Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations

Abstract

This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude Q multiplied by $e^{i\Phi }$, $\Phi$ being linear. Then, Q is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. In the other five cases, the solution is a traveling wave. It satisfies Abel-type ODE of the second kind, the first order ODE of the elliptic functions (the Weierstrass or Jacobi functions), the Airy equation, the Emden–Fawler equation, etc. At the end of the paper a short survey on the Jacobi elliptic and Weierstrass functions is included.

1. Introduction

As it is well known, many processes of mathematical physics are described by nonlinear partial differential equations (PDEs). Good examples are given by conservation laws in physics. Certainly, the equations originate from different models characterizing the corresponding processes. Recently, the concatenation model from optics with Kerr’s law of self-phase modulation (SPM) was proposed. It involves three of the following widely studied nonlinear equations: the nonlinear Schrödinger equation (NLSE), the Lakshmanan-Porcezian-Daniel equation (LPD), Sasa–Satsuma equation (SSE). In the last decade the papers [1,2,3,4] and others appeared on the subject. The above-mentioned and similar ones were also investigated from the point of view of the power law of SPM (e.g., in [5]), the recovery of one-soliton solutions and the corresponding conservation laws (e.g., [6]), numerical analysis by the LADM (Laplace–Adomian decomposition method) (e.g., [7,8]). We point out that in those papers, mainly bright and dark optical solitons were found. We obtain in our paper several new results on the PDE (1) that belongs to the same model, including solutions into an explicit form. The detection of its soliton solutions’ traveling waves is reduced to the study of fourth-order ordinary differential equations (ODEs). In the case of perturbed concatenation model (see [9,10]), the bright soliton solutions are not exact due to the application of the semi-inverse variational principle (SVP).

The methods used in the literature for studying the concatenation model (the respective PDEs) and its perturbations include the SEM (simple equations method) [1,11,12], Painlevé analysis [13], Lie symmetry analysis [14,15,16] numerical implementations addressing th solitons with applications of LADM approach [7], SVP principle [9,10] and methods from the classical ODE theory ([17,18,19] applied in [20,21,22] and here). In the plane hydrodynamic jet theory (see W. Bickley and H. Schlichting [23]) an evolution-type third-order nonlinear ODE arises, and it is worth finding out all its traveling-wave solutions. The same problem concerns the Kadomtsev–Petviachvili PDE from hydrodynamics (solitons theory).

We propose here other third-order nonlinear PDEs in two variables, looking for their traveling-wave solutions. As the solvability problem is reduced to the investigation of special classes of second-, third- and fourth-order nonlinear ODEs, we emphasize the concrete-type solutions as ODEs [17,18]. Our analysis is a mathematical one, i.e., we do not focus only on solitons, kinks, rational or damping oscillating solutions. Our aim is either to find all the solutions or as many feasible ones as possible, given by formulae. The soliton solutions of some fourth-order semilinear autonomous ODE enable us to construct solutions into an explicit form of rather complicated nonautonomous second-order ODEs. We also mention that Abel’s equation of the second kind appears in some cases. During our study, we use the method of the first integral and a variant of the simplest equation method.

This is not a paper in physics or numerical analysis but on PDEs and their applications. In many cases our traveling-wave solutions are explicitly expressed by the Weierstrass function ℘, the Jacobi functions $sn,cn,dn$ and some hyperbolic functions ($sech$). There are tables of these special functions, and more specially for ℘, one can find a lot of information in Mathematica. Concerning the link between the KP equation, ℘ function and mathematics, one can see chapter 9 of [24]. However, here, the qualitative picture is obtained. The graphs of those special functions $z=f\left(x\right)$, their squares and homographic functions can be found easily (periodic, solitons, kinks, exploding, unbounded periodic). If $z=f(x-t)$ is the corresponding traveling wave, its 3D graph is evident. We include in the paper the 3D mapping of $u=sech(x-t)$ (see Figure 1) and point out that $u=sec{h}^2(x-t)$ has a similar graph. The numerical experiments are delicate because if some algebraic cubic equation, say (A1), has three simple real $e_3<e_2<e_1$ roots, the nonconstant bounded solutions u are periodic, $e_3\le u\le e_2$, while for $e_1\to e_2$ (i.e., $e_1=e_2>e_3$), u-bounded soliton solutions appear ($e_3\le u<e_2$, $u(\pm \infty )=e_2$). Obviously, they are topologically completely different.

A short survey on Weierstrass and Jacobi elliptic functions is proposed in the Appendix A. In fact, in some cases, first-order ODEs satisfied by the elliptic functions appear. The crucial point in the investigation of LPD equations is the solvability of a nonlinear algebraic system (18 equations, 20 unknowns). As this can be performed in Maple, we do not concentrate on that problem in our paper. In many cases, solitons do not exist.

We shall say several words about the SEM (see [1,11,12]). It means that we are looking for solutions of a given nonlinear ODE having the form of a polynomial of some special functions with constant (and unknown) coefficients. That function satisfies an ODE (mainly nonlinear of the first order) which is well studied (e.g., Riccati, Bernoulli, Airy, Weierstrass, Jacobi, Abel and many others). The balance law enables us to determine the order of the polynomial.

Our paper is organized as follows. Section 1 is the Introduction. In Section 2, we formulate our main results and propose some comments on them. The proofs are given in Section 3.

2. Formulation of the Main Results and Some Comments

1. First, we introduce the nonlinear PDEs that are studied in that paper. They are:

$$i{q}_t+a{q}_{xx}+{b\left|q\right|}^{2n}q+c_1({\delta }_1{q}_{xxxx}+{\delta }_4{\left|q\right|}^{2n}{q}_{xx})+i{c}_2({\delta }_7{q}_{xxx}+{\delta }_8{\left|q\right|}^{2n}{q}_x)=0,$$

(1)

(LPD-type equation; the coefficients are real; see [14])

$$\begin{array}{c}ci{q}_t+a{q}_{xx}+{b\left|q\right|}^{2}q+c_1({\sigma }_1{q}_{xxxx}+{\sigma }_2{q}_x^2\overline{q}+{\sigma }_3|q_x{|}^{2}q+{\sigma }_4{\left|q\right|}^2{q}_{xx}+{\sigma }_5{q}^2{\overline{q}}_{xx}+{\sigma }_6{\left|q\right|}^{2}q)+\\ i{c}_2({\sigma }_7{q}_{xxx}+{\sigma }_8{\left|q\right|}^2{q}_x+{\sigma }_9{q}^2{\overline{q}}_x{)=i[\lambda \left(\right|q|}^{2m}{q)}_x+{\Theta }_1{\left(\right|q|}^{2m}{)}_{x}q+{\Theta }_2{\left|q\right|}^{2m}{q}_x].\end{array}$$

(2)

If $\lambda ={\Theta }_j=0$, $j=1,2$, we have the unperturbed concatenation model from fiber optics (see [9]). $q\equiv 0$ is a solution of (2). For the unperturbed model (2), when $c_1=0$, we obtain the SSE, while for $c_2=0$, we have the LPD model. If $c_1=c_2=0$, $\lambda ={\Theta }_j=0$, $j=1,2$, (2) is the NLSE.

$$3c{u}_{xxx}+u{u}_{xx}+u_t^2=0,c\in {\mathbf{R}}^1\setminus 0.$$

(3)

Equation (3) appears in the plane hydrodynamic jet theory [23,25].

$${(u_t+6u_{x}u+u_{xxx})}_x+\alpha u_{yy}=0,{\alpha }^2=1.$$

(4)

This is the Kadomtsev–Petviashvili (KP) PDE with two space variables [24,26].

$$u_{xxx}+a{u}_t=f\left(u\right)u_x,a\in {\mathbf{R}}^1,$$

(5)

with $f\left(u\right)$ real-valued.

$$u_{xxx}+a{\left(u_t^2\right)}_t=f\left(u\right)u_x,a\in {\mathbf{R}}^1,$$

(6)

with $f\left(u\right)$ real-valued.

$$u_{xxx}+a{\left(u_x^2\right)}_x+u_t=f\left(u\right)u_x.$$

(7)

Equation (1) is with a power-law SPM; $i{q}_t$ represents a linear temporal evolution, $q(x,t)$ denotes the wave profile, t is the time variable, while x is the space variable, n is the power-law parameter, and b determines the power-law characteristic of the SPM. If $c_1=0$, we obtain the SSE; if $c_2=0$, we have the LPD model, etc.

We look for a solution of (1) (see [14]) having the form

$$q(x,t)=Q\left(h(x-vt)\right)e^{i\Phi },\Phi =-kx+\omega t,\xi =h(x-vt).$$

(8)

All the constants and the function $Q\left(\xi \right)$ are real-valued. Here $\Phi$ is the phase of q, $\omega$ and k denote the wave number and the frequency, Q is the amplitude part, and h and v are the wave width and the velocity of the wave solution. Usually, $Q>0$. Then, we use

$$Q=U^{{\displaystyle \frac{1}{n}}}\left(\xi \right)$$

(9)

and conclude that the real-valued function U satisfies a fourth-order nonlinear autonomous ODE. According to the SEM, we seek

$$U=b_0+b_{1}G+b_2{G}^2,$$

(10)

where $b_j$, $j=0,1,2$ are real coefficients and

$$G^{\prime }\left(\xi \right)=\sqrt{a_0+a_{1}G+a_2{G}^2+a_3{G}^3+a_4{G}^4}\equiv \sqrt{P_4\left(G\right)},$$

(11)

$a_j\in {\mathbf{R}}^1$, $j=0,1,2,3,4$, $P_4\left(G\right)>0$ in some interval of ${\mathbf{R}}^1$.

This way, for the real coefficients $a,b,c_1$, ${\delta }_1$, ${\delta }_4$, $c_2$, ${\delta }_7$, ${\delta }_8$, $k,\omega$, $h,v$, $b_0$, $b_1$, $b_2$, $a_0$, $a_1$, $a_2$, $a_3$, $a_4$, we obtain a nonlinear algebraic system of 18 equations.

This is the main result of our paper.

Theorem 1.

(1). Consider Equation (1) and look for its solution of the form (8), (9), (10) and (11). Then, the solvability of (1) is reduced to the solvability of an algebraic system of 18 equations with 20 unknowns. If the latter possesses a nontrivial solution, then the same is valid for (1).

(2). Consider (2) and look for a solution of the form $q(x,t)=g(x-vt)e^{i\Phi }$, $\Phi =-kx+\omega t$, with $g\left(\xi \right)$ real-valued [9]. Then, the solvability of (2) is reduced to the solvability of a second-order nonautonomous ODE written as $w^{″}+{\displaystyle \frac{3}{2}}a{w}^{-1/3}+{\displaystyle \frac{3}{2}}a{w}^{-5/3}P\left(g\right)=0$, where $w=w\left(g\right)$, and $P\left(g\right)$ is an even polynomial of g.

(3). Search for a traveling-wave solution of (3), i.e., $u=\phi (x\pm t)$. The solvability of (3) is reduced to the solvability of a Riccati ODE and the Airy ODE.

(4). The traveling-wave solutions of (4) are expressed by the Weierstrass function ℘.

(5). To find out the traveling-wave solution of (5), we must solve some Abel-type ODE of the second kind [27,28]. In special cases, the traveling-wave solution can be expressed by the solution of the Emden–Fawler ODE [27].

(6)., (7). The traveling-wave solution satisfies an ODE with separate variables and consequently can be written in quadratures.

Remark 1.

The dark and bright soliton solutions of (1) are found in [14]. We propose below several cases when soliton solutions do not exist at all. For the unperturbed concatenation model, soliton solutions of the form $Q=AsechBx$, $A\ne 0$, $B\ne 0$ can be found for special values of A, B, a and $C_j$ if $Q^{iv}+a{Q}^{″}=C_{1}Q+C_3{Q}^3+C_5{Q}^5$. If on the right-hand side, we have an odd polynomial of order greater than 5, such nontrivial solution does not exist. As $ab\ne 0$ in (1) and (2), the nontrivial solution U satisfies the conditions $|b_1|+|b_2|>0$ in (10) and $a_4\ne 0$ in (11).

3. Proof of Theorem 1

1. We begin with the consideration of Equation (1). Put (8) into Equation (1) and separate the real and imaginary parts of the corresponding expression. This way, we obtain the following ODE for the real part:

$$h^2{Q}^{iv}+A_6{Q}^{2n}{Q}^{″}+A_7{Q}^{2n+1}+Q^{″}{A}_3+A_{1}Q=0,$$

(12)

where

$$A_6={\displaystyle \frac{{\delta }_4}{{\delta }_1}},A_7={\displaystyle \frac{b-k^2{c}_1{\delta }_4+c_{2}k{\delta }_8}{h^2{c}_1{\delta }_1}},A_3={\displaystyle \frac{a-6k^2{c}_1{\delta }_1+3k{c}_2{\delta }_7}{c_1{\delta }_1}}$$

(13)

and

$$A_1={\displaystyle \frac{-\omega -a{k}^2+c_1{\delta }_1{k}^4-c_2{\delta }_7{k}^3}{h^2{c}_1{\delta }_1}}.$$

The imaginary part is

$$h{Q}^{\prime }(-v-2ak+4c_1{\delta }_1{k}^3-3k^2{c}_2{\delta }_7)+Q^{‴}{h}^3(-4k{c}_1{\delta }_1+c_2{\delta }_7)+Q^{2n}{Q}^{\prime }h(-2k{c}_1{\delta }_4+c_2{\delta }_8)=0$$

(14)

We assume that

$$\begin{array}{c}\hfill \left|\begin{array}{c}v=-2ak+4c_1{\delta }_1{k}^3-3k^2{c}_2{\delta }_7\hfill \\ c_2{\delta }_7=4k{c}_1{\delta }_1\hfill \\ c_2{\delta }_8=2k{c}_1{\delta }_4,\hfill \end{array}\right.\end{array}$$

(15)

that is, the imaging part vanishes.

Our next step is to take $Q=U^{{\displaystyle \frac{1}{n}}}$ and to compute $Q^{\prime }$, $Q^{″}$, $Q^{‴}$ and $Q^{iv}$. Substituting these expressions in (12), we obtain a fourth-order ODE for $U\left(\xi \right)$:

$$\begin{array}{c}c{h}^2{\displaystyle [}{\displaystyle \frac{1}{n}}({\displaystyle \frac{1}{n}}-1)({\displaystyle \frac{1}{n}}-2)({\displaystyle \frac{1}{n}}-3){\left(U^{\prime }\right)}^4+{\displaystyle \frac{3}{n}}{({\displaystyle \frac{1}{n}}-1)}^{2}U{\left(U^{\prime }\right)}^2{U}^{″}+{\displaystyle \frac{1}{n}}({\displaystyle \frac{1}{n}}-1)U^2(3{\left(U^{″}\right)}^2+4U^{\prime }{U}^{‴})+\\ {\displaystyle \frac{1}{n}}{U}^3{U}^{iv}]+A_6{\displaystyle \frac{U^2}{n}}({\displaystyle \frac{1}{n-1}}{U}^2{\left(U^{\prime }\right)}^2+U^3{U}^{″})+A_7{U}^6+{\displaystyle \frac{A_3}{n}}({\displaystyle \frac{1}{n-1}}{U}^2{\left(U^{\prime }\right)}^2+U^3{U}^{″})+A_1{U}^4=0.\end{array}$$

(16)

According to (11), we have that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{G}^{\prime }=\sqrt{P_4\left(G\right)},G^{″}={\displaystyle \frac{1}{2}}{P}_4^{\prime }\left(G\right)\hfill \\ G^{‴}={\displaystyle \frac{1}{2}}{P}_4^{″}\left(G\right)\sqrt{P_4\left(G\right)}\hfill \\ G^{iv}={\displaystyle \frac{1}{2}}{P}_4^{‴}{P}_4+{\displaystyle \frac{1}{4}}{P}_4^{\prime }{P}_4^{″}.\hfill \end{array}\right.\end{array}$$

(17)

Formula (10) leads to

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{U}^{\prime }=\sqrt{P_4\left(G\right)}(b_1+2b_{2}G)\hfill \\ U^{″}=2b_2{P}_4+{\displaystyle \frac{1}{2}}{P}_4^{\prime }(b_1+2b_{2}G)\hfill \\ U^{‴}=\sqrt{P_4}(3P_4^{\prime }+{\displaystyle \frac{1}{2}}{P}_4^{″}(b_1+2b_{2}G))\hfill \\ U^{iv}=(b_1+2b_{2}G)({\displaystyle \frac{1}{2}}{P}_4^{‴}{P}_4+{\displaystyle \frac{1}{4}}{P}_4^{″}{P}_4^{\prime })+{\displaystyle \frac{3}{2}}{b}_2{\left(P_4^{\prime }\right)}^4+4b_2{P}_4{P}_4^{″}.\hfill \end{array}\right.\end{array}$$

(18)

Easy computations show that all the expressions containing U and its derivatives in (16) are polynomials of G of the following orders: $deg{\left(U^{\prime }\right)}^4=12$, $degU{\left(U^{\prime }\right)}^2{U}^{″}=12$, $deg{U}^2{\left(U^{″}\right)}^2=12$, $deg{U}^3{U}^{iv}=12$, $deg{U}^4{\left(U^{\prime }\right)}^2=14$, $deg{U}^5{U}^{″}=14$, $deg{U}^6=12$ and $deg{U}^2{\left(U^{\prime }\right)}^2=10$, $deg{U}^3{U}^{″}=10$, $deg{U}^4=8$. (In fact, $U^{\prime }{U}^{‴}$ is a polynomial of G of order eight).

Conclusions: On the left-hand side of (16), a polynomial of G of order 14 is present. Equalizing to zero the corresponding coefficients, we obtain 15 nonlinear algebraic equations. Moreover, we have the relations in (15). This way, we obtain 18 nonlinear algebraic equations, which are satisfied by 20 real parameters $a,b,c_{1,2}$, ${\delta }_1,{\delta }_4$, ${\delta }_7,{\delta }_8$, $k,\omega ,h,v$, $b_0$, $b_1$, $b_2$, $a_0$, $a_1$, $a_2$, $a_3$ and $a_4$. We do not write down this long and complicated system. As mentioned above, the problem consists in finding nontrivial solutions of the system. This way, we can find the special solutions (8), (9), (10) and (11) of (1). The aim of this paper was not the investigation of some nonlinear algebraic system. In the special case $a_1=a_3=0$, it is solved in [14]—see formulae (33), (35) and (37) there. Then, $U\left(\xi \right)=b_{0}sec{h}^2\left(\xi \right)$, $b_0\ne 0$ and n is odd if $b_0<0$; $\xi =h(x-vt)$. For the graph of $Q=\sqrt{U(x-t)}$, $b_0=1$ see Figure 1.

Theoretically, (1) possesses solutions of the form (8) and (9) that satisfy (16). This fourth-order autonomous ODE can be reduced to the third order via the standard change: $U^{\prime }=p\left(U\right)$. Then, $U^{″}=p{\displaystyle \frac{dp}{dU}}$, $U^{‴}=p{\left({\displaystyle \frac{dp}{dU}}\right)}^2+p^2{\displaystyle \frac{d^{2}p}{d{U}^2}}$,

$$U^{iv}=p{\left({\displaystyle \frac{dp}{dU}}\right)}^3+4p^2{\displaystyle \frac{dp}{dU}}{\displaystyle \frac{d^{2}p}{d{U}^2}}+p^3{\displaystyle \frac{d^{3}p}{d{U}^3}}.$$

Unfortunately, this difficult third-order ODE for $p=p\left(U\right)$ is nonautonomous. We point out that the solutions of (11) can be written in the form of Weierstrass’s ℘ function or via the Jacobi elliptic functions. The necessary results on the subject are included in the Appendix A.

2. We now study the concatenation model (2), searching for a solution of the form $q(x,t)=g(x-vt)e^{i\Phi }$, $\Phi =-kx+\omega t$. By putting q in (2) and splitting the real and imaginary parts (g is real-valued), we come to the following ODEs:

$$\begin{array}{c}cg(-\omega -a{k}^2+{\sigma }_1{c}_1{k}^4-c_2{\sigma }_7{k}^3)+g^3[b+c_{2}k({\sigma }_8-{\sigma }_9)+k^2{c}_1(-{\sigma }_2+{\sigma }_3-{\sigma }_4-{\sigma }_5)]+\\ +g^5{c}_1{\sigma }_6+g^{″}(a-6{\sigma }_1{k}^2{c}_1+3c_2{\sigma }_{7}k)+g^{iv}{c}_1{\sigma }_1+{\left(g^{\prime }\right)}^{2}g{c}_1({\sigma }_2+{\sigma }_3)+\\ g^{″}{g}^2{c}_1({\sigma }_4+{\sigma }_5)=k(\lambda +{\Theta }_2)g^{2m+1},\end{array}$$

(19)

$$\begin{array}{c}c{g}^{\prime }(-v-2ak+4{\sigma }_1{c}_1{k}^3-3c_2{\sigma }_7{k}^2)+g^{‴}(c_2{\sigma }_7-4{\sigma }_1{c}_{1}k)+g^2{g}^{\prime }[2c_{1}k(-{\sigma }_2-{\sigma }_4+{\sigma }_5)+\\ c_2({\sigma }_8+{\sigma }_9)]=g^{2m}{g}^{\prime }(\lambda (2m+1)+2m{\Theta }_1+{\Theta }_2).\end{array}$$

(20)

We impose the following conditions on the coefficients:

$$\begin{array}{c}\hfill \left|\begin{array}{c}{\sigma }_4+{\sigma }_5=0\hfill \\ {\sigma }_2+{\sigma }_3=0\hfill \end{array}\right.\end{array}$$

(21)

for the real part and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}v=-k(2a-4{\sigma }_1{c}_1{k}^2+3k{c}_2{\sigma }_7)\hfill \\ c_2({\sigma }_8+{\sigma }_9)=2c_{1}k({\sigma }_2+{\sigma }_4-{\sigma }_5)\hfill \\ c_2{\sigma }_7=4{\sigma }_1{c}_{1}k\hfill \\ \lambda (2m+1)+2m{\Theta }_1+{\Theta }_2=0\hfill \end{array}\right.\end{array}$$

(22)

for the imaginary one, i.e., it vanishes. This way, (19) can be rewritten as

$$P_{1}g-P_2{g}^3-P_3{g}^5-P_4{g}^{″}-P_5{g}^{iv}=k(\lambda +{\Theta }_2)g^{2m+1},$$

(23)

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{P}_1=-\omega -a{k}^2-3c_1{\sigma }_1{k}^2\hfill \\ P_2=-b-c_{2}k({\sigma }_8-{\sigma }_9)-k^2{c}_1({\sigma }_3-{\sigma }_2)\hfill \\ P_3=-c_1{\sigma }_6\hfill \\ P_4=-a+6{\sigma }_1{c}_1{k}^2-3c_2{\sigma }_{7}k=-a-6c_1{\sigma }_1{k}^2,\hfill \end{array}\right.\end{array}$$

(24)

as $c_2{\sigma }_7=4k{\sigma }_1{c}_1$.

Multiplying (23) by $g^{\prime }$ and integrating with respect to $\xi$, we obtain:

$$P_1{\displaystyle \frac{g^2}{2}}-{\displaystyle \frac{P_2}{4}}{g}^4-{\displaystyle \frac{P_3}{6}}{g}^6-{\displaystyle \frac{P_4{\left(g^{\prime }\right)}^2}{2}}-P_5(g^{\prime }{g}^{‴}-{\displaystyle \frac{1}{2}}{\left(g^{″}\right)}^2)-k{\displaystyle \frac{(\lambda +{\Theta }_2)}{2m+2}}{g}^{2m+2}={\displaystyle \frac{C}{12}}=const.,$$

i.e.,

$$6P_1{g}^2-3P_2{g}^4-2P_3{g}^6-6P_4{\left(g^{\prime }\right)}^2-g^{\prime }{g}^{‴}12P_5+6P_5{\left(g^{″}\right)}^2-{\displaystyle \frac{6k}{m+1}}(\lambda +{\Theta }_2)g^{2m+2}=C;g=g\left(\xi \right).$$

(25)

Simplifying, we come to the ODE:

$$g^{\prime }{g}^{‴}-{\displaystyle \frac{1}{2}}{\left(g^{″}\right)}^2+a{\left(g^{\prime }\right)}^2+P\left(g\right)=0,a=-{\displaystyle \frac{6P_4}{P_5}},$$

(26)

where $P\left(g\right)$ is a polynomial of g of order $2m+2$, with P even.

First, we make the standard change $g^{\prime }=p\left(g\right)$ in the autonomous fourth-order ODE and then the change $p=z^{2/3}\left(g\right)$. For the sake of simplicity, we assume that $g^{\prime }>0\Rightarrow p>0\Rightarrow z^2=p^3$. Thus, $g^{\prime }\left(\xi \right)=p\left(g\right)=z^{2/3}\left(g\right)$, and z satisfies the ODE (nonautonomous):

$${\displaystyle \frac{2}{3}}{z}^{5/3}{z}^{″}+a{z}^{4/3}+P\left(g\right)=0.$$

(27)

Equation (27) has a unique local solution (even analytic) for $z\left(0\right)=z_0\ne 0$, $z^{\prime }\left(0\right)=z_1$.

Consider now the unperturbed Equation (2). Then, ODE (23) takes the form

$$y^{iv}+a{y}^{″}=C_{1}y+C_2{y}^3+C_3{y}^5,$$

(28)

a, $C_j$, $j=1,2,3$ being real constants. One can easily see that $y=AsechBx={\displaystyle \frac{A}{chBx}}\in (0,A)$ satisfies (28) if

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_1=B^2(B^2+a)\hfill \\ C_2=-2B^2({\displaystyle \frac{10B^2}{A^2}}+{\displaystyle \frac{a}{A^2}})\hfill \\ C_3=24{\displaystyle \frac{A^4}{B^4}}.\hfill \end{array}\right.\end{array}$$

(29)

In fact, $y^{″}=-A{B}^2(-sechBx+2sec{h}^{3}Bx)$, $y^{iv}=-A{B}^4(-sechBx+20sec{h}^{3}Bx-24sec{h}^{5}Bx)$.

Moreover, $y^{\left(2k\right)}$ is an odd polynomial of $sechBx$ of order $2k+1$. Therefore, ODE (28) perturbed with $y^{2n+1}$, $n\ge 3$ does not possess an $AsechBx$ solution. From (28), it follows that

$$y^{\prime }{y}^{‴}-{\displaystyle \frac{1}{2}}{\left(y^{″}\right)}^2+{\displaystyle \frac{a{\left(y^{\prime }\right)}^2}{2}}=C_1{\displaystyle \frac{y^2}{2}}+C_2{\displaystyle \frac{y^4}{4}}+C_3{\displaystyle \frac{y^6}{6}}\equiv P_6\left(y\right),$$

(30)

as y is a soliton solution (vanishing with its derivatives at $\pm \infty$). Suppose that $A>0$, $B>0$ and $x<0\Rightarrow y^{\prime }=-ABthBxsechBx<0$. Thus, the change $y^{\prime }=z^{2/3}\Rightarrow {\left(y^{\prime }\right)}^{{\displaystyle \frac{3}{2}}}=\pm z\left(y\right)=$${\left(AB\right)}^{3/2}{(-tghBx)}^{3/2}{\left(sechBx\right)}^{3/2}$. Keeping in mind that ${\displaystyle \frac{y}{A}}=sechBx>0$, we obtain $x={\displaystyle \frac{1}{B}}Arcsech{\displaystyle \frac{y}{A}}$, $x<0$, i.e., $z\left(y\right)={\left(AB\right)}^{3/2}{[-tgh\left(Arcsech{\displaystyle \frac{y}{A}}\right)]}^{3/2}{\left[sech\left(Arcsech{\displaystyle \frac{y}{A}}\right)\right]}^{3/2}$. For $\alpha <0$ $th\alpha ={\displaystyle \frac{sh\alpha }{ch\alpha }}=-{\displaystyle \frac{\sqrt{c{h}^2\alpha -1}}{ch\alpha }}$$=-\sqrt{1-sec{h}^2\alpha }$ which implies

$$z\left(y\right)={\left(AB\right)}^{3/2}{\left({\displaystyle \frac{y}{A}}\right)}^{3/2}{(1-\left({{\displaystyle \frac{y}{A}}}^2\right))}^{3/4},0<y<A.$$

(31)

Evidently, (31) with $z\left(y\right)\sim B^{3/2}{y}^{3/2}$ for $y\to +0$, $z\left(y\right)\sim {\left(AB\right)}^{3/2}{2}^{3/4}{(1-{\displaystyle \frac{y}{A}})}^{3/4}$ for $y\to A-0$ satisfies the nonlinear nonautonomous ODE

$${\displaystyle \frac{2}{3}}{z}^{5/3}{z}^{″}+{\displaystyle \frac{a}{2}}{z}^{4/3}=P_6\left(y\right),0<y<A.$$

(32)

The polynomial $P_6\left(y\right)$ is given by (30) and (29) (see Figure 2); z is bounded, $z\left(0\right)=0$, $z\left(A\right)=0$ and $z^{\prime }\left(A\right)=-\infty$.

Remark 2.

We discuss briefly the non-existence of soliton solutions for some fourth-order ODE. The soliton $y\left(x\right)\in (0,B]$, $y^{\left(\alpha \right)}(\pm \infty )=0$, $\alpha =0,1,2,3,4$, and the integrals containing $y\left(x\right)$ from $(-\infty ,\infty )$ are convergent. Thus, let

$$y^{iv}+a{y}^{″}=P\left(y\right),P\left(0\right)=0,$$

(33)

P being polynomial of order $k\ge 1$. A necessary condition for the existence of a soliton solution of (33) is

$${\int }_{-\infty }^{\infty }P\left(y\left(x\right)\right)dx=0.$$

(34)

If $P\left(y\right)\ne 0$ for $y\ne 0$ and conserves its sign, or if the first positive root $y_0>0$, $P\left(y_0\right)=0$ is such that $0<B\le y_0$, then (34) is violated.

Consider now the ODE (12) with $h\ne 0$ and assume that $Q\left(\xi \right)$ is its soliton solution, defined as in the previous case. Integrating (12) within $(-\infty ,\infty )$, we obtain:

$${\int }_{-\infty }^{\infty }Q(-2n{\left(Q^{\prime }\right)}^2{Q}^{2n-2}{A}_6+A_7{Q}^{2n}+A_1)d\xi =0,$$

(35)

$Q\in (0,B]$. The assumption that $A_6<0$ and $A_7{Q}^{2n}+A_1\ge 0$ on $(0,B]$ and $A_6>0$, $A_7{Q}^{2n}+A_{1}x\le 0$ on $[0,B]$, respectively, implies the non-existence of a positive solution $0<Q\le B$.

3. Our next step is to study Equation (3). Its traveling-wave solution $u=\phi (x\pm t)$, $x\pm t=\xi$ satisfies the ODE

$$3c{\phi }^{‴}+\phi {\phi }^{″}+{\left({\phi }^{\prime }\right)}^2=0,c=const\ne 0,$$

(36)

i.e.,

$$3c{\phi }^{″}+\int \phi {\phi }^{″}d\xi +\int {\left({\phi }^{\prime }\right)}^{2}d\xi ={\displaystyle \frac{1}{2}}C=const$$

(37)

and consequently,

$$3c{\phi }^{″}+\phi {\phi }^{\prime }={\displaystyle \frac{1}{2}}C,$$

(38)

i.e.,

$$6c{\phi }^{\prime }+{\phi }^2=C\xi +D,D=const.$$

(39)

where (39) is a Riccati-type ODE for $C\ne 0$.

Let $C=0$. Then, for ${\phi }^2<D:$ $e^{{\displaystyle \frac{\xi +E}{3c}}\sqrt{D}}={\displaystyle \frac{\sqrt{D}+\phi }{\sqrt{D}-\phi }}$ $\Rightarrow \phi =\sqrt{D}tgh\left({\displaystyle \frac{\xi +E}{6c}}\sqrt{D}\right)$. If $D<0$, $E=const$ $\Rightarrow \phi =-\sqrt{\left|D\right|}tg\left({\displaystyle \frac{\xi +E}{6c}}\sqrt{\left|D\right|}\right)$. The case $C\ne 0$ is more complicated. As it is well known, the second-order linear ODE $y^{″}+P\left(\xi \right)y^{\prime }+Q\left(\xi \right)y=0$, $y=y\left(\xi \right)$ via the change ${\displaystyle \frac{y^{\prime }}{y}}=z$, i.e., $y=e^{\int zd\xi }$ becomes a Riccati-type ODE: $z^{\prime }+z^2+Pz+Q=0$. Conversely, the equation $y^{\prime }+p\left(\xi \right)y^2+q\left(\xi \right)y+r\left(\xi \right)=0$ via the change $y={\displaystyle \frac{1}{p\left(\xi \right)}}{\displaystyle \frac{z^{\prime }}{z}}$ becomes the linear ODE $z^{″}+z^{\prime }({\displaystyle \frac{-p^{\prime }}{p}}+q)+rpz=0$, $p\ne 0$.

In our case,

$${\phi }^{\prime }+{\displaystyle \frac{{\phi }^2}{6c}}-{\displaystyle \frac{C\xi }{6c}}=0(D=0).$$

(40)

Therefore, the change $\phi =6c{\displaystyle \frac{z^{\prime }}{z}}$ transforms (40) into

$$z^{″}\left(\xi \right)-{\displaystyle \frac{C\xi }{36c^2}}z\left(\xi \right)=0.$$

(41)

where (41) is an Airy ODE (see [19,25,27,28]). The change $\xi =\eta .k$, $k^3={\displaystyle \frac{\left|C\right|}{36c^2}}$ leads to

$$z_{\eta \eta }^{″}-sgnC\eta z\left(\eta \right)=0\Rightarrow 6c{\displaystyle \frac{z^{\prime }}{z}}={\displaystyle \frac{6kc{z}^{\prime }\left(\xi \right)}{z\left(\xi \right)}}=\phi .$$

(42)

If $C>0\Rightarrow z^{″}\left(\eta \right)-\eta z\left(\eta \right)=0$ is satisfied by the Airy function $Ai\left(\eta \right);z\left(t\right)=\sqrt{t}[A{J}_{1/3}\left({\displaystyle \frac{2}{3}}{t}^{3/2}\right)+B{J}_{-1/3}\left({\displaystyle \frac{2}{3}}{t}^{3/2}\right)]$, $t=-\eta$, $J_{\pm 1/3}$ being Bessel functions.

This is the asymptotic form of $Ai\left(\eta \right)$ that can be differentiated (see [19]):

$$Ai\left(\eta \right)\sim {\displaystyle \frac{1}{2\sqrt{\pi }}}{t}^{-1/4}{e}^{-{\displaystyle \frac{2}{3}}{t}^{3/2}},t\to +\infty$$

$$Ai\left(\eta \right)\sim {\displaystyle \frac{1}{\sqrt{\pi }}}{\left|t\right|}^{-1/4}\left[cos\right({\displaystyle \frac{2}{3}}{\left|t\right|}^{3/2}+{\displaystyle \frac{\pi }{4}}{)+O(\left|t\right|}^{-3/2}],t\to -\infty .$$

$Ai\left(\eta \right)$ possesses infinitely many zeroes ${\eta }_k$ for $t<0$ that tend to $-\infty$. ${\displaystyle \frac{A{i}^{\prime }\left(\eta \right)}{Ai\left(\eta \right)}}$ has simple poles at ${\eta }_k$, and $\left|\phi \right|$ tends to $+\infty$ in one side neighborhood of ${\eta }_k$, and it is flat in the other side neighborhood of ${\eta }_k$.

4. We are looking for a solution of (4) (see [26,29])

$$u=\phi (x+y+t\omega ),\xi =x+y+t\omega ,\omega \in {\mathbf{R}}^1.$$

Equation (4) takes the form

$${\phi }^{″}(\omega +\alpha )+{\phi }^{iv}+6{\left(\phi {\phi }^{\prime }\right)}^{\prime }=0,\phi {\phi }^{\prime }={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{d\xi }}{\phi }^2.$$

Integrating twice with constants zero, we obtain

$$\phi (\omega +\alpha )+{\phi }^{″}+3{\phi }^2=0.$$

(43)

We multiply (43) by ${\phi }^{\prime }$ and integrate with respect to $\xi$. Consequently,

$${\left({\phi }^{\prime }\right)}^2+2{\phi }^3+{\phi }^2(\omega +\alpha )-C=0,C=const.$$

(44)

It is convenient to write down $\phi =-\psi$. Thus,

$${\left({\psi }^{\prime }\right)}^2=2{\psi }^3-{\psi }^2(\omega +\alpha )+C.$$

(45)

Recall that the Weierstrass $\wp (\xi ,g_2,g_3)$ function is a solution of

$${\left({\wp }^{\prime }\right)}^2=4{\wp }^3-g_2\wp -g_3.$$

(46)

One can easily see that $W=a\rho +b$; $a\ne 0,b=const.$ is a solution of the ODE

$${\left(W^{\prime }\right)}^2={\displaystyle \frac{4}{a}}{W}^3-{\displaystyle \frac{12b}{a}}{W}^2+({\displaystyle \frac{12b^2}{a}}-a{g}_2)W-{\displaystyle \frac{4b^3}{a}}+ab{g}_2-a^2{g}_3\equiv P_3\left(W\right).$$

(47)

The parameters $a,b,g_2,g_3$ are included in (47).

Comparing (45) and (47), we conclude that $a=2$, $b={\displaystyle \frac{\omega +\alpha }{6}}$, $g_2=3b^2={\displaystyle \frac{{(\omega +\alpha )}^2}{12}}$, $C=0\Rightarrow$$g_3={\displaystyle \frac{-b^3}{2}}+{\displaystyle \frac{b}{2}}{g}_2$; ${\alpha }^2=1$. Therefore, the solution $\phi \left(\xi \right)$ of (4) is given by $\phi =-W=-2\wp -b$.

The necessary information for $\wp (\xi ,g_2,g_3)$ can be found in the Appendix A.

5. We make the change ${\phi }^{\prime }=p\left(\phi \right)$ in the ODE

$${\phi }^{‴}+a{\phi }^{\prime }=f\left(\phi \right){\phi }^{\prime },\phi =\phi \left(\xi \right),$$

(48)

which is satisfied by the traveling-wave solution $u=\phi (x+t)$ of the nonlinear PDE (5). Thus,

$${\phi }^{″}+a\phi =G\left(\phi \right)=\int f\left(\phi \right)d\phi ,$$

(49)

i.e.,

$$p{\displaystyle \frac{dp}{d\phi }}+ap=G\left(\phi \right);G^{\prime }\left(\phi \right)=f\left(\phi \right).$$

(50)

Equation (50) is an Abel-type ODE of the second kind [27,28]. Without loss of generality, we can take $a=-1$, as if $p=-r\Rightarrow r{\displaystyle \frac{dr}{d\phi }}=p{\displaystyle \frac{dp}{d\phi }}$. Thus,

$$p=p{p}^{\prime }-G\left(\phi \right)\Rightarrow p(1-p^{\prime })=-G\left(\phi \right)\Rightarrow p={\displaystyle \frac{G\left(\phi \right)}{p^{\prime }-1}}.$$

We take $p^{\prime }$ as as parameter such as $p^{\prime }=q\ne 1$, i.e.,

$$p={\displaystyle \frac{G\left(\phi \right)}{q-1}}.$$

(51)

Thus,

$${\displaystyle \frac{dp}{d\phi }}=q={\displaystyle \frac{G^{\prime }\left(\phi \right)}{q-1}}-{\displaystyle \frac{G\left(\phi \right)\frac{dq}{d\phi }}{{(q-1)}^2}}.$$

Therefore,

$${\displaystyle \frac{dq}{d\phi }}={\displaystyle \frac{G^{\prime }}{G}}(q-1)-{\displaystyle \frac{q{(q-1)}^2}{G}}.$$

(52)

If $G\left(\phi \right)=-\phi$ in (50), $ln|p^2+{\phi }^2+ap\phi |$ $-{\displaystyle \frac{2a}{\sqrt{4-a^2}}}arctg{\displaystyle \frac{2p+a\phi }{\phi \sqrt{4-a^2}}}=C=const$ $for \left|a\right|<2$, while $a^2=4\to (\phi \pm p)e^{{\displaystyle \frac{\phi }{\phi \pm p}}}=C$ (see [27]).

Keeping in mind that $G^{\prime }\left(\phi \right)=f\left(\phi \right)$, we obtain $f\left(\phi \right)=-1$ in (48). Consequently, (48) is a third-order linear ODE with constant coefficients and characteristic equation ${\lambda }^3+(a+1)\lambda =0$. Another interesting example for (50) is the equation with $G\left(\phi \right)={\displaystyle \frac{1-a^2}{4}}\phi -b{\phi }^n$ [27]. The change $z=\int {\displaystyle \frac{d\phi }{p\left(\phi \right)}}\Rightarrow \phi =\phi \left(z\right)$ leads to the ODE

$${\displaystyle \frac{d^2\phi }{d{z}^2}}+a{\displaystyle \frac{d\phi }{dz}}+{\displaystyle \frac{a^2-1}{4}}\phi +b{\phi }^n=0.$$

(53)

Another change $\phi ={\mu }^{\alpha }\eta \left(\mu \right)$, $\mu =e^z$, $\alpha ={\displaystyle \frac{1-a}{2}}$ in (53) gives us

$$\mu {\eta }^{″}\left(\mu \right)+2{\eta }^{\prime }+b{\mu }^{\alpha n-\alpha -1}{\eta }^n=0.$$

(54)

Equation (54) is an equation of Emden–Fawler type [27,28].

Usually, the Emden–Fawler equation is written as

$$x{y}^{″}\left(x\right)+2y^{\prime }+x^{\nu }{y}^n=0$$

(55)

and $n\ge 0$, $\nu >-1$.

It is known that for each $C>0$, for $x>0$, there exists only one solution of (54) such that $y\left(x\right)\to 0$ if $x\to +0$. Moreover, $y\left(x\right)>0$ for $x\in (0,{\left[C^{1-n}\nu (\nu +1)\right]}^{{\displaystyle \frac{1}{\nu +1}}})$ and $x^2{y}^{\prime }\to 0$ for $x\to +0$. If $2\nu -n+3>0$, the equation $y\left(x\right)=0$ possesses at least one positive solution $x_0$; $2\nu -n+3\le 0\Rightarrow y\left(x\right)>0$ for each $x>0$ and $y\to 0$ for $x\to +\infty$.

6. Let $u=\phi (x+t)$ in (6), $\xi =x+t$. Thus, $\phi =\phi \left(\xi \right)$,

$${\phi }^{‴}+a{\left({\left({\phi }^{\prime }\right)}^2\right)}^{\prime }=f\left(\phi \right){\phi }^{\prime },$$

i.e.,

$${\phi }^{″}+a{\left({\phi }^{\prime }\right)}^2=G\left(\phi \right)=\int f\left(\phi \right)d\phi .$$

Let ${\phi }^{\prime }=p\left(\phi \right)$. Then,

$${\displaystyle \frac{dq}{d\phi }}+2aq-2G\left(\phi \right)=0,q\left(\phi \right)=p^2\left(\phi \right).$$

(56)

Equation (56) is a linear first-order ODE. Therefore,

$$p^2=q=e^{-2a\phi }(C+2\int e^{2a\phi }G\left(\phi \right)d\phi ),$$

i.e.,

$${\displaystyle \frac{d\phi }{d\xi }}=p=\pm e^{-a\phi }\sqrt{C+\int e^{2a\phi }G\left(\phi \right)d\phi },C=const,$$

(57)

the function under the square root being non-negative on some interval $({\phi }_1,{\phi }_2)$. Taking only the positive sign, we obtain

$$F\left(\phi \right)=\int {\displaystyle \frac{e^{a\phi }d\phi }{\sqrt{C+\int e^{2a\phi }G\left(\phi \right)d\phi }}}=\xi +D,D=const.$$

(58)

If $f\left(\phi \right)$ is a polynomial of $\phi$, then $G\left(\phi \right)$ is also a polynomial. Let $degG\left(\phi \right)=k$. Evidently, $\int e^{2a\phi }G\left(\phi \right)=H\left(\phi \right)e^{2a\phi }$, the polynomial H is of the same order k. In the special case when $C=0$, we have $F\left(\phi \right)=\int {\displaystyle \frac{d\phi }{\sqrt{H\left(\phi \right)}}}=\xi +D$. Therefore, $F\left(\phi \right)$ can be computed for $k=0,1,2,3,4$ and formally, $\phi =F^{-1}(\xi +D)$. Elliptic functions appear if $k=3,4$.

To find the solution $\phi \left(\xi \right)$ of our ODE, we rewrite (58) as a definite integral:

$$F\left(\phi \right)={\int }_{{\phi }_0}^{\phi }{\displaystyle \frac{e^{a\lambda }}{\sqrt{C+{\int }_{{\phi }_0}^{\lambda }{e}^{2a\mu }G\left(\mu \right)d\mu }}}=\xi -{\xi }_0,$$

G being polynomial of order $k\ge 1$, $G\left(\mu \right)=A{\mu }^k+O\left({\mu }^{k-1}\right)$, $\mu \to \infty$; $A\ne 0$. Then, ${\int }_{{\phi }_0}^{\lambda }{e}^{2a\mu }G\left(\mu \right)d\mu =e^{2a\mu }{H\left(\mu \right)|}_{\mu ={\phi }_0}^{\mu =\lambda }$, and the polynomial $H\left(\mu \right)={\displaystyle \frac{A}{2a}}{\mu }^k+O\left({\mu }^{k-1}\right)$, $\mu \to \infty$. Thus,

$$F\left(\phi \right)={\int }_{{\phi }_0}^{\phi }{\displaystyle \frac{e^{a\lambda }d\lambda }{\sqrt{C^{\prime }+e^{2a\lambda }H\left(\lambda \right)}}},$$

where $C^{\prime }=C-e^{2a{\phi }_0}H\left({\phi }_0\right)$.

As the case $C^{\prime }=0$ is rather simple, let $C^{\prime }>0$, for example. Thus,

$$F\left(\phi \right)={\int }_{{\phi }_0}^{\phi }{\displaystyle \frac{d\lambda }{\sqrt{e^{-2a\lambda }{C}^{\prime }+H\left(\lambda \right)}}}=\xi -{\xi }_0.$$

Case 1. $a<0$, and ${\phi }_0$ is such that $T\left(\phi \right)=e^{-2a\phi }{C}^{\prime }+H\left(\phi \right)>0$ for $\phi \ge {\phi }_0$. Then, $F^{\prime }\left(\phi \right)>0$ for $\phi \ge {\phi }_0$, $F\left({\phi }_0\right)=0\Rightarrow \xi ={\xi }_0$, and $F(+\infty )=\overline{C}>0$ as the integral is convergent for $\phi =+\infty$. Consequently, $F\left(\phi \right)\in [0,\overline{C}]$, and $\overline{\xi }={\xi }_0+\overline{C}\Rightarrow F\left(\overline{\xi }\right)=+\infty$. The strictly monotonically solution $\phi \left(\xi \right)$ is such that $\phi \left({\xi }_0\right)={\phi }_0$, $\xi \in [{\xi }_0,{\xi }_0+\overline{C})$ and $\phi$ blows up for $\overline{\xi }={\xi }_0+\overline{C}$.

Case 2. $a>0$, $A>0$, and if $e^{-2a\phi }{C}^{\prime }+H\left(\phi \right)>0$ for $\phi \ge {\phi }_0$, we have that $F(+\infty )=\infty$ for $k=1,2$, while $F(+\infty )=\overline{C}>0$ for $k\ge 3$. Therefore, $k=1,2\Rightarrow F$ is monotonically increasing on $[{\xi }_0,\infty )$, $\phi (+\infty )=\infty$; $k\ge 3\Rightarrow \phi$ is defined on $[{\xi }_0,{\xi }_0+\overline{C})$ and blows up at $\overline{\xi }$; $0=C^{\prime }\Rightarrow \xi -{\xi }_0={\int }_{{\phi }_0}^{\phi }{\displaystyle \frac{d\lambda }{\sqrt{H\left(\lambda \right)}}}$. This case is studied in detail in [17] (mathematical pendulum).

Case 3. $C^{\prime }<0$, $a>0$. Consider the equation $0<-C^{\prime }{e}^{-2a\lambda }=H\left(\lambda \right)$ with k even and $A>0$. From geometrical reasons, it is clear that it possesses at least one real solution. Denote by $\overline{\phi }$ the biggest one and suppose that ${\phi }_0>\overline{\phi }$. Then, $T\left(\phi \right)>0$ for $\phi >\overline{\phi }$, $F^{\prime }\left(\phi \right)>0$ for $\phi >\overline{\phi }$, $F\left({\phi }_0\right)=0$, $F(+\infty )=\infty$ for $k=1,2$, and $F\left(\phi \right)=\overline{c}>0$ for $k\ge 3$. As $T\left(\overline{\phi }\right)=0$, it can be a simple or a multiple root. If $T^{\prime }\left(\overline{\phi }\right)\ne 0\Rightarrow F\left(\phi \right)<0$ for $\overline{\phi }<\phi <{\phi }_0$, $F\left(\overline{\phi }\right)=\overline{\overline{c}}<0$, while $F^{\prime }\left(\overline{\phi }\right)=+\infty$; ${\xi }_0+\overline{\overline{c}}=\overline{\xi }$. Therefore, the solution $\phi$ is strictly monotone on $({\xi }_0+\overline{\overline{c}},\infty )$ ($k=1,2$) and on $({\xi }_0+\overline{\overline{c}},{\xi }_0+\overline{c})$ ($k\ge 3)$. In both cases, ${\phi }^{\prime }({\xi }_0+\overline{\overline{c}})=0$, $\phi \left(\overline{\overline{\xi }}\right)=\overline{\overline{c}}$. This solution can be prolonged smoothly by the formula: $\phi (\overline{\overline{\xi }}+l)=\phi (\overline{\overline{\xi }}-l)$, $l\ge 0$. If $\phi$ is a multiple root, then $F\left(\overline{\overline{\xi }}\right)=-\infty$ etc.

7. In our last PDE, we take $u=\phi (t+x)$, $\xi =t+x$. Thus,

$${\phi }^{‴}+a{\left({\left({\phi }^{\prime }\right)}^2\right)}^{\prime }+{\phi }^{\prime }=f\left(\phi \right){\phi }^{\prime }$$

which implies, after integration with respect to $\xi$,

$${\phi }^{″}+a{\left({\phi }^{\prime }\right)}^2+\phi =G\left(\phi \right)=\int f\left(\phi \right){\phi }^{\prime }d\xi =\int f\left(\phi \right)d\phi .$$

The changes ${\phi }^{\prime }\left(\xi \right)=p\left(\phi \right)$, $p^2=q\left(\phi \right)$ lead to

$${\displaystyle \frac{dq}{d\phi }}+2aq=2(G\left(\phi \right)-\phi ).$$

(59)

The solution of the linear ODE (59) is

$$p^2=q\left(\phi \right)=e^{-2a\phi }(C+\int e^{2a\phi }(G\left(\phi \right)-\phi )d\phi ).$$

Consequently,

$$p={\displaystyle \frac{d\phi }{d\xi }}=\pm e^{a\phi }\sqrt{C+\int e^{2a\phi }(G\left(\phi \right)-\phi )d\phi }.$$

Taking the sign “+” in the previous formula, we obtain

$$F\left(\phi \right)=\int {\displaystyle \frac{e^{a\phi }d\phi }{\sqrt{C+\int e^{2a\phi }(G\left(\phi \right)-\phi )d\phi }}}=\xi +D;G\left(\phi \right)\ge \phi \mathrm{in}({\phi }_1,{\phi }_2).$$

(60)

Suppose that $C=0$ $degG=4$, and $\alpha$ is a double root of $G\left(\phi \right)-\phi =0$, i.e., $\sqrt{G\left(\phi \right)-\phi }=\pm (\phi -\alpha )\sqrt{P_2\left(\phi \right)}$; $deg{P}_2\left(\phi \right)=2$, $P_2\left(\phi \right)\ge 0$ near $\alpha$.

Evidently, the change $\phi -\alpha =\psi$ leads to $F(\psi +\alpha )=\int {\displaystyle \frac{d\psi }{\psi \sqrt{P_2(\alpha +\psi )}}}$, which is computed in [30], formula 380.111. Thus, $\phi =F^{-1}(\xi +D)$. Details on the solvability of the pendulum equation ${\phi }^{″}=f\left(\phi \right)$ can be found in [17] (see also [20,21,22,31]).

4. Discussion and Open Problems

It is well known that the second-order autonomous ODE $y^{″}=f\left(y\right)$ (the so-called equation of the mathematical pendulum, describing mechanical problems with one degree of freedom) possesses smooth solutions of the following types: periodic, solitons, kinks, blowup, etc. If $f\left(y\right)$ is a polynomial of order two or three, then $y\left(x\right)$ can be expressed by the elliptic functions. Therefore, the investigation of the fourth-order ODE

$$y^{iv}+a{y}^{″}=f\left(y\right)$$

(61)

that appears in (1) and (2) is interesting. Suppose that $f\left(y\right)$ is some polynomial as in (23). It is easy to reduce (61) to a second-order nonautonomous ODE containing only the second derivative and the unknown function. Unfortunately, the latter is not studied for the existence of solutions written into explicit form.

Our first problem is to find global solutions of (61) at least for $f\left(y\right)$ as a polynomial. For a special odd polynomial of order five, soliton solutions are found in the form $y=AsechBx$. Are there other solitons, kinks or periodic solutions of (61)? Is it possible to construct them via successive approximations or other fixed-point theorems? What about the existence of the so-called double- or multiple-soliton solutions?

Our second problem concerns the slight generalization of PDE (3), i.e., ODE (36):

$$c{\phi }^{‴}+d\phi {\phi }^{″}+{\left({\phi }^{\prime }\right)}^2=0,c\ne 0,d\ne o.$$

(62)

Can we find a global solution of (62) into explicit form for the arbitrary c and d? If not, can we prove the existence of global or blowup solutions? What about the properties of $\phi \left(x\right)$ (zeroes, possible asymptotes at the end points of the existence interval)?

5. Conclusions

• First, we considered two complex-valued nonlinear PDEs of the fourth order in two variables arising in nonlinear fiber optics looking for solutions of the form $u=Q{e}^{i\Phi }$, the real amplitude Q being of traveling-wave type and the phase $\Phi$ being a real linear function. Moreover, $Q=U^{{\displaystyle \frac{1}{n}}}$, while U was a quadratic polynomial of some (unknown) elliptic function of Jacobi or Weierstrass type. The existence of such u was reduced to the solvability of some nonlinear algebraic system. If it possessed a nontrivial solution, the function u was found. Certainly, it can be a soliton, periodic or blowup one. The amplitude Q satisfied a fourth-order semilinear autonomous ODE that could be brought to some second order nonautonomous ODE. The soliton solutions of the fourth-order ODE gave rise to some special explicit-form solutions of a second-order ODE.
• The solutions of (3) taken from hydrodynamics were reduced to some Riccati-type ODE (traveling-wave solution). The latter was solved for the two possible cases: autonomous and nonautonomous. It is interesting to point out that in the nonautonomous case, the solution was expressed by the Airy function ${\displaystyle \frac{A{i}^{\prime }\left(\lambda \right)}{Ai\left(\lambda \right)}}$.

The traveling-wave solutions of the KP equation were written by the Weierstrass $\wp (\xi ,g_2,g_3)$ function $A\wp +B$, $A,B=const.$ The traveling-wave solutions of (5) satisfied an Abel-type ODE of the second kind. In some cases, the Abel equation was reduced to the Emden–Fawler second-order ODE for which a precise theory was developed.

The solutions of PDEs (6) and (7) were written into an integral form as they satisfied ODEs with separate variables. They were strictly monotonic and usually blow up.

In Appendix A, we proposed a rather complete survey on the solvability of ${\left(y^{\prime }\right)}^2=P_4\left(y\right)$, $deg{P}_4=4$ and ${\left(y^{\prime }\right)}^2=P_3\left(y\right)$, $deg{P}_3=3$, expressing the corresponding solutions $y\left(x\right)$ by the Weierstrass or Jacobi elliptic functions.