**3. Basic equations**

First we will show that the kernel 121212 *K cc* (, ,, ,,) (as in one-dimensional case [18]) in the region 1 2 :{| | 1,| | 1} for arbitrary 1*c* 0 and 2*c* 0 is a positive kernel of the integral operator

$$Af(\xi\_{1'}\xi\_2) = \iint\_{\Omega} K(\xi\_{1'}\xi\_2, \xi\_{1'}'\xi\_{2'}'c\_{1'}c\_2) f(\xi\_{1'}'\xi\_2') d\xi\_1' d\xi\_2'... $$

Numerical Algorithms of Finding the Branching Lines

(, ,,) *c c* , we shall eliminate this function from

121212 1212 1 2

(, ,,) *c c* will not be the eigenfunction of

*c c* , , (18)

*i Tc c c*

follows from the continuity of the

, *<sup>i</sup>* 1,2 and

, we will reduce

*i E cc c*

> 

as

1 2

*nm nm*

(16)

 

(17)

, *<sup>i</sup>* 1,2 , and

*E cc ccdd* ( , , , , , )( , , , )

 

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 293

*c c* , then the equation (15) will be reduced to the integral

 

*c c*

(, ,,) *c c* . So, we obtain a self-adjoint generalized

 

 

0121201212

01212

To find the solutions distinct from 01212

1212 12 1212

From Schmidt's lemma [20] it follows, that 01212

, *i j* , 1,2 at arbitrary points 1 2 *c c*

corresponds to the function 01212

 

12 12 *L TI* ( , ) ( , ) 0,

parameters 1*c* and 2*<sup>c</sup>* . The existence of partial Frechet derivatives 1 2 (,)

and continuity in the derivatives <sup>121212</sup> (, ,, ,,)

Using the property of degeneracy of the kernel 121212 *K cc* (, ,, ,,)

1 2

*M M*

*n Mm M*

1 2

<sup>121212</sup> 1212 1212 (, ,, ,,) (, ,,) (, ,,)

equation (16) to an equivalent system of algebraic equations.

Using the formula (6), we write the kernel 121212 *E cc* (, ,, ,,)

*E cc* 

 

( , , , ) ( , )( , , , ) *c c Tc c c c*

the kernel 121212 (, ,, ,,) 

equation

 

with a symmetric kernel

eigenvalue problem

2

2

1 2 (,) *i j Tc c c c*

kernel 121212 *E cc* (, ,, ,,) 

> <sup>121212</sup> (, ,, ,,) *i j E cc c c*

 1 1

1 1

121212 1212 1212 *E cc w ccw cc* ( , , , , , ) ( , , , )( , , , )

121212 2

equation (16) anymore. That is, we have eliminated a continuum set of eigenvalues from a spectrum of the operator (16), which coincides with the first quadrant of the plane <sup>2</sup> *R* that

(, ,,)(, ,,) (, ,, ,,) . || ( , , , )|| *f cc f cc K cc*

 1 12 2 

with the operator 1 2 *Tc c* (,) which is a continuously differentiable with respect to the

, 

, *i j* , 1,2 which because of their bulky form, are not presented.

 *K cc K cc* 

on the set of its variables in the region and the existence

 

To this end, consider the scalar product

$$f(Af, f) = \iiint\limits\_{\Omega} K(\xi\_1', \xi\_2', \xi\_1', \xi\_2', c\_1, c\_2) f(\xi\_1', \xi\_2') \overline{f(\xi\_1', \xi\_2')} d\xi\_1' d\xi\_2' \, d\xi\_1 d\xi\_2.$$

Substituting the expression for 121212 *K cc* (, ,, ,,) (6), we obtain

$$\begin{split} f(Af,f) &= \iint \iint \left( \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_1} e^{i\left[ \begin{matrix} \sum\_1 (\xi\_1'-\xi\_1') + mc\_2 \langle \xi\_2 - \xi\_2' \rangle \right] \right]} \right) f(\xi\_1', \xi\_2') \overline{f(\xi\_1', \xi\_2')} d\xi\_1' d\xi\_2' \, d\xi\_1 d\xi\_2' = 0 \\ &= \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} \left\{ \iiint \int (\xi\_1', \xi\_2') e^{-i(nc\_1\xi\_1' + mc\_2\xi\_2')} \overline{f(\xi\_1', \xi\_2')} e^{i(nc\_1\xi\_1 + mc\_2\xi\_2)} d\xi\_1' d\xi\_2' \, d\xi\_1 d\xi\_2' \right\} = \\ &= \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} \left\| \iint f(\xi\_1', \xi\_2') e^{-i(nc\_1\xi\_1' + mc\_2\xi\_2')} d\xi\_1' d\xi\_2' \right\|^2 = \frac{4\pi^2}{c\_1 c\_2} \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} |I\_{nm}|^2 \ge 0. \end{split}$$

Obviously, the last inequality transforms into equality only when 11 22 0, , , , *nm I n MM m MM* . From this it follows that 121212 *K cc* (, ,, ,,) is positive, and positive operator *A* leaves invariant a cone **K** ( ) *A***K K** of the continuous nonnegative functions on . As a result, we obtain that 01212 *f* (, ,,) *c c* is positive on function. Taking it into account, we shall reduce the operator (12) to a selfajoint form by a standard method. Introducing a new function

$$\mathfrak{gl}(\underline{\xi}\_{1'}, \underline{\xi}\_{2'}c\_{1'}c\_2) = \sqrt{\mathfrak{w}(\underline{\xi}\_{1'}, \underline{\xi}\_{2'}c\_{1'}c\_2)} \text{ } \mathfrak{u}(\underline{\xi}\_{1'}\underline{\xi}\_{2'}c\_{1'}c\_2) \text{.} \tag{14}$$

where 1212 12 01212 *w cc F f cc* ( , , , ) ( , )/ ( , , , ) , we obtain the integral equation

$$\mathfrak{gl}(\underline{\xi}\_{1}, \underline{\xi}\_{2}, \underline{c}\_{1}, \underline{c}\_{2}) = \iint\_{\Omega} \mathfrak{d}(\underline{\xi}\_{1}, \underline{\xi}\_{2}, \underline{\xi}\_{1}', \underline{\xi}\_{2}', \underline{c}\_{1}, \underline{c}\_{2}) \mathfrak{gl}(\underline{\xi}\_{1}', \underline{\xi}\_{2}', \underline{c}\_{1}, \underline{c}\_{2}) d\underline{\xi}\_{1}' d\underline{\xi}\_{2}' \tag{15}$$

with a symmetric kernel

$$\Phi(\tilde{\xi}\_{1}, \tilde{\xi}\_{2}, \tilde{\xi}\_{1}', \tilde{\xi}\_{2}', c\_{1}, c\_{2}) = \mathbf{K}(\tilde{\xi}\_{1}, \tilde{\xi}\_{2}, \tilde{\xi}\_{1}', \tilde{\xi}\_{2}', c\_{1}, c\_{2}) \sqrt{\mathbf{w}(\tilde{\xi}\_{1}, \tilde{\xi}\_{2}, c\_{1}, c\_{2})} \mathbf{w}(\tilde{\xi}\_{1}', \tilde{\xi}\_{2}', c\_{1}, c\_{2}) .$$

Since at arbitrary 1*c* 0 and 2*c* 0 the function 01212 *f* (, ,,) *c c* is the eigenfunction of the equation (12), then with regard for (14), the eigenfunction of the equation (15) at arbitrary <sup>1</sup>*c* 0 and 2*c* 0 will be the function

$$
\varphi\_0(\xi\_{1'}\xi\_2',c\_{1'}c\_2) = \sqrt{F(\xi\_{1'}\xi\_2)f\_0(\xi\_{1'}\xi\_{2'}c\_{1'}c\_2)}\,\,\_2
$$

which corresponds to a spectrum of the operator (15), coinciding with the first quadrant of the plane <sup>2</sup> *R* .

To find the solutions distinct from 01212 (, ,,) *c c* , we shall eliminate this function from the kernel 121212 (, ,, ,,) *c c* , then the equation (15) will be reduced to the integral equation

$$\rho(\xi\_1, \xi\_2, c\_1, c\_2) = \mathcal{T}(c\_1, c\_2)\rho(\xi\_1, \xi\_2, c\_1, c\_2) \equiv \int \prod\_{-1-1}^{1} \mathcal{E}(\xi\_1, \xi\_2, \xi\_1', \xi\_2', c\_1, c\_2)\rho(\xi\_1', \xi\_2', c\_1, c\_2)d\xi\_1' d\xi\_2' \tag{16}$$

with a symmetric kernel

292 Nonlinearity, Bifurcation and Chaos – Theory and Applications

To this end, consider the scalar product

( ,) *Af f* 1 2

1 2

1 2

*M M*

*n Mm M*

 

*M M*

*n Mm M*

Substituting the expression for 121212 *K cc* (, ,, ,,)

standard method. Introducing a new function

where 1212 12 01212 *w cc F f cc* ( , , , ) ( , )/ ( , , , )

<sup>1</sup>*c* 0 and 2*c* 0 will be the function

 *c c*

with a symmetric kernel

the plane <sup>2</sup> *R* .

 

> 

12 121212 12 1 2 *Af K cc f dd* ( , ) ( , , , , , )( , ) .

121212 12 12 1 2 1 2 ( ,) ( , , , , , )( , )( , ) . *Aff K cc f f dd dd*

11 1 22 2

( )( )

 

 

 

(6), we obtain

11 22 11 22

() () 1 2 1 2 12 1 (,) (,)

*f e f e dd dd*

*i nc mc i nc mc*

 

*f e dd I*

 

Obviously, the last inequality transforms into equality only when 11 22 0, , , , *nm I n MM m MM* . From this it follows that 121212 *K cc* (, ,, ,,)

positive, and positive operator *A* leaves invariant a cone **K** ( ) *A***K K** of the continuous

function. Taking it into account, we shall reduce the operator (12) to a selfajoint form by a

1212 1212 1212

1212 121212 1212 1 2

121212 121212 1212 1212 ( , , , , , ) ( , , , , , ) ( , , , ) ( , , , ).

equation (12), then with regard for (14), the eigenfunction of the equation (15) at arbitrary

01212 1201212

which corresponds to a spectrum of the operator (15), coinciding with the first quadrant of

 ( , , , ) ( , ) ( , , , ), *cc F f cc* 

*cc K cc w ccw cc*

( , , , ) ( , , , , , )( , , , )

, we obtain the integral equation

 ( , , , ) ( , , , ) ( , , , ), *cc w cc u cc* 

 

 

*e f f dd dd*

<sup>2</sup> <sup>2</sup> ( ) 2

<sup>4</sup> (,) | | 0.

1 2

*c c*

 

 

> 

 

*f* (, ,,) 

*cc ccdd*

(15)

 

 

 

*f* (, ,,) 

> 

 

*c c* is the eigenfunction of the

12 12 12 12 ( , )( , )

 

 

*nm*

*c c* is positive on

is

(14)

*i nc mc*

 

1 2 1 2 11 22 1 2 2

 

*M M M M i nc mc*

*n Mm M n Mm M*

1 2 1 2

 

nonnegative functions on . As a result, we obtain that 01212

 

Since at arbitrary 1*c* 0 and 2*c* 0 the function 01212

 

1 2

 

 

$$\operatorname{E}(\xi\_{1},\xi\_{2},\xi\_{1}',\xi\_{2}',c\_{1},c\_{2}) = \sqrt{\operatorname{uv}(\xi\_{1},\xi\_{2}',c\_{1},c\_{2})\operatorname{uv}(\xi\_{1}',\xi\_{2}',c\_{1},c\_{2})} \times$$

$$\times \left[ \operatorname{K}(\xi\_{1},\xi\_{2}',\xi\_{1}',\xi\_{2}',c\_{1},c\_{2}) - \frac{f\_{0}(\xi\_{1},\xi\_{2}',c\_{1},c\_{2})f\_{0}(\xi\_{1}',\xi\_{2}',c\_{1},c\_{2})}{|\|\,\rho\_{0}(\xi\_{1},\xi\_{2}',c\_{1},c\_{2})\|\|^{2}} \right]. \tag{17}$$

From Schmidt's lemma [20] it follows, that 01212 (, ,,) *c c* will not be the eigenfunction of equation (16) anymore. That is, we have eliminated a continuum set of eigenvalues from a spectrum of the operator (16), which coincides with the first quadrant of the plane <sup>2</sup> *R* that corresponds to the function 01212 (, ,,) *c c* . So, we obtain a self-adjoint generalized eigenvalue problem

$$L(\mathcal{A}\_1, \mathcal{A}\_2)\boldsymbol{\varphi} \equiv \left(\mathrm{T}(\mathcal{A}\_1, \mathcal{A}\_2) - I\right)\boldsymbol{\varphi} = \boldsymbol{0}, \ \mathcal{A}\_1 = \mathbf{c}\_1, \ \mathcal{A}\_2 = \mathbf{c}\_2. \tag{18}$$

with the operator 1 2 *Tc c* (,) which is a continuously differentiable with respect to the parameters 1*c* and 2*<sup>c</sup>* . The existence of partial Frechet derivatives 1 2 (,) *i Tc c c* , *<sup>i</sup>* 1,2 , and

2 1 2 (,) *i j Tc c c c* , *i j* , 1,2 at arbitrary points 1 2 *c c* , follows from the continuity of the kernel 121212 *E cc* (, ,, ,,) on the set of its variables in the region and the existence and continuity in the derivatives <sup>121212</sup> (, ,, ,,) *i E cc c* , *<sup>i</sup>* 1,2 and

$$\frac{\partial^2 \mathbb{E}(\tilde{\varepsilon}\_1, \tilde{\varepsilon}\_2, \tilde{\varepsilon}\_1', \tilde{\varepsilon}\_2', c\_1, c\_2)}{\partial c\_i \partial c\_j}, \text{ i.e.}\\j = 1, 2 \text{ which because of their bulky form, are not presented.}$$

Using the property of degeneracy of the kernel 121212 *K cc* (, ,, ,,) , we will reduce equation (16) to an equivalent system of algebraic equations.

Using the formula (6), we write the kernel 121212 *E cc* (, ,, ,,) as

$$E(\xi\_1, \xi\_2, \xi\_1', \xi\_2', c\_1, c\_2) = \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} K\_{nm}^1(\xi\_1', \xi\_2', c\_1, c\_2) \cdot K\_{nm}^2(\xi\_1', \xi\_2', c\_1, c\_2) - 1$$

$$-\left(\sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} K\_{nm}^1(\xi\_1, \xi\_2, c\_1, c\_2) \boldsymbol{q}\_{nm}^1\right) \left(\sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} K\_{nm}^2(\xi\_1', \xi\_2', c\_1, c\_2) \boldsymbol{q}\_{nm}^2\right) \boldsymbol{q}\_{nm}^1$$

Numerical Algorithms of Finding the Branching Lines

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 295

 

(19)

*<sup>N</sup>* **D** (20)

*u* (21)

is the real ( ) *n n* matrix whose elements depend

. In order to detail how the method [15] is applied to

*u u* **T T n n** (22)

. Then, obviously, the eigenvalues of problem (21) are

at the given value of parameter

, and, therefore, the eigenvalues

 

*k k c d*

**[ ]** and

.

.

by the expression

of dimension *N N* , *N***I** is the identity matrix of

1 2

*M M*

*s Mt M*

 

2 2 2 2 1 1212 1212 1212 <sup>1</sup> (, ,,) (, ,,) (, ,,) .

*a cc K cc*

matrix eigenvalue problem equivalent to (18)

 

 (,) ( (,) ) 0 *N N N NN* **D b A Ib** 

> *c c* , .

in [14, 15] to compute all eigenvalues of the nonlinear matrix spectral problem

 

and consider the appropriate one-parameter problem

of the problem are in some given range of change of parameter ,

 

( , ) det ( , ) 0,

i.e. the eigenvalues of problem (19) are zeros of function

**4. Algorithm of finding the eigenvalue curves** 

belonging to some given range of the spectral parameter

 and 

One should determine how many zeros of the function *f*( )

*<sup>n</sup> u* , and ( , ) *<sup>n</sup>* **T**

 and 

Thus, we replace in the problem (21), for example, the parameter

In the problem (21) *<sup>n</sup>*

at the given fixed values

calculate each of them.

zeros of function

where ( ) *<sup>n</sup>* **T**

 

nonlinearly on the parameters

 

with symmetric matrix ( , ) *<sup>N</sup>* **A**

dimension *N N* , *<sup>N</sup>* **<sup>b</sup>***<sup>N</sup> <sup>R</sup>* , 1 2

*nm nm st st nm*

So, we have obtained the two-parameter nonlinear (with respect to the spectral parameters )

 

Thus, the problem of finding lines the branching of solutions of equation (7) is reduced to finding the eigenvalues curves of nonlinear two-parameter spectral problem (19).

The main calculational part of algorithm proposed is the implementation method proposed

( , ) 0, *n n* **T** 

the problem under consideration in this paper, we present the necessary results from [15].

( ) ( , , ) 0, *n n*

 

( ) det ( ) 0, *<sup>n</sup> f*

 **T**

is a real ( ) *n n* matrix whose elements depend nonlinearly on the parameter

 

> (,) .

Obviously, in order the problem (19) to have a nonzero solution it is necessary that

1 2

*K cc q q*

where

$$\begin{split} K\_{\text{nm}}^{1}(\xi\_{1},\xi\_{2},c\_{1},c\_{2}) &= \frac{\sqrt{c\_{1}c\_{2}}}{2\pi} \sqrt{\frac{F(\xi\_{1},\xi\_{2}^{\prime})}{f\_{0}(\xi\_{1},\xi\_{2}^{\prime}c\_{1},c\_{2})}} \cdot e^{i(c\_{1}n\tilde{\varepsilon}\_{1} + c\_{2}m\tilde{\varepsilon}\_{2})}, \\ K\_{\text{nm}}^{2}(\xi\_{1}^{\prime},\xi\_{2}^{\prime},c\_{1},c\_{2}) &= \frac{\sqrt{c\_{1}c\_{2}}}{2\pi} \sqrt{\frac{F(\xi\_{1}^{\prime},\xi\_{2}^{\prime})}{f\_{0}(\xi\_{1}^{\prime},\xi\_{2}^{\prime},c\_{1},c\_{2})}} \cdot e^{-i(c\_{1}n\tilde{\varepsilon}\_{1}^{\prime} + c\_{2}m\tilde{\varepsilon}\_{2}^{\prime})}, \\ &\quad q\_{\text{nm}}^{1} = \iint F(\xi\_{1}^{\prime},\xi\_{2}^{\prime}) e^{-i(c\_{1}n\tilde{\varepsilon}\_{1}^{\prime} + c\_{2}m\tilde{\varepsilon}\_{2}^{\prime})} d\xi\_{1}^{\prime} d\tilde{\varepsilon}\_{2}^{\prime}, \\ &\quad q\_{\text{nm}}^{2} = \iint F(\xi\_{1}^{\prime},\xi\_{2}^{\prime}) e^{i(c\_{1}n\tilde{\varepsilon}\_{1} + c\_{2}m\tilde{\varepsilon}\_{2}^{\prime})} d\xi\_{1} d\tilde{\varepsilon}\_{2}^{\prime}. \end{split}$$

Then equation (16) takes the form

$$\rho(\xi\_{1'}\xi\_{2'}c\_{1'}c\_2) = \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} a\_{nm}^1(\xi\_{1'}\xi\_{2'}c\_{1'}c\_2)b\_{nm'}$$

where

$$\begin{aligned} b\_{nm} &= \iint \mathcal{K}^2\_{nm}(\xi\_1', \xi\_2', c\_1, c\_2) q(\xi\_1', \xi\_2', c\_1, c\_2) d\xi\_1' d\xi\_2', \\\\ a\_{nm}^1(\xi\_1, \xi\_2, c\_1, c\_2) &= \mathcal{K}^1\_{nm}(\xi\_1', \xi\_2', c\_1, c\_2) - \frac{1}{\mathcal{Y}} \left( \sum\_{s=-M\_1}^{M\_1} \sum\_{t=-M\_2}^{M\_2} \mathcal{K}^1\_{st}(\xi\_1', \xi\_2', c\_1, c\_2) q^1\_{st} \right) q^2\_{nm}, \\\\ \mathcal{Y} &= \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} q^1\_{nm} q^2\_{nm} \end{aligned}$$

and the unknown coefficients *nm b* are determined as solutions of a homogeneous system of linear algebraic equations

$$b\_{kl} = \sum\_{n=-M\_1}^{M\_1} \sum\_{m=-M\_2}^{M\_2} \alpha\_{nm}^{(kl)}(c\_1, c\_2) b\_{nm'} \quad k = \overline{-M\_1, M\_1}, l = \overline{-M\_2, M\_2}.$$

where

$$\alpha\_{nm}^{(kl)}(c\_1, c\_2) = \iint \!\!\! a\_{kl}^2(\xi\_1, \xi\_2, c\_1, c\_2) a\_{nm}^1(\xi\_1, \xi\_2, c\_1, c\_2) d\xi\_1 d\xi\_2 d\dots$$

Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 295

$$\log^2\left(\xi\_1, \xi\_2, c\_1, c\_2\right) = K\_{nm}^2\left(\xi\_1, \xi\_2, c\_1, c\_2\right) - \frac{1}{\mathcal{Y}} \left(\sum\_{s=-M\_1}^{M\_1} \sum\_{t=-M\_2}^{M\_2} K\_{st}^2(\xi\_1, \xi\_2, c\_1, c\_2) q\_{st}^2\right) q\_{nm}^1.$$

So, we have obtained the two-parameter nonlinear (with respect to the spectral parameters ) matrix eigenvalue problem equivalent to (18)

$$\mathbf{D}\_{\mathcal{N}}(\mathcal{k},\mu)\mathbf{b}\_{\mathcal{N}} \equiv (\mathbf{A}\_{\mathcal{N}}(\mathcal{k},\mu) - \mathbf{I}\_{\mathcal{N}})\mathbf{b}\_{\mathcal{N}} = \mathbf{0} \tag{19}$$

with symmetric matrix ( , ) *<sup>N</sup>* **A** of dimension *N N* , *N***I** is the identity matrix of dimension *N N* , *<sup>N</sup>* **<sup>b</sup>***<sup>N</sup> <sup>R</sup>* , 1 2 *c c* , .

Thus, the problem of finding lines the branching of solutions of equation (7) is reduced to finding the eigenvalues curves of nonlinear two-parameter spectral problem (19). Obviously, in order the problem (19) to have a nonzero solution it is necessary that

$$\psi(\lambda,\mu) \equiv \det \mathbf{D}\_N(\lambda,\mu) = 0,\tag{20}$$

i.e. the eigenvalues of problem (19) are zeros of function (,) .
