular matrix**

**Remark 5.1** *If the perturbation U x*ð Þ *in Eq. (1) does not satisfy conditions (3)-(5) or condition (6), then the statement of Theorem 4.2 ceases to be true, which is shown by the*

*y* ¼ *λy*, 0≤ *x*< ∞, *y* ¼ *y*1

*y*ð Þ¼ 0 0*:* (94)

*y*1ð Þ¼ 0 0, *y*2ð Þ¼ 0 0*:* (97)

*y*2

*y*1 ¼ *λy*1, (95)

*y*2 ¼ *λy*2 (96)

(93)

1, *L*~ 2

2 are semi-

for *λ* ¼

1, *L*~

1 are *λn* ¼ 4*n* þ 3. The

is the eigenfunction of

2

conditions under which the spectrum is real and discrete are obtained.

*Recent Developments in the Solution of Nonlinear Differential Equations*

*x*2 *q x*ð Þ 0 *π*2*x*2

> *l*1 *y*1 ¼ �*y*1

*l*2 *y*2

¼ �*y*00

are the closures of the operators *L*1, *L*2, respectively. The operators *L*~

2*n* þ 1. Since *H*2*n*þ1ð Þ¼ 0 0, the eigenvalues of the operator *L*~

The Eq. (95) (cf. (49)) has the solution *y*1,*n*ð Þ¼ *x Hn*ð Þ� *x exp* � *x*2

0 *y*22ð Þ *x*, *λ*

the operator *L*, corresponding to the eigenvalue *λ*0, i.e. *λ*0 ∈*σ*ð Þ *L* . Moreover,

bounded; let us denote their spectra by *σ*1 ¼ *σ L*~

Denote by *Y x*ð Þ¼ , *λ y*11ð Þ *x*, *λ y*12ð Þ *x*, *λ*

satisfying the initial conditions *Y*ð Þ¼ 0, *λ* 0, *Y*0

operator *L*~1, then the vector function *y x*ð Þ¼ , *λ*0

Together with the problem (93), (94), consider the separated system

00 þ *x*2

2 þ *π*2 *x*2

As above, denote by *L*0 the differential operator generated by the differential expression *l*½ � *y* (93) and the boundary condition (94), and by *L*1, *L*2 denote the minimal symmetric operators on *L*2ð Þ 0; ∞ , generated by the differential expressions

and the boundary conditions (97). Their self-adjoint extensions *L*~

Denote by *L* the extension of the operator *L*0 generated by the requirement on the functions from the domain of the operator *L* to belong to *L*2ð Þ *H*2, 0; ð Þ ∞ , and by

1

ð Þ¼ 0, *λ I*.

*y x*ð Þ , *λ*0 0

, and *y x*ð Þ , *λ*0 - is the corresponding eigenfunction of the

is the eigenvalue of the operator *L* if and only if the solution *y*12ð Þ *x*, *λ*0 of

, *σ*2 ¼ *σ L*~

2 .

the matrix solution of the Eq. (93),

**coefficients**

*following example.*

*l*1 *y*1 , *l*2 *y*2

Example 5.1 Consider the equation:

*l*½ �¼� *y y*00 þ

with the boundary condition

with the boundary conditions.

sets *σ*1 and *σ*2 do not intersect.

*σ*ð Þ *L* its spectrum.

*λ*0 ∈*σ L*~2

**58**

the equation

If some *λ*0 ∈*σ L*~1

with the boundary conditions

$$
\cos a\_{11} y\_1'(0) - \sin a\_{11} y\_1(0) = 0,\tag{106}
$$

The function *Gb*ð Þ *x*, *t*, *λ* is the Green function of the operator *L*0

(105), (107), *y*2ð Þ¼ *b* 0 respectively. Eigenvalues of the operators *L*0

to ones of the operators *L*~1 and *L*~2 respectively as *b* ! ∞, Φ*b*ð Þ! *x*, *λ* Φð Þ *x*, *λ* ,

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

of the operators *L*0

*W Y*~, Φ*b*

Φ~ *b*ð Þ! *x*, *λ* Φ~ ð Þ *x*, *λ* , and

� � ¼ cos *A* � Φ*b*

*Gb*ð Þ! *x*, *t*, *λ G x*ð Þ¼ , *t*, *λ*

of the determinant Δð Þ*λ* ≔ detΩð Þ*λ* , where.

�

Consider the operator *Rλ*,*b* defined on *L*2ð Þ *H*2, 0; ð Þ *b* by.

*Gb*ð Þ *x*, *t*, *λ f t*ð Þ*dt* ¼ � ð

*Y x*ð Þ , *λ W* Φ~ *b*, *Y* � � � � �1

Since the matrices cos *A*, sin *A*, Φð Þ 0, *λ* , Φ0

/Δð Þ¼ *λ* Δ1ð Þ� *λ* Δ2ð Þ*λ* , where Δ*k*ð Þ¼ *λ* cos *αkk* � *φ*0

*x*¼0 ¼ cos *A* � Φ0

Ωð Þ¼ *λ W Y*~, Φ � ��

ð *b*

0

þ ð *b*

*x*

sequence of finite continuous vector functions *f n*ð Þ *x*

**Remark 5.2** *As in Example 5.1, if λ*0 ∈*σ L*~2

operators *L*~1 and *L*~2.

*Rλ*,*bf* � �ð Þ¼ *x*

*Rλf* � �ð Þ¼ *x* Ð ∞

**61**

by the formula (112).

*b*,1, *L*0

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

0

¼ *W Y*~, Φ � �,*W* Φ~ *b*, *Y* � � ! *W* Φ~ , *Y* � �,

8 <

problem (102), (103), *y b*ð Þ¼ 0, which spectrum coincides with the union of spectra

ð Þ� 0, *λ* sin *A* � Φ*b*ð Þ! 0, *λ* cos *A* � Φ0

*Y x*ð Þ , *λ W* Φ~ , *Y* � � � � �1

�Φð Þ *x*, *λ W Y*~, Φ � � � � �1

Poles of the Green function *G x*ð Þ , *t*, *λ* of the operator *L* coincide with the zero set

On the other hand, zeros of the function Δ*k*ð Þ*λ* are eigenvalues of the self-adjoint operator *L*~*k*. Hence the poles of the Green function *G x*ð Þ , *t*, *λ* of the operator *L* are situated on the real axis, and their set coincides with the union of spectra of the

*x*

Φ*b*ð Þ *x*, *λ W Y*~, Φ*b*

� � � � �1

Φ~ ð Þ *t*, *λ f t*ð Þ*dt:* (114)

n o ð Þ *n* ¼ 1, 2, … converging in

� � *and φ*12ð Þ *x*, *λ*0 ∉ *L*2ð Þ 0, ∞ *, then λ*0 *is*

0

One can directly verify that the operator *Rλ*,*b* is the resolvent of the operator *L*0

Let *f x*ð Þ be an arbitrary vector function square integrable on 0, ½ Þ ∞ . Choose a

mean square to *f x*ð Þ. Substituting *f n* for *f* in (114) and letting first *b* ! ∞ and then *n* ! ∞, we obtain the following formula for the resolvent *Rλ* of the operator *L*:

**Theorem 5.1** *The operator Rλ is the resolvent of the operator L. The resolvent's poles*

*the pole of the resolvent Rλ of the operator L but it is not the eigenvalue of this operator,*

*coincide with the union of the spectra of the self-adjoint operators L*~1 *and L*~2*.*

*i.e., λ*0 *is the point of the spectral singularity of the operator L.*

0 *G x*ð Þ , *t*, *λ f t*ð Þ*dt*, where the Green function of the operator *L* is defined

*b*,2 generated by the problems (104), (106), *y*1ð Þ¼ *b* 0 and

*b* generated by the

*b*,2 tend

(111)

*b*,1 and *L*0

ð Þ� 0, *λ* sin *A* � Φð Þ¼ 0, *λ*

Φ~ ð Þ *t*, *λ* 0≤*x*≤*t*

*Y t* ~ ð Þ , *λ t*≤ *x*

ð Þ� 0, *λ* sin *A* � Φð Þ 0, *λ :* (113)

*kk*ð Þ� 0, *λ* sin *αkk* � *φkk*ð Þ 0, *λ* , *k* ¼ 1, 2.

*Y t* ~ð Þ , *λ f t*ð Þ*dt*þ

*b* .

ð Þ 0, *λ* are triangle, we have

: *:* (112)

$$
\cos a\_{22} y\_2'(\mathbf{0}) - \sin a\_{22} y\_2(\mathbf{0}) = \mathbf{0}.\tag{107}
$$

Let *L*0 be the differential operator generated by the differential expression *l*½ � *y* (103) and the boundary condition (104), and let *L*1, *L*2 be minimal symmetric operators on *L*2ð Þ 0, ∞ generated by the differential expressions *l*1 *y*1 � �, *l*2 *y*2 � � and the boundary conditions (106), (108) respectively. Denote by *L*~ 1, *L*~ 2 the self-adjoint extensions of the operators *L*1, *L*2 respectively. The operators *L*~ 1, *L*~ 2 are semibounded; let us denote their spectra by *σ*1 and *σ*2 respectively. Denote by *L* the extension of the operator *L*0 and by *σ*ð Þ *L* its spectrum.

Let *u x*ð Þ , *λ* , *v x*ð Þ , *λ* be the solutions of the Eq. (104) with the boundary conditions *u*ð Þ¼ 0, *λ* 0, *u*0 ð Þ¼ 0, *λ* 1, *v*ð Þ¼� 0, *λ* 1, *v*0 ð Þ¼ 0, *λ* 0*:* The general solution of the Eq. (104) has the form *φ*ð Þ¼ *x*, *λ u x*ð Þþ , *λ lv x*ð Þ , *λ* up to a constant. Choose an *l* such that the condition *φ*ð Þ¼ *b*, *λ* 0 holds true. This equality is valid for *l* ¼ *l b*ð Þ¼ , *λ*

� *u b*ð Þ , *λ v b*ð Þ , *λ* (the solution *v x*ð Þ , *λ* has finitely many zeros for a fixed *λ*, hence *v b*ð Þ , *λ* 6¼ 0

whenever *b* is sufficiently large). Put *φ*ð Þ *b* 11 ð Þ¼ *x*, *λ u x*ð Þþ , *λ l b*ð Þ , *λ v x*ð Þ , *λ* . Since for the operator *L*1 there is the case of a limit point, then, as is known, *l b*ð Þ , *λ* has a unique limit *m*ð Þ*λ* as *b* ! ∞, and the solution of the Eq. (104) satisfies *φ*11ð Þ¼ *x*, *λ u x*ð Þþ , *λ m*ð Þ*λ v x*ð Þ , *λ* ∈*L*2ð Þ 0, ∞ . Similarly we obtain that the solution of the Eq. (105) satisfies *φ*22ð Þ *x*, *λ* ∈*L*2ð Þ 0, ∞ .

Denote by Φ*b*ð Þ¼ *x*, *λ φ*ð Þ *b* 11 ð Þ *x*, *λ φ*ð Þ *b* 12 ð Þ *x*, *λ* 0 *φ*ð Þ *b* 22 ð Þ *x*, *λ* ! the matrix solution of the

Eq. (103) satisfying the initial conditions Φ*b*ð Þ¼ *b*, *λ* 0, Φ*b* 0 ð Þ¼ *b*, *λ I*. We have *φ*ð Þ *b* 11 ð Þ! *x*, *λ φ*11ð Þ *x*, *λ* ∈*L*2ð Þ 0, ∞ ; *φ*ð Þ *b* 22 ð Þ! *x*, *λ φ*22ð Þ *x*, *λ* ∈*L*2ð Þ 0, ∞ as *b* ! ∞.

The solution *φ*ð Þ *b* 12 ð Þ *x*, *λ* is given by *φ*ð Þ *b* 12 ð Þ¼ *x*, *λ* Ð *x* 0 *q t*ðÞ� *C x*ð Þ� , *t*, *λ φ*ð Þ *b* 22 ð Þ *t*, *λ dt*, where *C x*ð Þ¼ , *t*, *λ u x*ð Þ , *λ v t*ð Þ� , *λ v x*ð Þ , *λ u t*ð Þ , *λ* is the Cauchy function of the Eq. (104).

Further, we have *φ*ð Þ *b* 12 ð Þ! *x*, *λ* Ð *x* 0 *q t*ð Þ� *C x*ð Þ� , *t*, *λ φ*22ð Þ *t*, *λ dt* ≔ *φ*12ð Þ *x*, *λ* as *b* ! ∞.

Put Φð Þ¼ *x*, *λ φ*11ð Þ *x*, *λ φ*12ð Þ *x*, *λ* 0 *φ*22ð Þ *x*, *λ* � �.

Together with the Eq. (102), we consider the left equation.

$$
\tilde{l}[\tilde{\mathbf{y}}] = -\tilde{\mathbf{y}}'' + \tilde{\mathbf{y}}V(\mathbf{x}) = \lambda \tilde{\mathbf{y}}, \quad \tilde{\mathbf{y}} = \begin{pmatrix} y\_1 \ y\_2 \end{pmatrix}. \tag{108}
$$

The matrix solutions of the Eq. (108) will be denoted by Φ~ *b*ð Þ *x*, *λ* and Φ~ ð Þ *x*, *λ* .

Denote by *Y x*ð Þ , *λ* and *Y x* ~ð Þ , *λ* the solutions of the Eqs. (102) and (108) respectively satisfying the initial conditions

$$Y(\mathbf{0}, \boldsymbol{\lambda}) = \cos A,\\ Y'(\mathbf{0}, \boldsymbol{\lambda}) = \sin A,\\ \tilde{Y}(\mathbf{0}, \boldsymbol{\lambda}) = \cos A,\\ \tilde{Y}'(\mathbf{0}, \boldsymbol{\lambda}) = \sin A, \ \boldsymbol{\lambda} \in \mathbb{C}.\tag{109}$$

Put

$$G\_{b}(\mathbf{x},t,\lambda) = \begin{cases} \begin{array}{ll} Y(\mathbf{x},\lambda) \left(W(\tilde{\Phi}\_{b},Y)\right)^{-1} \tilde{\Phi}\_{b}(t,\lambda) & 0 \le \mathbf{x} \le t \\\ -\Phi\_{b}(\mathbf{x},\lambda) \left(W(\tilde{Y},\Phi\_{b})\right)^{-1} \tilde{Y}(t,\lambda) & t \le \mathbf{x} \le b \end{array} \end{cases} . \tag{110}$$

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

The function *Gb*ð Þ *x*, *t*, *λ* is the Green function of the operator *L*0 *b* generated by the problem (102), (103), *y b*ð Þ¼ 0, which spectrum coincides with the union of spectra of the operators *L*0 *b*,1, *L*0 *b*,2 generated by the problems (104), (106), *y*1ð Þ¼ *b* 0 and (105), (107), *y*2ð Þ¼ *b* 0 respectively. Eigenvalues of the operators *L*0 *b*,1 and *L*0 *b*,2 tend to ones of the operators *L*~1 and *L*~2 respectively as *b* ! ∞, Φ*b*ð Þ! *x*, *λ* Φð Þ *x*, *λ* , Φ~ *b*ð Þ! *x*, *λ* Φ~ ð Þ *x*, *λ* , and

$$\mathcal{W}(\tilde{Y}, \Phi\_b) = \cos A \cdot \Phi\_b'(0, \lambda) - \sin A \cdot \Phi\_b(0, \lambda) \to \cos A \cdot \Phi'(0, \lambda) - \sin A \cdot \Phi(0, \lambda) =$$

$$= \mathcal{W}(\tilde{Y}, \Phi), \mathcal{W}(\tilde{\Phi}\_b, Y) \to \mathcal{W}(\tilde{\Phi}, Y),$$

$$\mathbf{G}\_{b}(\mathbf{x},t,\lambda) \to \mathbf{G}(\mathbf{x},t,\lambda) = \begin{cases} \begin{array}{ll} Y(\mathbf{x},\lambda) \left(\mathcal{W}(\bar{\Phi},\mathbf{Y})\right)^{-1} \tilde{\Phi}(t,\lambda) & \mathbf{0} \le \mathbf{x} \le t \\\\ -\Phi(\mathbf{x},\lambda) \left(\mathcal{W}(\bar{\mathbf{Y}},\Phi)\right)^{-1} \tilde{\mathcal{Y}}(t,\lambda) & t \le \mathbf{x} \end{array} \end{cases} . \tag{112}$$

(111)

Poles of the Green function *G x*ð Þ , *t*, *λ* of the operator *L* coincide with the zero set of the determinant Δð Þ*λ* ≔ detΩð Þ*λ* , where.

$$\left.\Omega(\lambda) = W(\tilde{Y}, \Phi)\right|\_{\mathbf{x}=0} = \cos A \cdot \Phi'(0, \lambda) - \sin A \cdot \Phi(0, \lambda). \tag{113}$$

Since the matrices cos *A*, sin *A*, Φð Þ 0, *λ* , Φ0 ð Þ 0, *λ* are triangle, we have /Δð Þ¼ *λ* Δ1ð Þ� *λ* Δ2ð Þ*λ* , where Δ*k*ð Þ¼ *λ* cos *αkk* � *φ*0 *kk*ð Þ� 0, *λ* sin *αkk* � *φkk*ð Þ 0, *λ* , *k* ¼ 1, 2. On the other hand, zeros of the function Δ*k*ð Þ*λ* are eigenvalues of the self-adjoint operator *L*~*k*. Hence the poles of the Green function *G x*ð Þ , *t*, *λ* of the operator *L* are situated on the real axis, and their set coincides with the union of spectra of the operators *L*~1 and *L*~2.

Consider the operator *Rλ*,*b* defined on *L*2ð Þ *H*2, 0; ð Þ *b* by.

$$\left(R\_{\lambda,b}\overline{\hat{f}}\right)(\mathbf{x}) = \int\_{0}^{b} G\_{b}(\mathbf{x},t,\lambda)\overline{\hat{f}}(t)dt = -\int\_{0}^{\mathbf{x}} \Phi\_{b}(\mathbf{x},\lambda) \left(W(\hat{Y},\Phi\_{b})\right)^{-1} \hat{Y}(t,\lambda)\overline{\hat{f}}(t)dt + \int\_{\mathbf{x}}^{b} Y(\mathbf{x},\lambda) \left(W(\hat{\Phi}\_{b},Y)\right)^{-1} \hat{Y}(t,\lambda)\overline{\hat{f}}(t)dt. \tag{114}$$

$$+ \int\_{\mathbf{x}}^{b} Y(\mathbf{x},\lambda) \left(W(\hat{\Phi}\_{b},Y)\right)^{-1} \tilde{\Phi}(t,\lambda)\overline{\hat{f}}(t)dt. \tag{114}$$

One can directly verify that the operator *Rλ*,*b* is the resolvent of the operator *L*0 *b* . Let *f x*ð Þ be an arbitrary vector function square integrable on 0, ½ Þ ∞ . Choose a sequence of finite continuous vector functions *f n*ð Þ *x* n o ð Þ *n* ¼ 1, 2, … converging in mean square to *f x*ð Þ. Substituting *f n* for *f* in (114) and letting first *b* ! ∞ and then *n* ! ∞, we obtain the following formula for the resolvent *Rλ* of the operator *L*: *Rλf* � �ð Þ¼ *x* Ð ∞ 0 *G x*ð Þ , *t*, *λ f t*ð Þ*dt*, where the Green function of the operator *L* is defined by the formula (112).

**Theorem 5.1** *The operator Rλ is the resolvent of the operator L. The resolvent's poles coincide with the union of the spectra of the self-adjoint operators L*~1 *and L*~2*.*

**Remark 5.2** *As in Example 5.1, if λ*0 ∈*σ L*~2 � � *and φ*12ð Þ *x*, *λ*0 ∉ *L*2ð Þ 0, ∞ *, then λ*0 *is the pole of the resolvent Rλ of the operator L but it is not the eigenvalue of this operator, i.e., λ*0 *is the point of the spectral singularity of the operator L.*

with the boundary conditions

*u*ð Þ¼ 0, *λ* 0, *u*0

� *u b*ð Þ , *λ*

*φ*ð Þ *b*

cos *α*11*y*0

*Recent Developments in the Solution of Nonlinear Differential Equations*

cos *α*22*y*0

boundary conditions (106), (108) respectively. Denote by *L*~

extension of the operator *L*0 and by *σ*ð Þ *L* its spectrum.

whenever *b* is sufficiently large). Put *φ*ð Þ *b*

Eq. (105) satisfies *φ*22ð Þ *x*, *λ* ∈*L*2ð Þ 0, ∞ . Denote by Φ*b*ð Þ¼ *x*, *λ φ*ð Þ *b*

11 ð Þ! *x*, *λ φ*11ð Þ *x*, *λ* ∈*L*2ð Þ 0, ∞ ; *φ*ð Þ *b*

The solution *φ*ð Þ *b*

Further, we have *φ*ð Þ *b*

Put Φð Þ¼ *x*, *λ φ*11ð Þ *x*, *λ φ*12ð Þ *x*, *λ*

tively satisfying the initial conditions

8 <

*Y*ð Þ¼ 0, *λ* cos *A*, *Y*0

*Gb*ð Þ¼ *x*, *t*, *λ*

Put

**60**

ð Þ¼ 0, *λ* 1, *v*ð Þ¼� 0, *λ* 1, *v*0

extensions of the operators *L*1, *L*2 respectively. The operators *L*~

operators on *L*2ð Þ 0, ∞ generated by the differential expressions *l*1 *y*1

Let *L*0 be the differential operator generated by the differential expression *l*½ � *y* (103) and the boundary condition (104), and let *L*1, *L*2 be minimal symmetric

bounded; let us denote their spectra by *σ*1 and *σ*2 respectively. Denote by *L* the

Let *u x*ð Þ , *λ* , *v x*ð Þ , *λ* be the solutions of the Eq. (104) with the boundary conditions

Eq. (104) has the form *φ*ð Þ¼ *x*, *λ u x*ð Þþ , *λ lv x*ð Þ , *λ* up to a constant. Choose an *l* such that the condition *φ*ð Þ¼ *b*, *λ* 0 holds true. This equality is valid for *l* ¼ *l b*ð Þ¼ , *λ*

*v b*ð Þ , *λ* (the solution *v x*ð Þ , *λ* has finitely many zeros for a fixed *λ*, hence *v b*ð Þ , *λ* 6¼ 0

12 ð Þ *x*, *λ*

22 ð Þ *x*, *λ*

12 ð Þ¼ *x*, *λ* Ð

*C x*ð Þ¼ , *t*, *λ u x*ð Þ , *λ v t*ð Þ� , *λ v x*ð Þ , *λ u t*ð Þ , *λ* is the Cauchy function of the Eq. (104).

~*l*½ �¼� ~*y* ~*y*00 þ ~*yV x*ð Þ¼ *λ*~*y*, ~*y* ¼ *y*1 *y*2

ð Þ¼ 0, *λ* sin *A*, *Y*~ð Þ¼ 0, *λ* cos *A*, *Y*~0

� � � � �1

The matrix solutions of the Eq. (108) will be denoted by Φ~ *b*ð Þ *x*, *λ* and Φ~ ð Þ *x*, *λ* . Denote by *Y x*ð Þ , *λ* and *Y x* ~ð Þ , *λ* the solutions of the Eqs. (102) and (108) respec-

the operator *L*1 there is the case of a limit point, then, as is known, *l b*ð Þ , *λ* has a unique limit *m*ð Þ*λ* as *b* ! ∞, and the solution of the Eq. (104) satisfies *φ*11ð Þ¼ *x*, *λ u x*ð Þþ , *λ m*ð Þ*λ v x*ð Þ , *λ* ∈*L*2ð Þ 0, ∞ . Similarly we obtain that the solution of the

11 ð Þ *x*, *λ φ*ð Þ *b*

*x* 0

Together with the Eq. (102), we consider the left equation.

*Y x*ð Þ , *λ W* Φ~ *b*, *Y* � � � � �1

�Φ*b*ð Þ *x*, *λ W Y*~, Φ*b*

.

Eq. (103) satisfying the initial conditions Φ*b*ð Þ¼ *b*, *λ* 0, Φ*b*

12 ð Þ *x*, *λ* is given by *φ*ð Þ *b*

12 ð Þ! *x*, *λ* Ð

0 *φ*22ð Þ *x*, *λ* � �

0 *φ*ð Þ *b*

!

1ð Þ� 0 sin *α*11*y*1ð Þ¼ 0 0, (106)

2ð Þ� 0 sin *α*22*y*2ð Þ¼ 0 0*:* (107)

1, *L*~

ð Þ¼ 0, *λ* 0*:* The general solution of the

11 ð Þ¼ *x*, *λ u x*ð Þþ , *λ l b*ð Þ , *λ v x*ð Þ , *λ* . Since for

the matrix solution of the

ð Þ¼ *b*, *λ I*. We have

� �*:* (108)

ð Þ¼ 0, *λ* sin *A*, *λ*∈ *:*

(109)

22 ð Þ *t*, *λ dt*, where

0

*q t*ðÞ� *C x*ð Þ� , *t*, *λ φ*ð Þ *b*

*q t*ð Þ� *C x*ð Þ� , *t*, *λ φ*22ð Þ *t*, *λ dt* ≔ *φ*12ð Þ *x*, *λ* as *b* ! ∞.

Φ~ *b*ð Þ *t*, *λ* 0≤*x*≤*t*

~ ð Þ , *λ t*≤*x*≤*b*

*Y t*

: *:* (110)

22 ð Þ! *x*, *λ φ*22ð Þ *x*, *λ* ∈*L*2ð Þ 0, ∞ as *b* ! ∞.

*x* 0

1, *L*~

� �, *l*2 *y*2

2 the self-adjoint

2 are semi-

� � and the

Theorem 5.1 implies that, if the rate of the coefficient's growth *q x*ð Þ of the Eq. (102) is subordinated to one of *p x*ð Þ and *r x*ð Þ, then the operator *L* has no spectral singularities, and its spectrum is real and coincides with the union of the spectra of the operators *L*~1 and *L*~2.

**References**

+438 p.

[1] Rofe-Beketov F. S., Kholkin A. M. Spectral analysis of differential operators. Interplay between spectral and oscillatory properties. New Jersye, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai: World Scientific publishing co.; 2005.-xxiii

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

[10] Rofe-Beketov F.S., Zubkova E.I. Inverse scattering problem on the axis for the triangular 2 2 matrix potential a system with or without a virtual level. Azerbaijan Journal of Mathematic. 2011;

[11] Kholkin A.M. The fundamental solutions of differential operator with block-triangular operator coefficients. J. of Advances in Mathematics. 2015; **10**

[12] Kholkin A.M. Resolvent for Nonself-Adjoint Differential Operator with Block-Triangular Operator Potential. Abstract and Applied Analysis. 2016;

[13] Kholkin A.M. Asymptotics of the fundamental system solutions of differential equation with blocktriangular operator potential. Far East J. of Mathematical Sciences. 2017; **102**

[14] Kholkin A.M. Spectral singularities of differential operators with triangular

Functional Analysis and Topology. 2013;

matrix coefficients. Methods of

[15] Titchmarsh EC. Eigenfunction expansions associated with second – order differential equations. Vol. 2: Oxford: Clarendon Press; 1958.-xi+404 p.

[16] Naimark M.A. Investigation of

[17] Schwartz J. T. Some nonselfadjoint operators. Comm. Pure and Appl. Math.

spectrum and expansion in eigenfunctions of nonselfadjoint differential operators of second - order on the semiaxis. Tr. Mosk. Mat. Obs.

**1**(2): 3–69.

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

(6): 3555–3561.

**2016**: 1–6.

(10): 2359–2366.

**19**(3): 260–267.

1954; **3**: 181–270.

1960; **13**: 609–639.

[2] Kholkin A.M, Rofe-Beketov F.S. Sturm type theorems for equations with block-triangular matrix coefficients. Methods of Functional Analysis and Topology. 2012; **18**(2): 176–188.

[3] Naimark M.A. Linear differential operators. Part I: New York: Frederick Ungar Publishing Co.; 1968.-xiii+144 p.; Part II: New York: Frederick Ungar Publishing Co.; 1969.-xv+352 p.

[4] Kholkin A.M, Rofe-Beketov F.S. On spectrum of differential operator with block-triangular matrix coefficients. Journal Mathematical Physics, Analysis,

Geometry. 2014; **10**(1): 44–63.

[5] Marchenko V.A. Sturm-Liouville Operators and its applications. Basel: Birkhauser Verlag; 1986.-xii+367 p.

[6] Lyantse V.E. On non-self-adjoint second-order differential operators on the semi-axis*.* Doklady Akademii Nauk SSSR. 1964; **154**(5): 1030–1033.

[7] Rofe-Beketov F.S. Expansion in eigenfunctions of infinite systems of differential equations in the non-selfadjoint and self- adjoint cases. Mat. Sb.

[8] Schwartz J.T. Some nonselfadjoint operators. Comm. for pure and appl.

[9] Bondarenko E.I., Rofe-Beketov F.S. Phase equivalent matrix potential*.* Telecommun And Radio Eng. 2000; **56**

1960; **51**(3): 293–342.

(8 and 9): 4–29.

**63**

Math. 1960; **XIII**: 609–639.

For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed. A special role of these points was found first by M.A. Naimark in [16]. The notion "spectral singularity" was introduced later due to J. Schwartz [17] (see also Supplement I in the monograph [3]).

#### **6. Conclusion**

We consider the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential. For him, built a fundamental system of solutions, one of which is decreasing at infinity, and the second is growing. The asymptotics of these solutions at infinity is defined. For non-self-adjoint operator generated by such differential expression obtained the Green's function. A resolvent of such an operator is constructed. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained.

#### **Author details**

Aleksandr Kholkin

Department of Higher and Applied Mathematics, Pryazovkyi State Technical University, Mariupol, Ukraine

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

#### **References**

Theorem 5.1 implies that, if the rate of the coefficient's growth *q x*ð Þ of the Eq. (102) is subordinated to one of *p x*ð Þ and *r x*ð Þ, then the operator *L* has no spectral singularities, and its spectrum is real and coincides with the union of the spectra of

*Recent Developments in the Solution of Nonlinear Differential Equations*

tial growing at infinity, an example of operator having spectral singularities is constructed. A special role of these points was found first by M.A. Naimark in [16]. The notion "spectral singularity" was introduced later due to J. Schwartz [17] (see

For a non-self-adjoint Sturm-Liouville operator with a triangular matrix poten-

We consider the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential. For him, built a fundamental system of solutions, one of which is decreasing at infinity, and the second is growing. The asymptotics of these solutions at infinity is defined. For non-self-adjoint operator generated by such differential expression obtained the Green's function. A resolvent of such an operator is constructed. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained.

Department of Higher and Applied Mathematics, Pryazovkyi State Technical

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

the operators *L*~1 and *L*~2.

**6. Conclusion**

**Author details**

Aleksandr Kholkin

**62**

University, Mariupol, Ukraine

provided the original work is properly cited.

\*Address all correspondence to<EMAIL_ADDRESS>
also Supplement I in the monograph [3]).

[1] Rofe-Beketov F. S., Kholkin A. M. Spectral analysis of differential operators. Interplay between spectral and oscillatory properties. New Jersye, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai: World Scientific publishing co.; 2005.-xxiii +438 p.

[2] Kholkin A.M, Rofe-Beketov F.S. Sturm type theorems for equations with block-triangular matrix coefficients. Methods of Functional Analysis and Topology. 2012; **18**(2): 176–188.

[3] Naimark M.A. Linear differential operators. Part I: New York: Frederick Ungar Publishing Co.; 1968.-xiii+144 p.; Part II: New York: Frederick Ungar Publishing Co.; 1969.-xv+352 p.

[4] Kholkin A.M, Rofe-Beketov F.S. On spectrum of differential operator with block-triangular matrix coefficients. Journal Mathematical Physics, Analysis, Geometry. 2014; **10**(1): 44–63.

[5] Marchenko V.A. Sturm-Liouville Operators and its applications. Basel: Birkhauser Verlag; 1986.-xii+367 p.

[6] Lyantse V.E. On non-self-adjoint second-order differential operators on the semi-axis*.* Doklady Akademii Nauk SSSR. 1964; **154**(5): 1030–1033.

[7] Rofe-Beketov F.S. Expansion in eigenfunctions of infinite systems of differential equations in the non-selfadjoint and self- adjoint cases. Mat. Sb. 1960; **51**(3): 293–342.

[8] Schwartz J.T. Some nonselfadjoint operators. Comm. for pure and appl. Math. 1960; **XIII**: 609–639.

[9] Bondarenko E.I., Rofe-Beketov F.S. Phase equivalent matrix potential*.* Telecommun And Radio Eng. 2000; **56** (8 and 9): 4–29.

[10] Rofe-Beketov F.S., Zubkova E.I. Inverse scattering problem on the axis for the triangular 2 2 matrix potential a system with or without a virtual level. Azerbaijan Journal of Mathematic. 2011; **1**(2): 3–69.

[11] Kholkin A.M. The fundamental solutions of differential operator with block-triangular operator coefficients. J. of Advances in Mathematics. 2015; **10** (6): 3555–3561.

[12] Kholkin A.M. Resolvent for Nonself-Adjoint Differential Operator with Block-Triangular Operator Potential. Abstract and Applied Analysis. 2016; **2016**: 1–6.

[13] Kholkin A.M. Asymptotics of the fundamental system solutions of differential equation with blocktriangular operator potential. Far East J. of Mathematical Sciences. 2017; **102** (10): 2359–2366.

[14] Kholkin A.M. Spectral singularities of differential operators with triangular matrix coefficients. Methods of Functional Analysis and Topology. 2013; **19**(3): 260–267.

[15] Titchmarsh EC. Eigenfunction expansions associated with second – order differential equations. Vol. 2: Oxford: Clarendon Press; 1958.-xi+404 p.

[16] Naimark M.A. Investigation of spectrum and expansion in eigenfunctions of nonselfadjoint differential operators of second - order on the semiaxis. Tr. Mosk. Mat. Obs. 1954; **3**: 181–270.

[17] Schwartz J. T. Some nonselfadjoint operators. Comm. Pure and Appl. Math. 1960; **13**: 609–639.

**Chapter 4**

**Abstract**

Deformed Sine-Gordon Models,

Solitons and Anomalous Charges

We study certain deformations of the integrable sine-Gordon model (DSG). It is found analytically and numerically several towers of infinite number of anomalous charges for soliton solutions possessing a special space–time symmetry. Moreover, it is uncovered exact conserved charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kinkantikink (even parity) scatterings with equal and opposite velocities. Moreover, we provide a linear formulation of the modified SG model and a related tower of infinite number of exact non-local conservation laws. We back up our results with extensive numerical simulations for kink-kink, kink-antikink and breather config-

*q*2 tan 2 *w*

2 1 � sin *w*

 *q* 2

2

,ð Þ *q*∈ *R* ,

*Harold Blas, Hector F. Callisaya, João P.R. Campos,*

*Bibiano M. Cerna and Carlos Reyes*

urations of the Bazeia *et al.* potential *Vq*ð Þ¼ *w* 64

charges, non-local charges

**1. Introduction**

**65**

which contains the usual SG potential *V*2ð Þ¼ *w* 2 1½ � � cos 2ð Þ *w* .

**Keywords:** quasi-integrability, solitons, deformed sine-Gordon, anomalous

deformations of integrable models such as the sine-Gordon (SG), nonlinear Schrödinger (NLS), Korteweg-de Vries (KdV) and Toda models have been introduced, such that they exhibit soliton-type solutions with some properties resembling to their counterparts of the truly integrable ones. In this context the so-called quasi-integrability concept has been put forward [4]. These properties have been examined in the frameworks of the anomalous zero-curvature [4–7] and the

Riccati-type pseudo-potential approaches [8–10], respectively.

Solitons can be regarded as isolated waves that travel without loss of energy. The solitons emerge with their velocities and shapes completely unchanged after collision to each other, the only outcome being their phase shifts. The soliton solution is the main feature of the integrable models [1–3]. However, certain non-linear models in physics, with solitary wave solutions, are not integrable. Recently, certain

The main developments have been focused on the construction of infinite number of quasi-conservation laws which give rise to asymptotically conserved charges, i.e. conserved charges, such that their values vary during the scattering of the solitons only. The main observation in the both approaches to quasi-integrability is that, in general, the conserved charges of the standard integrable systems turn out to be the so-called asymptotically conserved charges in the deformed models. In fact, the exact conservation laws of the usual integrable systems become quasi-conservation laws of

#### **Chapter 4**

## Deformed Sine-Gordon Models, Solitons and Anomalous Charges

*Harold Blas, Hector F. Callisaya, João P.R. Campos, Bibiano M. Cerna and Carlos Reyes*

#### **Abstract**

We study certain deformations of the integrable sine-Gordon model (DSG). It is found analytically and numerically several towers of infinite number of anomalous charges for soliton solutions possessing a special space–time symmetry. Moreover, it is uncovered exact conserved charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kinkantikink (even parity) scatterings with equal and opposite velocities. Moreover, we provide a linear formulation of the modified SG model and a related tower of infinite number of exact non-local conservation laws. We back up our results with extensive numerical simulations for kink-kink, kink-antikink and breather configurations of the Bazeia *et al.* potential *Vq*ð Þ¼ *w* 64 *q*2 tan 2 *w* 2 1 � sin *w* 2 *q* 2 ,ð Þ *q*∈ *R* , which contains the usual SG potential *V*2ð Þ¼ *w* 2 1½ � � cos 2ð Þ *w* .

**Keywords:** quasi-integrability, solitons, deformed sine-Gordon, anomalous charges, non-local charges

#### **1. Introduction**

Solitons can be regarded as isolated waves that travel without loss of energy. The solitons emerge with their velocities and shapes completely unchanged after collision to each other, the only outcome being their phase shifts. The soliton solution is the main feature of the integrable models [1–3]. However, certain non-linear models in physics, with solitary wave solutions, are not integrable. Recently, certain deformations of integrable models such as the sine-Gordon (SG), nonlinear Schrödinger (NLS), Korteweg-de Vries (KdV) and Toda models have been introduced, such that they exhibit soliton-type solutions with some properties resembling to their counterparts of the truly integrable ones. In this context the so-called quasi-integrability concept has been put forward [4]. These properties have been examined in the frameworks of the anomalous zero-curvature [4–7] and the Riccati-type pseudo-potential approaches [8–10], respectively.

The main developments have been focused on the construction of infinite number of quasi-conservation laws which give rise to asymptotically conserved charges, i.e. conserved charges, such that their values vary during the scattering of the solitons only. The main observation in the both approaches to quasi-integrability is that, in general, the conserved charges of the standard integrable systems turn out to be the so-called asymptotically conserved charges in the deformed models. In fact, the exact conservation laws of the usual integrable systems become quasi-conservation laws of

the deformed integrable models. The non-homogeneous terms of the quasiconservation laws are dubbed as anomalies such that they vanish when integrated on the space–time plane, provided that the fields satisfy a special space–time symmetry.

*∂ξ∂η w* þ *V*ð Þ1 ð Þ¼ *w* 0, (1)

16 ½ � 1 � cos 4ð Þ *w :* (3)

*w* 1 � j j sin *w q* ½ �2

*dw V w*ð Þ.

, (2)

where *w* is a real scalar field *w*, *V w*ð Þ is the scalar potential and *V*ð Þ1 ð Þ� *w d*

The family of potentials *V w*ð Þ will represent certain deformations of the usual SG model. The theory (1) has been studied using the techniques of integrable field theories, such as the anomalous zero-curvature [4, 11] and deformed Riccati-type pseudopotential formulations [8], respectively. In our simulations we will consider [4, 24].

where *q* is a real parameter such that for *q* ¼ 2 the potential reduces to the SG

So, we introduce the deformation parameter *ε* as *q* ¼ 2 þ *ε*, such that in the limit

The quasi-integrability has been introduced for deformed sine-Gordon models

P�P*s*T *d*, P*s* : *x*~ ! �*x*~, T *d* : ~*t* ! �~~*t*, *x*~ � *x* � *x*Δ, ~*t* ¼ *t* � *t*Δ, (5)

defined around a given point ð Þ *x*Δ, *t*Δ . Moreover, let us consider the space-

Several towers of quasi-conservation laws, with anomaly terms possessing odd parities under (6)–(8), have been found [8, 11]. Next, we consider those quasi-conservation laws and examine their anomalies in view of the symmetries

and assume that the scalar field is an eigenstate of the operator *Px*

In addition, consider an even potential *V* under P*x*

P : *w* ! �*w* þ const*:*; *V w*ð Þ! *V w*ð Þ, (4)

P*x* : *x* \$ �*x*, (6)

P*x*ð Þ¼ *V V:* (8)

P*x* : *w* ! ϱ *w*, ϱ ¼ �1*:* (7)

such that the field *w* and the potential *V* satisfy the symmetry [4, 8, 11].

The model (1) possesses several towers of anomalous charges associated to quasi-conservation laws [4, 8, 11]. In [11] it has been introduced a subset of exactly conserved charges associated to space-reflection eigenstates as kink-antikink, kinkkink and breather configurations, respectively. New types of two sets of dual towers of asymptotically conserved charges have been uncovered [8]. Remarkably, even the usual sine-Gordon models possesses anomalous charges. So far, it is attributed to the space–time symmetry properties of the solitons. Those charges can be relevant in the study of soliton gases and formation of certain structures, such as soliton

2 *q*2 tan 2

*V w*ð Þ¼ , 2 1

*V w*ð Þ¼ , *q*

*Deformed Sine-Gordon Models, Solitons and Anomalous Charges*

*DOI: http://dx.doi.org/10.5772/intechopen.95432*

turbulence, soliton gas dynamics and rogue waves [16].

under the special space–time reflection

reflection transformation

(4)–(5) and (6)–(8), respectively.

**67**

*ε* ¼ 0 one reproduces the SG model.

potential

The properties of the soliton-like configurations in the quasi-integrable models are, so far, largely unknown. We summarize the main results. First, the one-soliton sectors exhibit infinite conserved charges. Second, the space–time integration of the anomalies vanish when one-soliton like solutions are located far away from each other. The anomalies are significant around the space–time regions of their interaction. Third, a sufficient condition for the vanishing of the space–time integrated anomalies is that the *N*�soliton possesses definite parity under a shifted parity and delayed time reversion (P*s*T *d*) symmetry. When the anomaly densities possess odd parities the space–time integration of them vanish, which imply the existence of anomalous charges. Fourth, the conserved charges of the usual integrable systems turn out to be the anomalous charges upon deformation. Fifth, there exist infinite towers of infinitely many anomalous charges, different in form from the ones of the usual integrable models. New towers of anomalous charges have been uncovered in [8–10]. Remarkably, even the usual integrable models possess quasi-conservation laws with anomalous charges for analytical *N*� soliton with *C*P*s*T *d* symmetry [9, 10]. For the standard SG theory it has been discussed for the 2-soliton sector of the theory [8]. Sixth, there is a subset of exact conserved charges for soliton eigenstates simply of the shifted space-reflection P*s*. The deformed NLS model for two-soliton solutions [6, 7] and the deformed sine-Gordon model [11] for two-kink and breather solutions exhibit this property.

In the context of the Riccati-type method there have been shown that the deformed SG, KdV and NLS models [8–10], respectively, possess linear system formulations and that they exhibit infinite towers of exact non-local conservation laws. The NLS-type, KdV-type and SG-type models share the same importance due to their potential applications, since they are ubiquitous in all areas of nonlinear physics, such as Bose-Einsten condensation and superconductivity [12–14], soliton gas and soliton turbulence in fluid dynamics [15–20], the Alice-Bob physics [21, 22] and the understanding of a kind of triality among the gauge theories, integrable models and gravity theories [23].

Here, we discuss the previous results in the field by utilizing a deformed sine-Gordon model. We will introduce the relationship between the space–time parity and asymptotically conserved charges. Next, we clarified on the space-reflection parity related to the linear combination of the dual sets of anomalous quantities. In addition, it is focused on the space-reflection symmetry of some two-soliton solutions of deformed sine-Gordon models. Then one proceeds to construct a tower of exactly conserved charges for each solution possessing a definite space-reflection parity. Lastly, by considering linear combinations of the anomalous conserved charges it is showed, through analytical and numerical methods, that there is a subset of exactly conserved charges.

A modified SG model and the space–time symmetries are presented in the next section. In Section 3, the towers of quasi-conservation laws are presented. In Section 4 our numerical simulations are described. The linear formulation and the non-local conservation laws are discussed in the Riccati-type pseudo-potential approach in Section 5. Finally, in Section 6 we present some conclusions.

#### **2. A deformation of the sine-Gordon model**

Let us consider the relativistic field theories in 1ð Þ þ 1 -dimensions with equation of motion1

1 In the *x* and *t* laboratory coordinates: *η* ¼ *t*þ*x* 2 , *ξ* ¼ *t*�*x* 2 , *∂η* ¼ *∂t* þ *∂x*, *∂ξ* ¼ *∂t* � *∂x*, *∂η∂ξ* ¼ *∂*2 *t* � *∂*2 *x*

*Deformed Sine-Gordon Models, Solitons and Anomalous Charges DOI: http://dx.doi.org/10.5772/intechopen.95432*

the deformed integrable models. The non-homogeneous terms of the quasi-

*Recent Developments in the Solution of Nonlinear Differential Equations*

model [11] for two-kink and breather solutions exhibit this property.

Section 5. Finally, in Section 6 we present some conclusions.

**2. A deformation of the sine-Gordon model**

1 In the *x* and *t* laboratory coordinates: *η* ¼ *t*þ*x*

of motion1

**66**

models and gravity theories [23].

In the context of the Riccati-type method there have been shown that the deformed SG, KdV and NLS models [8–10], respectively, possess linear system formulations and that they exhibit infinite towers of exact non-local conservation laws. The NLS-type, KdV-type and SG-type models share the same importance due to their potential applications, since they are ubiquitous in all areas of nonlinear physics, such as Bose-Einsten condensation and superconductivity [12–14], soliton gas and soliton turbulence in fluid dynamics [15–20], the Alice-Bob physics [21, 22] and the understanding of a kind of triality among the gauge theories, integrable

Here, we discuss the previous results in the field by utilizing a deformed sine-Gordon model. We will introduce the relationship between the space–time parity and asymptotically conserved charges. Next, we clarified on the space-reflection parity related to the linear combination of the dual sets of anomalous quantities. In addition, it is focused on the space-reflection symmetry of some two-soliton solutions of deformed sine-Gordon models. Then one proceeds to construct a tower of exactly conserved charges for each solution possessing a definite space-reflection parity. Lastly, by considering linear combinations of the anomalous conserved charges it is showed, through analytical and numerical methods, that there is a subset of exactly conserved charges. A modified SG model and the space–time symmetries are presented in the next section. In Section 3, the towers of quasi-conservation laws are presented. In Section 4 our numerical simulations are described. The linear formulation and the non-local conservation laws are discussed in the Riccati-type pseudo-potential approach in

Let us consider the relativistic field theories in 1ð Þ þ 1 -dimensions with equation

2 , *∂η* ¼ *∂t* þ *∂x*, *∂ξ* ¼ *∂t* � *∂x*, *∂η∂ξ* ¼ *∂*2

*t* � *∂*2 *x*

2 , *ξ* ¼ *t*�*x*

conservation laws are dubbed as anomalies such that they vanish when integrated on the space–time plane, provided that the fields satisfy a special space–time symmetry. The properties of the soliton-like configurations in the quasi-integrable models are, so far, largely unknown. We summarize the main results. First, the one-soliton sectors exhibit infinite conserved charges. Second, the space–time integration of the anomalies vanish when one-soliton like solutions are located far away from each other. The anomalies are significant around the space–time regions of their interaction. Third, a sufficient condition for the vanishing of the space–time integrated anomalies is that the *N*�soliton possesses definite parity under a shifted parity and delayed time reversion (P*s*T *d*) symmetry. When the anomaly densities possess odd parities the space–time integration of them vanish, which imply the existence of anomalous charges. Fourth, the conserved charges of the usual integrable systems turn out to be the anomalous charges upon deformation. Fifth, there exist infinite towers of infinitely many anomalous charges, different in form from the ones of the usual integrable models. New towers of anomalous charges have been uncovered in [8–10]. Remarkably, even the usual integrable models possess quasi-conservation laws with anomalous charges for analytical *N*� soliton with *C*P*s*T *d* symmetry [9, 10]. For the standard SG theory it has been discussed for the 2-soliton sector of the theory [8]. Sixth, there is a subset of exact conserved charges for soliton eigenstates simply of the shifted space-reflection P*s*. The deformed NLS model for two-soliton solutions [6, 7] and the deformed sine-Gordon

$$
\partial\_{\xi} \partial\_{\eta} w + V^{(1)}(w) = 0,\tag{1}
$$

where *w* is a real scalar field *w*, *V w*ð Þ is the scalar potential and *V*ð Þ1 ð Þ� *w d dw V w*ð Þ. The family of potentials *V w*ð Þ will represent certain deformations of the usual SG model. The theory (1) has been studied using the techniques of integrable field theories, such as the anomalous zero-curvature [4, 11] and deformed Riccati-type pseudopotential formulations [8], respectively. In our simulations we will consider [4, 24].

$$V(w,q) = \frac{2}{q^2} \tan^2 w \left[1 - |\sin w|^q\right]^2,\tag{2}$$

where *q* is a real parameter such that for *q* ¼ 2 the potential reduces to the SG potential

$$V(w,2) = \frac{1}{16}[1 - \cos(4w)].\tag{3}$$

So, we introduce the deformation parameter *ε* as *q* ¼ 2 þ *ε*, such that in the limit *ε* ¼ 0 one reproduces the SG model.

The model (1) possesses several towers of anomalous charges associated to quasi-conservation laws [4, 8, 11]. In [11] it has been introduced a subset of exactly conserved charges associated to space-reflection eigenstates as kink-antikink, kinkkink and breather configurations, respectively. New types of two sets of dual towers of asymptotically conserved charges have been uncovered [8]. Remarkably, even the usual sine-Gordon models possesses anomalous charges. So far, it is attributed to the space–time symmetry properties of the solitons. Those charges can be relevant in the study of soliton gases and formation of certain structures, such as soliton turbulence, soliton gas dynamics and rogue waves [16].

The quasi-integrability has been introduced for deformed sine-Gordon models such that the field *w* and the potential *V* satisfy the symmetry [4, 8, 11].

$$\mathcal{P}: w \to -w + \text{const}; \ V(w) \to V(w), \tag{4}$$

under the special space–time reflection

$$\mathcal{P} \equiv \mathcal{P}\_{\mathfrak{s}} \mathcal{T}\_{d}, \; \mathcal{P}\_{\mathfrak{s}} : \tilde{\mathfrak{x}} \to -\tilde{\mathfrak{x}}, \; \mathcal{T}\_{d} : \tilde{t} \to -\tilde{\tilde{t}}, \; \tilde{\mathfrak{x}} \equiv \mathfrak{x} - \mathfrak{x}\_{\Delta}, \; \tilde{t} = t - t\_{\Delta}, \tag{5}$$

defined around a given point ð Þ *x*Δ, *t*Δ . Moreover, let us consider the spacereflection transformation

$$\mathcal{P}\_{\mathfrak{x}} : \mathfrak{x} \leftrightarrow -\mathfrak{x}, \tag{6}$$

and assume that the scalar field is an eigenstate of the operator *Px*

$$\mathcal{P}\_{\mathfrak{x}} : w \to \mathfrak{g} \, w, \; \mathfrak{g} = \pm \mathbf{1}. \tag{7}$$

In addition, consider an even potential *V* under P*x*

$$\mathcal{P}\_x(V) = V.\tag{8}$$

Several towers of quasi-conservation laws, with anomaly terms possessing odd parities under (6)–(8), have been found [8, 11]. Next, we consider those quasi-conservation laws and examine their anomalies in view of the symmetries (4)–(5) and (6)–(8), respectively.

#### **3. Quasi-conservation laws of the deformed SG model**

We will discuss some of the infinite towers of quasi-conservation laws of the deformed SG model (1).

#### **3.1 First type of towers: The SG-type quasi-conservation laws**

The usual SG charges turn out to be the anomalous charges of the DSG. So, one has the infinite set of quasi-conservation laws [4, 11].

$$\frac{d}{dt}q\_{\mu}^{(2n+1)} = \int d\mathbf{x} \,\beta^{(2n+1)}, \ \mathbf{n} = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots \tag{9}$$

with the charges *q*

where *E* ¼ *q*

*β* ð Þ3 � ¼ �

1

*β*ð Þ5 � ¼ �

*β*ð Þ 2*n*þ1

**69**

ð Þ1

1 2

therefore, the anomalies vanish *β*ð Þ3

2 *Z* 24ð Þ *∂ξw*

The properties of the quantities *q*

of motion (1) is used to substitute *∂*2

*f* ð Þ3

þ ð Þ� *x*, *t V*00

function, and thus the *x*�integrated anomaly *α*ð Þ3

where we have defined the anomaly density *f*

Following analogous procedure as above one has

*α*ð Þ3 � ¼ �4

surface terms one has

ð Þ 2*n*þ1

*DOI: http://dx.doi.org/10.5772/intechopen.95432*

*q* ð Þ 2*n*þ1 *a*,� � ∓

*Deformed Sine-Gordon Models, Solitons and Anomalous Charges*

*β* ð Þ 2*n*þ1 *a*,� � ∓

order *n* ¼ 0, and the linear combinations of the charges *q*ð Þ1

ðþ∞ �∞

þ is the energy and *P* ¼ *q*ð Þ1

The first non-trivial anomalies become [11].

*Z ∂ξ* ð Þ *∂ξw*

2 *∂*2 *ξw* þ *∂*4 *ξw* � �*∂ξw* � 24 *∂ηw* � �2

*dx* 1 2 ð Þ *∂tw* 2 þ 1 2 ð Þ *∂xw*

> ðþ∞ �∞

Notice that for the SG potential (3) the factor *Z* above vanishes identically;

symmetry properties of the solitons, in particular on the space-reflection symmetry of

� , as we will see below. So, let us examine the space-reflection symmetry of them. Let us write the anomalies in terms of the *∂x* and *∂t* derivatives. So, once the eq.

*t w* ! *∂*2

ð *dx f*ð Þ3

ð *dx f* ð Þ3

þ 16*V* � � *∂x* ð Þ *∂tw*

potentials (8) and for definite parity (even or odd) fields *w* the density *f*

the exactly conserved charges of the standard SG model at this order.

*α*ð Þ3 þ � �2

ð Þ 2*n*þ1 � and Ð

*q*ð Þ1 � ¼

and *β*~ð Þ 2*n*þ1 in (10) have been used, respectively.

energy and momentum, respectively [11].

*q* ð Þ1 þ ¼

*a*,� and anomalies *β*ð Þ 2*n*þ1

1 16 *q*ð Þ 2*n*þ1

1

in which the quantities *q*ð Þ 2*n*þ1 and *β*ð Þ 2*n*þ1 defined in (9) and the quantities ~*q*ð Þ 2*n*þ1

Since the theory (1) is invariant under space–time translations one has that the energy momentum tensor is conserved. In fact, one has *β*ð Þ1 ¼ *β*~ð Þ1 ¼ 0 at the zero'th

� , respectively, defined as

� �, (12)

16 *β*ð Þ 2*n*þ1 � *β*~ð Þ 2*n*þ1 � � (13)

*a* and ~*q*ð Þ1

*dx ∂xw∂tw*, (15)

2 þ *V* � �, (14)

> *∂*2 *ηw* þ *∂*4 *ηw*

� �*∂ηw*

ð Þ *w* � �, as well as, neglecting

2 n o h i , (19)

þ ð Þ *x*, *t* , (18)

þ . Notice that for even parity

� ð Þ *x*, *t* , (20)

ð Þ3

þ is an odd

ð Þ3

� in (11) will depend on the

� turn out to be

� is the momentum.

2 h i∓*∂η ∂ηw* � �2 n o h i , *Z* � *V*ð Þ2 þ 16*V* � 1*:* (16)

h i*:* (17)

� ¼ 0, and the relevant charges *q*

*dxβ* ð Þ 2*n*þ1

*xw* � *V*0

2 h i þ *∂x* ð Þ *∂xw*

þ vanishes.

ð Þ3

*a*

*a* leads to the

*a* � ~*q*ð Þ 2*n*þ1 *a*

where the quantities *q*ð Þ 2*n*þ1 *a* define the anomalous charges, provided that the time-integrated anomalies Ð *dt*Ð *dxβ*ð Þ 2*n*þ1 vanish for solitons satisfying (4) and (5). This condition, when combined with Eq. (9), implies *q*ð Þ 2*n*þ1 *a* ð Þ¼ *t* ! þ∞ *q*ð Þ 2*n*þ1 *a* ð Þ *t* ! �∞ . So, we have that *q*ð Þ 2*n*þ1 *a* are anomalous for *n* ¼ 1, 2, 3, *:* … . The charges *q*ð Þ 2*n*þ1 *a* maintain the same form as the ones of the usual SG.

In 1ð Þ þ 1 -dimensional Lorentz invariant integrable field theories one has dual integrability conditions or Lax equations. Analogously, for the deformations of the SG model there exist a dual formulation for each equation as in (9) by interchanging *ξ* \$ *η* in the procedure to obtain the relevant quasi-conservation laws. So, one can get

$$\frac{d}{dt}\tilde{q}\_{d}^{(2n+1)} = \int d\mathbf{x} \,\tilde{\boldsymbol{\beta}}^{(2n+1)}, \; n = 1, 2, 3, \; \dots \tag{10}$$

where the quantities ~*q*ð Þ 2*n*þ1 *a* define the dual asymptotically conserved charges, provided that the time-integrated anomalies Ð *dt*Ð *dxβ*~ð Þ 2*n*þ1 vanish. Likewise, this result implies ~*q*ð Þ 2*n*þ1 *a* ð Þ¼ *t* ! þ∞ ~*q*ð Þ 2*n*þ1 *a* ð Þ *t* ! �∞ .

These towers of quasi-conservation laws reproduce the same polynomial form as in the usual sine-Gordon charge densities. In fact, the anomalies *β*ð Þ 2*n*þ1 and *β*~ð Þ 2*n*þ1 vanish identically provided that the deformed potential *V w*ð Þ recovers the form of the standard SG potential.

The importance and the relevance of such a dual construction will become clear below when the linear combinations of the charges in (9) and (10) give rise to infinite towers of exactly conserved charges, provided that the space-integral of the linear combination of the anomaly densities *β*ð Þ 2*n*þ1 and ~*β* ð Þ 2*n*þ1 vanish for special two-soliton solutions.

#### *3.1.1 Space-reflection parity and conserved charges*

The above dual sets of quasi-conservation laws are used to construct a sequence of conserved charges and vanishing anomalies. The space-reflection symmetry of some soliton solutions of the deformed SG model will imply the existence of an infinite tower of conserved charges. So, let us examine a linear combination, at each order *n* ¼ 1, 2, … , of the above two sets of quasi-conserved charges *q*ð Þ 2*n*þ1 *a* (9) and ~*q*ð Þ 2*n*þ1 *a* (10). Consider the new quasi-conservation laws

$$\frac{d}{dt}q\_{a,\pm}^{(2n+1)} = -\int d\mathbf{x} \,\beta\_{\pm}^{(2n+1)},\ \mathbf{n} = \mathbf{1}, \mathbf{2}, \dots,\tag{11}$$

*Deformed Sine-Gordon Models, Solitons and Anomalous Charges DOI: http://dx.doi.org/10.5772/intechopen.95432*

**3. Quasi-conservation laws of the deformed SG model**

*Recent Developments in the Solution of Nonlinear Differential Equations*

**3.1 First type of towers: The SG-type quasi-conservation laws**

ð

This condition, when combined with Eq. (9), implies *q*ð Þ 2*n*þ1 *a* ð Þ¼ *t* ! þ∞ *q*ð Þ 2*n*þ1 *a* ð Þ *t* ! �∞ . So, we have that *q*ð Þ 2*n*þ1 *a* are anomalous for *n* ¼ 1, 2, 3, *:* … . The

charges *q*ð Þ 2*n*þ1 *a* maintain the same form as the ones of the usual SG.

ð

has the infinite set of quasi-conservation laws [4, 11].

*dt*Ð

*d dt q*ð Þ 2*n*þ1 *a* ¼

*d dt* ~*q*ð Þ 2*n*þ1 *a* ¼

provided that the time-integrated anomalies Ð

*a* ð Þ¼ *t* ! þ∞ ~*q*ð Þ 2*n*þ1

linear combination of the anomaly densities *β*ð Þ 2*n*þ1 and ~*β*

*a* (10). Consider the new quasi-conservation laws

ð *dxβ* ð Þ 2*n*þ1

*3.1.1 Space-reflection parity and conserved charges*

*d dt q* ð Þ 2*n*þ1 *a*,� ¼ �

where the quantities ~*q*ð Þ 2*n*þ1

result implies ~*q*ð Þ 2*n*þ1

the standard SG potential.

two-soliton solutions.

~*q*ð Þ 2*n*þ1

**68**

deformed SG model (1).

time-integrated anomalies Ð

We will discuss some of the infinite towers of quasi-conservation laws of the

The usual SG charges turn out to be the anomalous charges of the DSG. So, one

where the quantities *q*ð Þ 2*n*þ1 *a* define the anomalous charges, provided that the

In 1ð Þ þ 1 -dimensional Lorentz invariant integrable field theories one has dual integrability conditions or Lax equations. Analogously, for the deformations of the SG model there exist a dual formulation for each equation as in (9) by interchanging *ξ* \$ *η* in the procedure to obtain the relevant quasi-conservation laws. So, one can get

*dxβ*~ð Þ 2*n*þ1

*a* ð Þ *t* ! �∞ . These towers of quasi-conservation laws reproduce the same polynomial form as in the usual sine-Gordon charge densities. In fact, the anomalies *β*ð Þ 2*n*þ1 and *β*~ð Þ 2*n*þ1 vanish identically provided that the deformed potential *V w*ð Þ recovers the form of

The importance and the relevance of such a dual construction will become clear

The above dual sets of quasi-conservation laws are used to construct a sequence of conserved charges and vanishing anomalies. The space-reflection symmetry of some soliton solutions of the deformed SG model will imply the existence of an infinite tower of conserved charges. So, let us examine a linear combination, at each order *n* ¼ 1, 2, … , of the above two sets of quasi-conserved charges *q*ð Þ 2*n*þ1 *a* (9) and

below when the linear combinations of the charges in (9) and (10) give rise to infinite towers of exactly conserved charges, provided that the space-integral of the

*dxβ*ð Þ 2*n*þ1 , *n* ¼ 1, 2, 3, … (9)

*dxβ*ð Þ 2*n*þ1 vanish for solitons satisfying (4) and (5).

, *n* ¼ 1, 2, 3, … (10)

*dxβ*~ð Þ 2*n*þ1 vanish. Likewise, this

ð Þ 2*n*þ1 vanish for special

� , *n* ¼ 1, 2, … , (11)

*a* define the dual asymptotically conserved charges,

*dt*Ð

with the charges *q* ð Þ 2*n*þ1 *a*,� and anomalies *β* ð Þ 2*n*þ1 � , respectively, defined as

$$q\_{a, \pm}^{(2n+1)} \equiv \mp \frac{1}{16} \left( q\_a^{(2n+1)} \pm \ddot{q}\_a^{(2n+1)} \right),\tag{12}$$

$$\boldsymbol{\beta}\_{a,\pm}^{(2n+1)} \equiv \mp \frac{1}{\mathbf{1}\mathbf{6}} \left( \boldsymbol{\beta}^{(2n+1)} \pm \boldsymbol{\tilde{\rho}}^{(2n+1)} \right) \tag{13}$$

in which the quantities *q*ð Þ 2*n*þ1 and *β*ð Þ 2*n*þ1 defined in (9) and the quantities ~*q*ð Þ 2*n*þ1 *a* and *β*~ð Þ 2*n*þ1 in (10) have been used, respectively.

Since the theory (1) is invariant under space–time translations one has that the energy momentum tensor is conserved. In fact, one has *β*ð Þ1 ¼ *β*~ð Þ1 ¼ 0 at the zero'th order *n* ¼ 0, and the linear combinations of the charges *q*ð Þ1 *a* and ~*q*ð Þ1 *a* leads to the energy and momentum, respectively [11].

$$q\_{+}^{(1)} = \int\_{-\infty}^{+\infty} d\mathbf{x} \left[\frac{1}{2} \left(\partial\_{l}\mathbf{w}\right)^{2} + \frac{1}{2} \left(\partial\_{\mathbf{x}}\mathbf{w}\right)^{2} + V\right],\tag{14}$$

$$q\_{-}^{(1)} = \int\_{-\infty}^{+\infty} d\mathbf{x} \,\partial\_{\mathbf{x}} w \partial\_{t} w,\tag{15}$$

where *E* ¼ *q* ð Þ1 þ is the energy and *P* ¼ *q*ð Þ1 � is the momentum. The first non-trivial anomalies become [11].

$$\mathcal{J}\_{\pm}^{(3)} = \pm \frac{1}{2} Z \left\{ \partial\_{\xi} \left[ \left( \partial\_{\xi} w \right)^{2} \right] \mp \partial\_{\eta} \left[ \left( \partial\_{\eta} w \right)^{2} \right] \right\}, \quad Z \equiv V^{(2)} + 1 \mathsf{6} V - 1. \tag{16}$$

$$\mathcal{J}\_{\pm}^{(5)} = \pm \frac{1}{2} Z \left[ \left( 24 (\partial\_{\xi} w)^{2} \partial\_{\xi}^{2} w + \partial\_{\xi}^{4} w \right) \partial\_{\xi} w \pm \left( 24 (\partial\_{\eta} w)^{2} \partial\_{\eta}^{2} w + \partial\_{\eta}^{4} w \right) \partial\_{\eta} w \right]. \tag{17}$$

Notice that for the SG potential (3) the factor *Z* above vanishes identically; therefore, the anomalies vanish *β*ð Þ3 � ¼ 0, and the relevant charges *q* ð Þ3 � turn out to be the exactly conserved charges of the standard SG model at this order.

The properties of the quantities *q* ð Þ 2*n*þ1 � and Ð *dxβ* ð Þ 2*n*þ1 � in (11) will depend on the symmetry properties of the solitons, in particular on the space-reflection symmetry of *β*ð Þ 2*n*þ1 � , as we will see below. So, let us examine the space-reflection symmetry of them.

Let us write the anomalies in terms of the *∂x* and *∂t* derivatives. So, once the eq. of motion (1) is used to substitute *∂*2 *t w* ! *∂*2 *xw* � *V*0 ð Þ *w* � �, as well as, neglecting surface terms one has

$$a\_+^{(3)} \equiv -2 \int d\mathbf{x} f\_+^{(3)}(\mathbf{x}, t),\tag{18}$$

$$f\_{+}^{(3)}(\mathbf{x},t) \equiv \left[V^{\prime} + \mathbf{1}\mathfrak{G}V\right] \left\{\partial\_{\mathbf{x}}\left[\left(\partial\_{t}w\right)^{2}\right] + \partial\_{\mathbf{x}}\left[\left(\partial\_{\mathbf{x}}w\right)^{2}\right]\right\},\tag{19}$$

where we have defined the anomaly density *f* ð Þ3 þ . Notice that for even parity potentials (8) and for definite parity (even or odd) fields *w* the density *f* ð Þ3 þ is an odd function, and thus the *x*�integrated anomaly *α*ð Þ3 þ vanishes.

Following analogous procedure as above one has

$$a\_{-}^{(3)} = -4 \int d\mathbf{x} f\_{-}^{(3)}(\mathbf{x}, t),\tag{20}$$

$$f\_{-}^{(3)}(\mathbf{x},t) \equiv \left[V^{\prime\prime} + 16V\right] \left\{\partial\_{l}w\partial\_{\mathbf{x}}^{2}w + \partial\_{\mathbf{x}}w\partial\_{\mathbf{x}}\partial\_{l}w\right\},\tag{21}$$

*γ*ð Þ *N* �

*Deformed Sine-Gordon Models, Solitons and Anomalous Charges*

*DOI: http://dx.doi.org/10.5772/intechopen.95432*

*Q*~ ð Þ *N a* � ð *dx* 1 2 *VN*�1

~*γ*ð Þ *N* �

towers will be simulated below for 2-soliton interactions.

ð *dx* 1

corresponding anomalies *γ*ð Þ *N* .

So, one has

anomalies ~*γ*ð Þ *N* .

and its deformations [9].

**4. Numerical simulations**

these field configurations.

**71**

*Mn*ðÞ¼ *t* Ð þ∞

ð *dx* 1 2 ð Þ *∂ξw* 2 *∂ηVN*�1

where we have introduced the asymptotically conserved charges *Q*^ ð Þ *N*

*d dt Q*~ ð Þ *N*

2 *∂ηw* � �2

where we have defined the dual asymptotically conserved charges *Q*~ ð Þ *N*

Similarly, the densities of the anomalies *γ*ð Þ *N* and ~*γ*ð Þ *N* in (30) and (33), respectively, possess odd parities under (4) and (5), so the quasi-conservation laws (28) and (31), respectively, allow the construction of asymptotically conserved charges. The relevant anomalies of the lowest order quasi-conservation laws of the above

Remarkably, the above charges turn out to be anomalous even for the standard sine-Gordon model. In fact, the relevant 2-soliton solutions have been constructed analytically [4, 11] which possess a definite parity under (4)–(5), such that the odd anomaly densities vanish upon space–time integration. The usual explanation for the appearance of novel anomalous charges in the standard sine-Gordon model is the symmetry argument. The anomalous charges also appear in the standard KdV

These charges have been computed for soliton collisions in the treatment of soliton gases and formation of some structures in integrable systems, such as integrable turbulence and rogue waves. In the context of the usual KdV model it has been analyzed the behavior of the statistical moments defined by (see e.g. [16, 17])

charges. It is remarkable that the moments, *M*3,4, respectively, in the interaction region of two-solitons, behave as the anomalous charges of the quasi-integrable KdV models [9]. In fact, in the quasi-integrable KdV models the moments *M*2,3 are in fact anomalous charges [9]. So, since the two-soliton collision is an important ingredient in the formation of soliton turbulence and the dynamics of soliton gases, we can expect they will be important in the quasi-integrable counterparts. In the case of the SG soliton ensemble, to our knowledge, it is needed a further theoretical research.

Here we will check numerically the lowest order expressions of the various towers of quasi-conservation laws presented above. For this purpose we will numerically solve the Eq. (1) with the particular deformed potential (2). In the **Figures 1** and **2** we plot the kink-kink and kink-antikink collisions, respectively. Moreover, we show the first conserved charges, i.e. the energy and momentum for

�∞ *vn dx*, *n*≥1; where *v* is the KdV field. The *M*1,2 cases are conserved

The interchange *η* \$ *ξ* allows us to reproduce the dual quasi-conservation laws.

ð Þ *∂ξw* 2 þ 1 *N VN* � �, (32)

*∂ξVN*�1

, *N* ≥2, (30)

, *N* ≥2, (33)

*a* ¼ ~*γ*ð Þ *N* , (31)

*a* and the

*a* and the

where we have defined the anomaly density *f* ð Þ3 � . Notice that for even parity potentials (8) and for definite parity (even or odd) fields *w* the density *f* ð Þ3 � is an even function, and thus the *x*�integrated anomaly *α*ð Þ3 � will not vanish solely by a space-reflection parity reason.

The anomalies *α*ð Þ3 � and Ð *dtα*ð Þ3 � in (18) and (20), will be computed numerically for two-solitons and breather-like solutions below.

By direct construction it has been found new towers of anomalous charges in [8]. In the next subsections we will discuss those charges and anomalies in relation to the symmetry (4) and (5).

#### **3.2 Second type of towers**

The quasi-conservation laws [8].

$$\frac{d}{dt}\mathbf{Q}\_{\mathbf{a}}^{(N)} = \mathbf{a}^{(N)},\tag{22}$$

$$Q\_a^{(N)} \equiv \int d\mathbf{x} \left[ \frac{\mathbf{1}}{N} (\partial\_\xi w)^N + V (\partial\_\xi w)^{N-2} \right],\tag{23}$$

$$\mathbf{a}^{(N)} \equiv \int d\mathbf{x} \, (N-2) (\partial\_{\xi} w)^{N-3} \partial\_{\xi}^2 w \, V, \ \ N \ge 3,\tag{24}$$

define the asymptotically conserved charges *Q*ð Þ *N a* and the corresponding anomalies að Þ *N* .

The dual quasi-conservation laws become

$$\frac{d}{dt}\tilde{\mathbf{Q}}\_{\mathfrak{a}}^{(N)} = \tilde{\mathbf{a}}^{(N)},\tag{25}$$

$$\tilde{\mathbf{Q}}\_a^{(N)} \equiv \int d\mathbf{x} \left[ \frac{\mathbf{1}}{N} \left( \partial\_\eta \boldsymbol{w} \right)^N + \mathbf{V} \left( \partial\_\eta \boldsymbol{w} \right)^{N-2} \right],\tag{26}$$

$$\tilde{\mathbf{a}}^{(N)} \equiv \int d\mathbf{x} \, (N-2) \left(\partial\_{\eta}\boldsymbol{w}\right)^{N-3} \partial\_{\eta}^2 \boldsymbol{w} \boldsymbol{V}, \ \ N \ge 3,\tag{27}$$

where we have introduced the dual asymptotically conserved charges *Q*~ ð Þ *N a* and the relevant anomalies ~að Þ *N* .

The densities of the anomalies að Þ *N* and ~að Þ *N* in (24) and (27), respectively, possess odd parities under (4) and (5), so the quasi-conservation laws (22) and (25), respectively, allow the construction of asymptotically conserved charges.

#### **3.3 Third type of towers**

Let us define the quasi-conservation laws [8].

$$\frac{d}{dt}\mathcal{Q}\_a^{(N)} = \mathcal{\eta}^{(N)},\tag{28}$$

$$Q\_a^{(N)} \equiv \int d\mathbf{x} \left[ \frac{\mathbf{1}}{2} V^{N-1} (\partial\_{\xi} w)^2 + \frac{\mathbf{1}}{N} V^N \right],\tag{29}$$

*Deformed Sine-Gordon Models, Solitons and Anomalous Charges DOI: http://dx.doi.org/10.5772/intechopen.95432*

*f* ð Þ3

space-reflection parity reason. The anomalies *α*ð Þ3

to the symmetry (4) and (5).

**3.2 Second type of towers**

anomalies að Þ *N* .

The quasi-conservation laws [8].

� ð Þ� *x*, *t V*00

where we have defined the anomaly density *f*

even function, and thus the *x*�integrated anomaly *α*ð Þ3

*dtα*ð Þ3

*Recent Developments in the Solution of Nonlinear Differential Equations*

� and Ð

for two-solitons and breather-like solutions below.

*Q*ð Þ *N a* � ð *dx* 1

ð

define the asymptotically conserved charges *Q*ð Þ *N*

að Þ *N* �

The dual quasi-conservation laws become

*Q*~ ð Þ *N a* � ð *dx* 1

ð

Let us define the quasi-conservation laws [8].

*Q*ð Þ *N a* � ð *dx* 1 2 *VN*�1

~að Þ *N* �

the relevant anomalies ~að Þ *N* .

**3.3 Third type of towers**

**70**

þ 16*V* � � *∂tw∂*2

By direct construction it has been found new towers of anomalous charges in [8]. In the next subsections we will discuss those charges and anomalies in relation

potentials (8) and for definite parity (even or odd) fields *w* the density *f*

*d dt Q*ð Þ *N*

*N* ð Þ *∂ξw*

*dx N*ð Þ � 2 ð Þ *∂ξw*

*d dt Q*~ ð Þ *N*

*dx N*ð Þ � 2 *∂ηw* � �*N*�3

where we have introduced the dual asymptotically conserved charges *Q*~ ð Þ *N*

The densities of the anomalies að Þ *N* and ~að Þ *N* in (24) and (27), respectively, possess odd parities under (4) and (5), so the quasi-conservation laws (22) and (25),

respectively, allow the construction of asymptotically conserved charges.

*d dt Q*ð Þ *N* *N* þ *V*ð Þ *∂ξw*

� �

*N*�3 *∂*2

*N ∂ηw* � �*N* þ *V ∂ηw* � �*N*�2 � �

> ð Þ *∂ξw* 2 þ 1 *N VN*

� �

*∂*2

*xw* þ *∂xw∂x∂tw* � �, (21)

� in (18) and (20), will be computed numerically

*a* ¼ að Þ *N* , (22)

, (23)

, (26)

, (29)

*a* and

*ηwV*, *N* ≥ 3, (27)

*ξwV*, *N* ≥3, (24)

*a* and the corresponding

*a* ¼ ~að Þ *N* , (25)

*a* ¼ *γ*ð Þ *N* , (28)

*N*�2

� . Notice that for even parity

� will not vanish solely by a

ð Þ3 � is an

ð Þ3

$$\gamma^{(N)} \equiv \int d\mathbf{x} \, \frac{1}{2} (\partial\_{\xi} \boldsymbol{w})^2 \partial\_{\eta} \boldsymbol{V}^{N-1}, \quad N \ge 2,\tag{30}$$

where we have introduced the asymptotically conserved charges *Q*^ ð Þ *N a* and the corresponding anomalies *γ*ð Þ *N* .

The interchange *η* \$ *ξ* allows us to reproduce the dual quasi-conservation laws. So, one has

$$\frac{d}{dt}\tilde{\mathbf{Q}}\_{a}^{(N)} = \tilde{\boldsymbol{\gamma}}^{(N)},\tag{31}$$

$$\tilde{Q}\_a^{(N)} \equiv \int d\mathbf{x} \left[ \frac{1}{2} \mathbf{V}^{N-1} (\partial\_{\xi} w)^2 + \frac{1}{N} \mathbf{V}^N \right],\tag{32}$$

$$\tilde{\gamma}^{(N)} \equiv \int d\mathbf{x} \, \frac{1}{2} \left( \partial\_{\eta} w \right)^{2} \partial\_{\xi} V^{N-1}, \quad N \ge 2,\tag{33}$$

where we have defined the dual asymptotically conserved charges *Q*~ ð Þ *N a* and the anomalies ~*γ*ð Þ *N* .

Similarly, the densities of the anomalies *γ*ð Þ *N* and ~*γ*ð Þ *N* in (30) and (33), respectively, possess odd parities under (4) and (5), so the quasi-conservation laws (28) and (31), respectively, allow the construction of asymptotically conserved charges.

The relevant anomalies of the lowest order quasi-conservation laws of the above towers will be simulated below for 2-soliton interactions.

Remarkably, the above charges turn out to be anomalous even for the standard sine-Gordon model. In fact, the relevant 2-soliton solutions have been constructed analytically [4, 11] which possess a definite parity under (4)–(5), such that the odd anomaly densities vanish upon space–time integration. The usual explanation for the appearance of novel anomalous charges in the standard sine-Gordon model is the symmetry argument. The anomalous charges also appear in the standard KdV and its deformations [9].

These charges have been computed for soliton collisions in the treatment of soliton gases and formation of some structures in integrable systems, such as integrable turbulence and rogue waves. In the context of the usual KdV model it has been analyzed the behavior of the statistical moments defined by (see e.g. [16, 17]) *Mn*ðÞ¼ *t* Ð þ∞ �∞ *vn dx*, *n*≥1; where *v* is the KdV field. The *M*1,2 cases are conserved charges. It is remarkable that the moments, *M*3,4, respectively, in the interaction region of two-solitons, behave as the anomalous charges of the quasi-integrable KdV models [9]. In fact, in the quasi-integrable KdV models the moments *M*2,3 are in fact anomalous charges [9]. So, since the two-soliton collision is an important ingredient in the formation of soliton turbulence and the dynamics of soliton gases, we can expect they will be important in the quasi-integrable counterparts. In the case of the SG soliton ensemble, to our knowledge, it is needed a further theoretical research.

#### **4. Numerical simulations**

Here we will check numerically the lowest order expressions of the various towers of quasi-conservation laws presented above. For this purpose we will numerically solve the Eq. (1) with the particular deformed potential (2). In the **Figures 1** and **2** we plot the kink-kink and kink-antikink collisions, respectively. Moreover, we show the first conserved charges, i.e. the energy and momentum for these field configurations.

#### **Figure 1.**

*Kink-kink with velocities v*2 ¼ �*v*1 ¼ 0*:*15 *and q* ¼ 2*:*01 *in (2), for initial (green), collision (blue) and final (red) times. Bottom, the energy (E) and momentum (P) charges of the kink-kink.*

#### **Figure 2.**

*Kink-antikink with velocities v*2 ¼ �*v*1 ¼ 0*:*15 *and q* ¼ 2*:*01 *in (2), for initial (green), collision (blue) and final (red) times. Bottom, the energy (E) and momentum (P) charges of the kink-antikink.*

#### **4.1 First non-trivial anomalies of the SG-type quasi-conservation laws**

We have checked our results by numerical simulation of the anomalies *α*ð Þ3 � (18)– (21) for kink-antikink, kink-kink and breather solutions of the model (2).

So, let us write (11) in the form

$$q\_{a,\pm}^{(3)}(t) - q\_{a,\pm}^{(3)}(t\_0) = -\int\_{t\_0}^t dt a\_{\pm}^{(3)}(t),\tag{34}$$

shown for three successive times (top figures). Their integration in space furnish

*density figure shows initial ti (green), collision tc (blue) and final t f (red) times of the kink-antikink.*

**Figure 4**, respectively. According to (34) our numerical simulations show the

In the **Figures 5** and **6** we show the results for kink-kink system with velocities

for three successive times (top figures). Their integration in space furnish vanishing

� ð Þ*t* (middle figures). The bottom figures show

*dt*0 *α*ð Þ3 � *t*

� *in (20)–(21) for kink-antikink with velocities v*2 ¼ �*v*1 ¼ 0*:*5 *and ε* ¼ 0*:*06*. The*

þ *in (18) and (19) for kink-antikink with velocities v*2 ¼ �*v*1 ¼ 0*:*5 *and ε* ¼ 0*:*06*. The*

*density figure shows initial ti (green), collision tc (blue) and final t f (red) times of the kink-antikink.*

*Deformed Sine-Gordon Models, Solitons and Anomalous Charges*

*DOI: http://dx.doi.org/10.5772/intechopen.95432*

� ð Þ*t* (middle figures). The bottom figures

*a*,� and the exact conservation of the

ð Þ3

0 ð Þ asymptotically vanishing in

� ð Þ *x*, *t vsx* are shown

þ ð Þ*t* and non-vanishing *α*ð Þ3

asymptotically conservation of the charge *q*ð Þ3

*a*,þ, within numerical accuracy.

0 ð Þ, vanishing in **Figure 3** and Ð

*v*2 ¼ �*v*1 ¼ 0*:*5 and *ε* ¼ 0*:*06. The plots of (19) and (21) as *f*

vanishing *α*ð Þ3

*dt*0 *α*ð Þ3 þ *t*

*dtα*ð Þ 3

ð Þ3

þ ð Þ*t* and non-vanishing *α*ð Þ3

*4.1.2 kink-kink*

show Ð

**Figure 4.** *f* ð Þ 3 � *, α*ð Þ 3 � *and* Ð

**Figure 3.** *f* ð Þ 3 þ *, α*ð Þ 3 þ *and* Ð

*dtα*ð Þ 3

charge *q*

*α*ð Þ3

**73**

where *α*ð Þ3 � ð Þ*t* were defined in (18) and (20) and *t*0 is the initial time.

The simulations of the kink-antikink, kink-kink and breather systems of the deformed SG model will consider, as the initial condition, two analytic solitary wave solutions presented in Eq. (1.2) of [4], located some distance apart and stitched together at the middle point.

#### *4.1.1 Kink-antikink*

In the **Figures 3** and **4** we show the results for kink-antikink system with velocities *v*2 ¼ �*v*1 ¼ 0*:*5 and *ε* ¼ 0*:*06. The plots of (19) and (21) as *f* ð Þ3 � ð Þ *x*, *t vsx* are *Deformed Sine-Gordon Models, Solitons and Anomalous Charges DOI: http://dx.doi.org/10.5772/intechopen.95432*

#### **Figure 3.**

*f* ð Þ 3 þ *, α*ð Þ 3 þ *and* Ð *dtα*ð Þ 3 þ *in (18) and (19) for kink-antikink with velocities v*2 ¼ �*v*1 ¼ 0*:*5 *and ε* ¼ 0*:*06*. The density figure shows initial ti (green), collision tc (blue) and final t f (red) times of the kink-antikink.*

#### **Figure 4.**

**4.1 First non-trivial anomalies of the SG-type quasi-conservation laws**

*final (red) times. Bottom, the energy (E) and momentum (P) charges of the kink-antikink.*

(21) for kink-antikink, kink-kink and breather solutions of the model (2).

ð Þ3

So, let us write (11) in the form

where *α*ð Þ3

**Figure 1.**

**Figure 2.**

*4.1.1 Kink-antikink*

**72**

together at the middle point.

*q* ð Þ3 *a*,�ðÞ�*t q*

We have checked our results by numerical simulation of the anomalies *α*ð Þ3

*Kink-antikink with velocities v*2 ¼ �*v*1 ¼ 0*:*15 *and q* ¼ 2*:*01 *in (2), for initial (green), collision (blue) and*

*Kink-kink with velocities v*2 ¼ �*v*1 ¼ 0*:*15 *and q* ¼ 2*:*01 *in (2), for initial (green), collision (blue) and final*

*(red) times. Bottom, the energy (E) and momentum (P) charges of the kink-kink.*

*Recent Developments in the Solution of Nonlinear Differential Equations*

*a*,�ð Þ¼� *t*0

� ð Þ*t* were defined in (18) and (20) and *t*0 is the initial time. The simulations of the kink-antikink, kink-kink and breather systems of the deformed SG model will consider, as the initial condition, two analytic solitary wave solutions presented in Eq. (1.2) of [4], located some distance apart and stitched

In the **Figures 3** and **4** we show the results for kink-antikink system with

velocities *v*2 ¼ �*v*1 ¼ 0*:*5 and *ε* ¼ 0*:*06. The plots of (19) and (21) as *f*

ð*t t*0 *dtα*ð Þ3 � (18)–

� ð Þ*t* , (34)

ð Þ3

� ð Þ *x*, *t vsx* are

*f* ð Þ 3 � *, α*ð Þ 3 � *and* Ð *dtα*ð Þ 3 � *in (20)–(21) for kink-antikink with velocities v*2 ¼ �*v*1 ¼ 0*:*5 *and ε* ¼ 0*:*06*. The density figure shows initial ti (green), collision tc (blue) and final t f (red) times of the kink-antikink.*

shown for three successive times (top figures). Their integration in space furnish vanishing *α*ð Þ3 þ ð Þ*t* and non-vanishing *α*ð Þ3 � ð Þ*t* (middle figures). The bottom figures show Ð *dt*0 *α*ð Þ3 þ *t* 0 ð Þ, vanishing in **Figure 3** and Ð *dt*0 *α*ð Þ3 � *t* 0 ð Þ asymptotically vanishing in **Figure 4**, respectively. According to (34) our numerical simulations show the asymptotically conservation of the charge *q*ð Þ3 *a*,� and the exact conservation of the charge *q* ð Þ3 *a*,þ, within numerical accuracy.

#### *4.1.2 kink-kink*

In the **Figures 5** and **6** we show the results for kink-kink system with velocities *v*2 ¼ �*v*1 ¼ 0*:*5 and *ε* ¼ 0*:*06. The plots of (19) and (21) as *f* ð Þ3 � ð Þ *x*, *t vsx* are shown for three successive times (top figures). Their integration in space furnish vanishing *α*ð Þ3 þ ð Þ*t* and non-vanishing *α*ð Þ3 � ð Þ*t* (middle figures). The bottom figures show

*4.1.3 Breather: kink-antikink bound state*

*DOI: http://dx.doi.org/10.5772/intechopen.95432*

and non-vanishing (periodic in time) *α*ð Þ3

**Figures 7** and **8** show the vanishing Ð

ð Þ3

*Deformed Sine-Gordon Models, Solitons and Anomalous Charges*

a fixed value and the exact conservation of the charge *q*

*ε* ¼ 0*:*06. The densities *f*

**Figure 7.** *f* ð Þ 3 þ *, α*ð Þ 3 þ *and* Ð

**Figure 8.** *f* ð Þ 3 � *, α*ð Þ 3 � *and* Ð

**75**

*t* ∈ *t f* � *T*0, *t f*

*t* ∈ *t f* � *T*0, *t f*

*dtα*ð Þ 3

*dtα*ð Þ 3

� �*, T*0 ¼ 7*:*025*. The long-lived breather for t f*≈105*.*

� �*, T*0 ¼ 7*:*025*. The long-lived breather for t f*≈105*.*

**Figures 7** and **8** show the results for breather (kink-antikink bound state) with

functions of *x* for three successive times (top figures). They show the vanishing *α*ð Þ3

We will compute the linear combinations of the lowest order anomalies of the

þ *in (18) and (19) for breather with ε* ¼ 0*:*06*. The density is shown for three times*

� *in (20) and (21) for breather with ε* ¼ 0*:*06*. The density is shown for three times*

*dt*0 *α*ð Þ3 þ *t*

According to (34) our numerical results show the oscillation of the charges *q*ð Þ3

**4.2 Lowest order anomalies of the second and third types of towers**

second and third types of towers in (22)–(27) and (28)–(33), respectively,

� ð Þ *x*, *t* in (19) and (21), respectively, have been plotted as

0 ð Þ and periodic Ð

ð Þ3

� ð Þ*t* (middle figures). The bottom figures of

*dt*0 *α*ð Þ3 � *t* þ ð Þ*t*

*a*,� around

0 ð Þ expressions.

*a*,þ, within numerical accuracy.

#### **Figure 5.**

*f* ð Þ 3 þ *, α*ð Þ 3 þ *and* Ð *dtα*ð Þ 3 þ *in (18) and (19) for kink-kink with velocities v*2 ¼ �*v*1 ¼ 0*:*5 *and ε* ¼ 0*:*06*. The density figure shows initial ti (green), collision tc (blue) and final t f (red) times of the kink-kink.*

Ð *dt*0 *α*ð Þ3 þ *t* 0 ð Þ, vanishing in **Figure 5** and Ð *dt*0 *α*ð Þ3 � *t* 0 ð Þ asymptotically vanishing in **Figure 6**. According to (34) our numerical results show the asymptotically conservation of the charge *q*ð Þ3 *a*,� and the exact conservation of the charge *q* ð Þ3 *a*,þ, within numerical accuracy.

So, one can conclude that for kink-antikink (kink-kink) solution the definite parity related to the space-reflection symmetry is a necessary condition in order to achieve a conserved *q* ð Þ3 *a*,þ charge, within numerical accuracy.

The both kink-antikink and kink-kink solitons of the SG model with opposite and different velocities do not possess the required parity symmetry. However, it has been shown that in the center-of-mass reference frame (*x*0 , *t* 0 ) the parity symmetries are recovered, as discussed in [11]. So, the simulations performed in these reference frames, in the both kink-antikink and kink-kink cases, will provide vanishing *α*ð Þ3 þ anomalies as shown above.

#### **Figure 6.**

*f* ð Þ 3 � *, α*ð Þ 3 � *and* Ð *dtα*ð Þ 3 � *in (20) and (21) for kink-kink with velocities v*2 ¼ �*v*1 ¼ 0*:*5 *and ε* ¼ 0*:*06*. The density figure shows initial ti (green), collision tc (blue) and final t f (red) times of the kink-kink.*

#### *4.1.3 Breather: kink-antikink bound state*

**Figures 7** and **8** show the results for breather (kink-antikink bound state) with *ε* ¼ 0*:*06. The densities *f* ð Þ3 � ð Þ *x*, *t* in (19) and (21), respectively, have been plotted as functions of *x* for three successive times (top figures). They show the vanishing *α*ð Þ3 þ ð Þ*t* and non-vanishing (periodic in time) *α*ð Þ3 � ð Þ*t* (middle figures). The bottom figures of **Figures 7** and **8** show the vanishing Ð *dt*0 *α*ð Þ3 þ *t* 0 ð Þ and periodic Ð *dt*0 *α*ð Þ3 � *t* 0 ð Þ expressions. According to (34) our numerical results show the oscillation of the charges *q*ð Þ3 *a*,� around a fixed value and the exact conservation of the charge *q* ð Þ3 *a*,þ, within numerical accuracy.

#### **4.2 Lowest order anomalies of the second and third types of towers**

We will compute the linear combinations of the lowest order anomalies of the second and third types of towers in (22)–(27) and (28)–(33), respectively,

#### **Figure 7.**

Ð *dt*0 *α*ð Þ3 þ *t*

**Figure 5.** *f* ð Þ 3 þ *, α*ð Þ 3 þ *and* Ð

0 ð Þ, vanishing in **Figure 5** and Ð

ð Þ3

has been shown that in the center-of-mass reference frame (*x*0

þ anomalies as shown above.

vation of the charge *q*ð Þ3

*dtα*ð Þ 3

numerical accuracy.

achieve a conserved *q*

vanishing *α*ð Þ3

**Figure 6.** *f* ð Þ 3 � *, α*ð Þ 3 � *and* Ð

**74**

*dtα*ð Þ 3

*dt*0 *α*ð Þ3 � *t*

*density figure shows initial ti (green), collision tc (blue) and final t f (red) times of the kink-kink.*

*Recent Developments in the Solution of Nonlinear Differential Equations*

**Figure 6**. According to (34) our numerical results show the asymptotically conser-

So, one can conclude that for kink-antikink (kink-kink) solution the definite parity related to the space-reflection symmetry is a necessary condition in order to

*a*,þ charge, within numerical accuracy. The both kink-antikink and kink-kink solitons of the SG model with opposite and different velocities do not possess the required parity symmetry. However, it

metries are recovered, as discussed in [11]. So, the simulations performed in these reference frames, in the both kink-antikink and kink-kink cases, will provide

� *in (20) and (21) for kink-kink with velocities v*2 ¼ �*v*1 ¼ 0*:*5 *and ε* ¼ 0*:*06*. The*

*density figure shows initial ti (green), collision tc (blue) and final t f (red) times of the kink-kink.*

*a*,� and the exact conservation of the charge *q*

þ *in (18) and (19) for kink-kink with velocities v*2 ¼ �*v*1 ¼ 0*:*5 *and ε* ¼ 0*:*06*. The*

0 ð Þ asymptotically vanishing in

, *t* 0 ð Þ3 *a*,þ, within

) the parity sym-

*f* ð Þ 3 þ *, α*ð Þ 3 þ *and* Ð *dtα*ð Þ 3 þ *in (18) and (19) for breather with ε* ¼ 0*:*06*. The density is shown for three times t* ∈ *t f* � *T*0, *t f* � �*, T*0 ¼ 7*:*025*. The long-lived breather for t f*≈105*.*

#### **Figure 8.**

*f* ð Þ 3 � *, α*ð Þ 3 � *and* Ð *dtα*ð Þ 3 � *in (20) and (21) for breather with ε* ¼ 0*:*06*. The density is shown for three times t* ∈ *t f* � *T*0, *t f* � �*, T*0 ¼ 7*:*025*. The long-lived breather for t f*≈105*.*

$$\mathbf{a}\_{\pm} \equiv \mathbf{a}^{(3)} \pm \tilde{\mathbf{a}}^{(3)},\tag{35}$$

$$
\gamma\_{\pm} \equiv \gamma^{(2)} \pm \tilde{\gamma}^{(2)}.\tag{36}
$$

#### *4.2.1 Second and third types of towers and lowest order anomalies*

The two anomalies in (35) can be written as

$$\mathbf{a}\_{+} = \int d\mathbf{x} \, 2 \left[ \partial\_{t}^{2} w + \partial\_{\mathbf{x}}^{2} w \right] V,\tag{37}$$