Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy

Abstract

This study concerns the research of rational solutions to the hierarchy of the nonlinear Schrödinger equation. In particular, we are interested in the equation of order 4. Rational solutions to the fourth equation of the NLS hierarchy are constructed and explicit expressions of these solutions are given for the first order. These solutions depend on multiple real parameters. We study the associated patterns of these solutions in the $(x,t)$ plane according to the different values of their parameters. This work allows us to highlight the phenomenon of rogue waves, such as those seen in the case of lower-order equations such as the nonlinear Schrödinger equation, the mKdV equation, or the Hirota equation.

1. Introduction

We consider the fourth equation of the NLS hierarchy of order 4 $\left(NLS4\right)$, which can be written as

$$u_t+u_{5x}+{10\left|u\right|}^2{u}_{3x}+20u_{2x}{u}_x\overline{u}+10\left(\right|u_x{|}^2{u)}_x+30{\left|u\right|}^4{u}_x=0,$$

(1)

with, as usual, the subscript meaning the partial derivatives and $\overline{u}$ meaning the complex conjugate of u.

As the NLS equation [1,2,3,4,5,6,7,8,9,10] is the first equation of this hierarchy, the mKdV equation [11,12,13,14] is the second, and the LPD equation [15,16,17,18,19,20,21] is the third, the $\left(NLS4\right)$ equation is the fourth equation of this NLS hierarchy.

Recent works have been published on this subject. We can quote [22], where the global existence of smooth solutions to the initial value problem in the Lax hierarchy of the nonlinear Schrodinger equation is established. We can also mention [23], where rogue wave solutions on a periodic background were constructed for the fourth-order nonlinear Schrödinger (NLS) equation in terms of Jacobi elliptic functions.

We recall some basic facts about the first three equations of this NLS hierarchy and the phenomenon of rogue waves.

The first equation is the well-known, one-dimensional focusing nonlinear Schrödinger equation (NLS), which can be written in the following form:

$$i{v}_t+v_{xx}+2{\left|v\right|}^{2}v=0,$$

(2)

with the subscripts having the same meaning as described in the beginning.

We do not give details on the history of the research on the impressive numbers in this equation but limit ourselves to precursors.

Zakharov and Shabat [1,24] solved the equation in 1972 using the inverse scattering method. Periodic and almost-periodic algebro-geometric solutions were first constructed by Its and Kotlyarov [25,26] in 1976.

Concerning the phenomenon of rogue waves, relatively recent results can be cited. Different types of representations of these solutions have been given in terms of Fredholm determinants, in terms of Wronskians, and in terms of degenerate determinants of the order N. Families of quasi-rational solutions $v_N$ have been explicitly constructed until order $N=13$ [27].

These solutions appear as deformations of the Peregrine breather $P_N$, as the last one can be obtained when all parameters are equal to 0; thus, we also obtain three particular representations of the Peregrine breather $P_N$. The $P_N$ breather can be expressed as a quotient of two polynomials of degree $N(N+1)$ of the variables x and t; the maximum of the modulus of the $P_N$ breather is equal to $2N+1$. The classification of patterns of the modulus of the deformations of the $P_N$ breather within the function of the parameters has been realized, highlighting the phenomenon of rogue waves [27]. The second equation of the hierarchy is the modified Korteweg–De Vries (mKdV) equation, which can be written in the form

$$u_t-6u^2{u}_x+u_{xxx}=0,$$

(3)

with $u_t={\partial }_{t}u$, $u_x={\partial }_{x}u$, and $u_{xxx}={\partial }_x^{3}u$.

The mKdV equation has many applications in various fields, such as modelling supercontinuum generation in optical fibres [28] or describing pulses consisting of a few optical cycles [29].

A lot of studies have been carried out to construct solutions to the mKdV equation. In particular, Hirota [30] constructed the exact soliton of this equation in 1972. The inverse scattering method was used by Tanaka [11] to solve the mKdV equation. Rational solutions were constructed by using particular types of polynomials and the bilinear Hirota method in [31].

The third equation of the NLS hierarchy is the Lakshmanan–Porsezian–Daniel (LPD), which can be written as

$$i{u}_t+u_{xxxx}+{8\left|u\right|}^2{u}_{xx}+2u^2{\overline{u}}_{xx}+6u_x^2\overline{u}+4u|u_x{|}^2+6{\left|u\right|}^{4}u=0,$$

(4)

where the subscripts mean the partial derivatives.

This model was introduced in the context of a Heisenberg spin chain [15]. This equation has spatio-temporal dispersion as well as group velocity dispersion and is used to obtain and describe solitons, which are some of the main focal points in the fields of mathematical physics, optical fibres, and nonlinear optics.

Many mathematical methods have been used to study this equation, including, in particular, the method of undetermined coefficients, which was applied in [32] to look for bright, dark, and singular soliton solutions.

Here, we are interested in the equation of order 4, and we construct the first orders of its solutions, which, to the best of our knowledge, have not yet been found.

In this paper, explicit solutions to the fouth equation of the NLS hierarchy have been completely expressed and depend on multiple real parameters. In order N, these solutions depend on $2N-2$ real parameters. The evolution of these solutions in function of different real parameters has been studied for the first three orders of this hierarchy. These solutions are important because they allow us to highlight the phenomenon of the appearance of rogue waves as a function of these parameters and in the case of the simple nonlinear Schrödinger equation.

The general result will be published at a later date as it has not yet been proven. We therefore limit ourselves to giving the expression of a few solutions which, on the other hand, are correct and have been verified.

The formulation of these solutions is further enriched by utilizing techniques found in special function theory, particularly those discussed in works such as [33]. These contributions to the generalized forms of special functions and their application in solving differential equations are significant in this context, providing insights into how these mathematical structures aid in constructing solutions to integrable systems.

2. Rational Solutions to Order 1 of the NLS4 Equation

Theorem 1.

The function $v(x,t)$ is defined by

$$\begin{array}{c}\hfill v(x,t)=1-{\displaystyle \frac{4}{1+{\left(2x-60t\right)}^2}}\end{array}$$

(5)

is a solution to $\left(NLS4\right)$, Equation (1):

$$u_t+u_{5x}+{10\left|u\right|}^2{u}_{3x}+20u_{2x}{u}_x\overline{u}+10\left(\right|u_x{|}^2{u)}_x+30{\left|u\right|}^4{u}_x=0.$$

Proof. 

It is sufficient to replace the expression of the solution given by (5) and check that (1) is verified. □

Three views of solutions with different areas of representationare given in Figure 1.

We obtain a smooth solution to Equation (1).

On the line $\left\{\right(x,t)/x-6t=0\}$, the solution u is constantly equal to $-3$.

When $x\to \infty$ or $t\to \infty$, the solution u is such that $u(x,t)\to 1$.

3. Rational Solutions to Order 2 of the NLS4 Equation Depending on Two Real Parameters

Theorem 2.

The function $v(x,t)$ is defined by

$$\begin{array}{c}\hfill v(x,t)={\displaystyle \frac{n(x,t)}{d(x,t)}}\end{array}$$

(6)

$$\begin{gathered} \mathit{with}n(x,t)=-(-64x^6+2304b_1{x}^5+11520t{x}^5+768i{a}_1{x}^4-864000t^2{x}^4+144x^4-768{a_1}^2{x}^4 \\ -345600b_{1}t{x}^4-34560{b_1}^2{x}^4-4992b_1{x}^3+276480{b_1}^3{x}^3+34560000t^3{x}^3+20736000b_1{t}^2{x}^3- \\ 92160i{a}_{1}t{x}^3-18432i{a}_1{x}^3{b}_1+4147200{b_1}^{2}t{x}^3+92160t{a_1}^2{x}^3-32640t{x}^3+18432b_1{a_1}^2{x}^3- \\ 777600000t^4{x}^2-3072{a_1}^4{x}^2-186624000{b_1}^2{t}^2{x}^2-165888{b_1}^2{a_1}^2{x}^2+58752{b_1}^2{x}^2+2160000t^2{x}^2 \\ +4147200i{a}_1{t}^2{x}^2-4147200t^2{a_1}^2{x}^2-24883200{b_1}^{3}t{x}^2+165888i{a}_1{b_1}^2{x}^2-622080000b_1{t}^3{x}^2+ \\ 6144i{a_1}^3{x}^2-1244160{b_1}^4{x}^2-1152i{a}_1{x}^2+5760{a_1}^2{x}^2+725760b_{1}t{x}^2+1658880i{a}_1{b}_{1}t{x}^2 \\ +180x^2-1658880b_{1}t{a_1}^2{x}^2+3732480000{b_1}^2{t}^{3}x-73728i{a_1}^{3}x{b}_1-368640i{a_1}^{3}tx+36864b_1{a_1}^{4}x+ \\ 74649600{b_1}^{4}tx-5184000{b_1}^{2}tx-663552i{a}_{1}x{b_1}^3-161280t{a_1}^{2}x-290304{b_1}^{3}x+82944000t^3{a_1}^{2}x \\ +184320t{a_1}^{4}x+2985984{b_1}^{5}x-4608i{a}_{1}x{b}_1-49766400i{a}_{1}x{b}_1{t}^2+9331200000t^{5}x+746496000 \\ {b_1}^3{t}^{2}x-5616b_{1}x-57024000t^{3}x+9953280{b_1}^{2}t{a_1}^{2}x-115200i{a}_{1}tx-45360tx+663552{b_1}^3{a_1}^{2}x \\ -50688b_1{a_1}^{2}x+49766400b_1{t}^2{a_1}^{2}x-82944000i{a}_1{t}^{3}x-9953280i{a}_{1}x{b_1}^{2}t+9331200000b_1{t}^{4}x- \\ 30067200b_1{t}^{2}x-89579520{b_1}^{5}t-27993600000{b_1}^2{t}^4-110592{b_1}^2{a_1}^4-995328{b_1}^4{a_1}^2- \\ 7464960000{b_1}^3{t}^3+221184i{a_1}^3{b_1}^2+531360000t^4-1119744000{b_1}^4{t}^2+12288i{a_1}^5+383616000 \\ {b}_1{t}^3+1872{a_1}^2+12026880{b_1}^{3}t+102643200{b_1}^2{t}^2-622080000t^4{a_1}^2-45+96768{b_1}^2{a_1}^2- \\ 345600t^2{a_1}^2+414720b_{1}t{a_1}^2-46656000000t^6+5529600i{a_1}^3{t}^2-720i{a}_1-2985984{b_1}^6-4096 \\ {a_1}^6+8448{a_1}^4+518400{b_1}^4-19906560{b_1}^{3}t{a_1}^2-497664000b_1{t}^3{a_1}^2-1105920b_{1}t{a_1}^4-149299200 \\ {b_1}^2{t}^2{a_1}^2+277200t^2+995328i{a}_1{b_1}^4+1536i{a_1}^3+19906560i{a}_1{b_1}^{3}t+149299200i{a}_1{b_1}^2{t}^2+ \\ 497664000i{a}_1{b}_1{t}^3+69120i{a}_1{b_1}^2+4492800i{a}_1{t}^2+2211840i{a_1}^3{b}_{1}t+18000{b_1}^2-55987200000b_1{t}^5 \\ +1244160i{a}_1{b}_{1}t-2764800t^2{a_1}^4+622080000i{a}_1{t}^4+191520b_{1}t){\mathrm{e}}^{2i{a}_1} \mathit{and} d(x,t)=-(-64x^6+ \\ 2304b_1{x}^5+11520t{x}^5+768i{a}_1{x}^4-864000t^2{x}^4+144x^4-768{a_1}^2{x}^4-345600b_{1}t{x}^4-34560{b_1}^2{x}^4 \\ -4992b_1{x}^3+276480{b_1}^3{x}^3+34560000t^3{x}^3+20736000b_1{t}^2{x}^3-92160i{a}_{1}t{x}^3-18432i{a}_1{x}^3{b}_1+ \\ 4147200{b_1}^{2}t{x}^3+92160t{a_1}^2{x}^3-32640t{x}^3+18432b_1{a_1}^2{x}^3-777600000t^4{x}^2-3072{a_1}^4{x}^2- \\ 186624000{b_1}^2{t}^2{x}^2-165888{b_1}^2{a_1}^2{x}^2+58752{b_1}^2{x}^2+2160000t^2{x}^2+4147200 \\ i{a}_1{t}^2{x}^2-4147200t^2{a_1}^2{x}^2-24883200{b_1}^{3}t{x}^2+165888i{a}_1{b_1}^2{x}^2-622080000b_1{t}^3{x}^2+6144i{a_1}^3{x}^2- \\ 1244160{b_1}^4{x}^2-1152i{a}_1{x}^2+5760{a_1}^2{x}^2+725760b_{1}t{x}^2+1658880i{a}_1{b}_{1}t{x}^2+180x^2 \\ -1658880 b_{1}t{a_1}^2{x}^2+3732480000{b_1}^2{t}^{3}x-73728i{a_1}^{3}x{b}_1-368640i{a_1}^{3}tx+36864b_1{a_1}^{4}x+74649600{b_1}^{4}tx- \\ 5184000{b_1}^{2}tx-663552i{a}_{1}x{b_1}^3-161280t{a_1}^{2}x-290304{b_1}^{3}x+82944000t^3{a_1}^{2}x+184320t{a_1}^{4}x+ \\ 2985984{b_1}^{5}x-4608i{a}_{1}x{b}_1-49766400i{a}_{1}x{b}_1{t}^2+9331200000t^{5}x+746496000{b_1}^3{t}^{2}x-5616b_{1}x- \\ 57024000t^{3}x+9953280{b_1}^{2}t{a_1}^{2}x-115200i{a}_{1}tx-45360tx+663552{b_1}^3{a_1}^{2}x-50688b_1{a_1}^{2}x+ \\ 49766400b_1{t}^2{a_1}^{2}x-82944000i{a}_1{t}^{3}x-9953280i{a}_{1}x{b_1}^{2}t+9331200000b_1{t}^{4}x-30067200b_1{t}^{2}x- \\ 89579520{b_1}^{5}t-27993600000{b_1}^2{t}^4-110592{b_1}^2{a_1}^4-995328{b_1}^4{a_1}^2-7464960000{b_1}^3{t}^3+221184 \\ i{a_1}^3{b_1}^2+531360000t^4-1119744000{b_1}^4{t}^2+12288i{a_1}^5+383616000b_1{t}^3+1872{a_1}^2+12026880 \\ {b_1}^{3}t+102643200{b_1}^2{t}^2-622080000t^4{a_1}^2-45+96768{b_1}^2{a_1}^2-345600t^2{a_1}^2+414720b_{1}t{a_1}^2- \\ 46656000000t^6+5529600i{a_1}^3{t}^2-720i{a}_1-2985984{b_1}^6-4096{a_1}^6+8448{a_1}^4+518400{b_1}^4- \\ 19906560{b_1}^{3}t{a_1}^2-497664000b_1{t}^3{a_1}^2-1105920b_{1}t{a_1}^4-149299200{b_1}^2{t}^2{a_1}^2+277200t^2+995328 \\ i{a}_1{b_1}^4+1536i{a_1}^3+19906560i{a}_1{b_1}^{3}t+149299200i{a}_1{b_1}^2{t}^2+497664000i{a}_1{b}_1{t}^3+69120i{a}_1{b_1}^2+ \\ 4492800i{a}_1{t}^2+2211840i{a_1}^3{b}_{1}t+18000{b_1}^2-55987200000b_1{t}^5+1244160i{a}_1{b}_{1}t-2764800t^2{a_1}^4+ \\ 622080000i{a}_1{t}^4+191520b_{1}t){\mathrm{e}}^{2i{a}_1} \mathit{is} a \mathit{solution} \mathit{to} \mathit{the} (\mathrm{NLS4}) \mathit{Equation} \left(1\right). \end{gathered}$$

Proof. 

By replacing the expression of the solution given by (6), we check that Relation (1) is verified. □

Contrary to the previous section, in all the following, we represent the modules of the solutions in the coordinate plane $(x;t)$. They are given in Figure 2 and Figure 3.

When $x\to \infty$ or $t\to \infty$, the solution u is such that $u(x,t)\to e^{2i{a}_1}$ or 1 if all parameters are equal to 0.

4. Rational Solutions to Order 3 of the NLS4 Equation

As the solution depending on four real parameters is too long, we present it in the Appendix A. We present here the solution without parameters.

Theorem 3.

The function $v(x,t)$ is defined by

$$\begin{array}{c}\hfill v(x,t)={\displaystyle \frac{n(x,t)}{d(x,t)}}\end{array}$$

(7)

$$\begin{gathered} \mathit{with} n(x,t)=4096x^{12}-1474560t{x}^{11}-18432x^{10}+243302400t^2{x}^{10}+10444800t{x}^9- \\ 24330240000t^3{x}^9+1642291200000t^4{x}^8-57600x^8-2073600000t^2{x}^8-78829977600000t^5{x}^7+ \\ 7925760t{x}^7+218972160000t^3{x}^7-172800x^6-14282956800000t^4{x}^6+671846400t^2{x}^6+ \\ 2759049216000000t^6{x}^6-183638016000t^3{x}^5+614515507200000t^5{x}^5-70946979840000000t^7{x}^5+ \\ 18938880t{x}^5-17871114240000000t^6{x}^4-14000256000t^2{x}^4+226800x^4+14251852800000t^4{x}^4+ \\ 1330255872000000000t^8{x}^4+349360128000000000t^7{x}^3-566590464000000t^5{x}^3- \\ 17736744960000000000t^9{x}^3+1248998400000t^3{x}^3-137808000t{x}^3+17448566400t^2{x}^2- \\ 39822451200000t^4{x}^2-4414030848000000000t^8{x}^2+113400x^2+12583683072000000t^6{x}^2+ \\ 159630704640000000000t^{10}{x}^2-1070915904000t^{3}x-81972000tx+477570816000000t^{5}x+ \\ 32651735040000000000t^{9}x-149015531520000000t^{7}x-870712934400000000000t^{11}x- \\ 4419424800t^2-107629793280000000000t^{10}+2176782336000000000000t^{12}+736175692800000000 \\ {t}^8+41194539360000t^4-14175-1203911424000000t^6 \mathrm{and} d(x,t)=4096x^{12}-1474560t{x}^{11}+ \\ 6144x^{10}+243302400t^2{x}^{10}+3072000t{x}^9-24330240000t^3{x}^9+1642291200000t^4{x}^8+34560x^8- \\ 1078272000t^2{x}^8-78829977600000t^5{x}^7-14192640t{x}^7+139345920000t^3{x}^7+149760x^6- \\ 10102579200000t^4{x}^6+2994278400t^2{x}^6+2759049216000000t^6{x}^6-322983936000t^3{x}^5+ \\ 464021913600000t^5{x}^5-70946979840000000t^7{x}^5-30274560t{x}^5-14108774400000000t^6{x}^4+ \\ 2298240000t^2{x}^4+54000x^4+19477324800000t^4{x}^4+1330255872000000000t^8{x}^4+ \\ 284862873600000000t^7{x}^3-692001792000000t^5{x}^3-17736744960000000000t^9{x}^3-438082560000 \\ {t}^3{x}^3+37756800t{x}^3-4054665600t^2{x}^2+33371481600000t^4{x}^2-3688436736000000000t^8{x}^2+ \\ 48600x^2+14464852992000000t^6{x}^2+159630704640000000000t^{10}{x}^2+825598656000t^{3}x- \\ 20023200tx-966898944000000t^{5}x+27814440960000000000t^{9}x-165139845120000000t^{7}x- \\ 870712934400000000000t^{11}x+5359413600t^2-93117911040000000000t^{10}+ \\ 2176782336000000000000t^{12}+796641868800000000t^8-948270240000t^4+2025+ \\ 9565786368000000t^6 \mathit{is} a \mathrm{solution} \mathrm{to} \mathrm{the} \left(NLS4\right) \mathit{Equation} (1). \end{gathered}$$

Proof. 

We have to check that Relation (1) is verified when we replace the expression of the solution given by (A1). □

In the following, we give the patterns of the modules of the solutions based on different parameter values in Figure 4, Figure 5, Figure 6 and Figure 7.

In the case where all parameters are equal to 0, the solution u can be written in the form $$\begin{gathered} u(x,t)=1-{\displaystyle \frac{N(x,t)}{D(x,t)}}, \mathrm{with} N(x,t)=16200+11578982400t^2+24(-675-69120000t^2){(2x-60t)}^2 \\ +24(1209600t+22118400000t^3)(2x-60t)+360{(2x-60t)}^8+5040{(2x-60t)}^6 \\ +276480t{(2x-60t)}^5+24(-450+34560000t^2){(2x-60t)}^4+19353600t {(2x-60t)}^3+24{(2x-60t)}^{10} \\ \mathrm{and} D(x,t)=2024+4802457600t^2+21233664000000t^4+(12144-91238400t^2){(2x-60t)}^2+ \\ 9600t{(2x-60t)}^9-46080t{(2x-60t)}^7+(-8553600t+345047040000t^3)(2x-60t)+120{(2x-60t)}^8+ \\ (2320+13824000t^2){(2x-60t)}^6-103680t {(2x-60t)}^5+(3360-13824000t^2){(2x-60t)}^4+ \\ (5529600t-44236800000t^3){(2x-60t)}^3+{(1+{(2x-60t)}^2)}^6. \end{gathered}$$

When $x\to \infty$ or $t\to \infty$, the solution u is such that $u(x,t)\to e^{2i(a_1-3a_2+10b_2)}$ or 1 if all parameters are equal to 0.

Contrary to the other equations belonging to this NLS hierarchy, such as the NLS equation [27], the mKdV equation [31], or the Lakshmanan–Porsezian–Daniel equation [34], there are no specific configurations which appear as a function of the different values of these parameters.

5. Conclusions

Rational solutions to the NLS4 equation have been given for its first few orders. For the first order, we get smoother solutions.

We observe the appearance of the lines $\left\{\right(x;t)/2x-60t=c\}$ in these solutions.

When $x\to \infty$ or $t\to \infty$, all these solutions u conform to $u(x,t)\to 1$.

In all these N-order solutions, we obtain a quotient of a polynomial of degree $N(N+1)$ in x and t for the numerator using a polynomial of degree $N(N+1)$ in x and t for the denominator.

This explains the asymptotic behaviour of these rational solutions.