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Time‐localized solutions for some soliton equations
Yasuhiro Ohta

Department of Mathematics, Kobe University
1Introduction“Rogue wave” is one of the recent topics in nonlinear water waves, which describes sudden
excitation of big amplitude in calm or moderate state of waves. The important character‐istic of rogue waves is that they appear from nowhere and disappear without a trace [ 1].

Such type of solutions have been constructed for various soliton equations among which

the focusing nonlinear Schrödinger (NLS) equation,

iu_{t}=u_{xx}+ \frac{1}{2}|u|^{2}u ,(1)is the most famous and fundamental example. There are two types of rogue wave solutions

for NLS eq. (1). One is the Akhmediev breather given by rational function of exponentials,

and the other is the Peregrine rogue wave and its higher order ones which are rational

solutions multiplied by a gauge factor [1]-[3] . Both of these solutions tend to the plane
wave with finite amplitude in the limit tarrow\pm\infty , and nontrivial wave structures appear
only in a limited time region. In this sense the rogue waves are the solutions which arelocalized in time.

The rogue wave solutions have been studied mostly for soliton equations of complex
field, such as the NLS equation, vector NLS equation, Yajima‐Oikawa equation, Davey‐Stewartson equation and so on. Especially the algebraic structures of rational rogue wave

solutions have been revealed in detail [4],[5]. On the other hand, rogue waves for equationsof real dependent variables are not well‐studied relatively. In this paper we consider
solutions localized in time for equations of real variables. The Boussinesq equation is oneof the real variable equations which admit time‐localized solutions. It should be notedthat the rational rogue wave solutions for the Boussinesq equation were constructed in[6]-[8] . In section 2 we first classify the Boussinesq equation into several types according to
the signs of coefficients and next consider the lowest order exponential rational solutions
and rational solutions for all types. In some cases we obtain the Akhmediev breathers andrational rogue waves, and in some other cases singular solutions are derived. In section
3 the determinant expressions of multi‐breather and higher order rational solutions arestudied. Finally we demonstrate a simple way of constructing equations and their time‐
localized solutions by using the bilinear method in section 4.

2

Boussinesq equations and their basic solutions

The Boussinesq equation is written as\pm u_{tt}=u_{xxxx}+3(u^{2})_{xx}+\sigma u_{xx} ,(2)where \sigma is a constant. We can not change the sign \pm in left‐hand side by scaling ofvariables. We call eq. (2) the Boussinesq I equation if the sign is + , and we call it the129
Boussinesq II equation if the sign is −, because they are derived from the KPI and KPII
equations,(u_{t}+u_{xxx}+6uu_{x})_{x}=\pm u_{yy},respectively, by the reductionu_{t}=\sigma u_{x}and rewriting y by t . We can change \sigma in eq. (2)by constant shift uarrow u+ const, thus the value of \sigma is linked with the boundary conditionof u . Let us normalize the boundary value of u by
uarrow 0

astarrow\pm\infty .

(3)

Then the absolute value of \sigma can be changed by scaling of variables but the sign of \sigma isunchangeable. So we can take \sigma=\pm 1 or 0 without loss of generality.
The Boussinesq I equation is transformed into the bilinear form,(D_{x}^{4}+aD_{x}^{2}-D_{t}^{2})f\cdot f=0 ,(4)

through the dependent variable transformation,

u=(2\log f)_{xx} .

(5)

f=1+e^{ikx+\omega t}+e^{-ikx+\omega t}+Ae^{2\omega t} ,(6)

Substituting the perturbative form,

into the bilinear eq. (4) and determining \omega and A , we obtain the Akhmediev breathersolution,

f-a\cosh\omega t-\cos kx ,(7)

\omega=k\sqrt{k^{2}-\sigma}, a=\sqrt{A}=\sqrt{\frac{4k^{2}-\sigma}{k^{2}- \sigma} ,where k is an arbitrary constant satisfying k^{2}>\sigma . Here=means equivalence by multi‐

plication of an exponential factor and constant shifts of independent variables. From (5)we getu=2k^{2} \frac{a\cosh\omega t\cos kx-1}{(a\cosh\omega t-\cos kx)^{2} ,(8)which is regular and localized in time. Therefore the Boussinesq I equation admits theAkhmediev breather solution for any \sigma . Time evolution of the solution (8) is shown in
Fig. 1. We note that the solution is symmetric in time reverse.

(a)Figure 1:\sigma=-1.

t=\pm 6(b)t=\pm 2(c)t=\pm 0.7

(x, u) ‐plots of Akhmediev breather solution (8) with

k=1(d)

t=0for Boussinesq I equation with

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The fundamental rational rogue wave solution is obtained by the limit
above solution. The limit can be taken for \sigma=-1 and we get

karrow 0 in thef arrow\wedge t^{2}+x^{2}+3, u=4\frac{t^{2}-x^{2}+3}{(t^{2}+x^{2}+3)^{2} .
This is a regular and time‐localized solution of Boussinesq I equation with \sigma=-1 . Fig. 2
shows time evolution of the above rogue wave solution which has time reversal symmetry.
In the case of \sigma=+1 , we can not take the limit karrow 0 because of the condition k^{2}>\sigma,

(a)

(b)

t=\pm 20t=\pm 6(c)(d)

t=\pm 2t=0

Figure 2: (x, u) ‐plots of rational rogue wave solution for Boussinesq I equation with \sigma=-1.

however we can start from the ansatz,f=t^{2}+Ax^{2}+B ,(9)and determine constants A and B from the bilinear eq. (4). Then we obtain a singularrational solution for\sigma=+1,f=t^{2}-x^{2}+3, u=-4 \frac{t^{2}+x^{2}+3}{(t^{2}-x^{2}+3)^{2} ,which describes repulsive interaction of a pair of singularities. Time evolution of thissolution is shown in Fig. 3. For \sigma=0 , it is easy to check that there is no rational solution(a)t=\pm 15(b)t=\pm 10(c)(d)

t=\pm 5t=0Figure 3: (x, u) ‐plots of singular rational solution for Boussinesq I equation with \sigma=+1.

of the form of (9).
By using the variable transformation (5), the Boussinesq II equation is transformedinto(D_{x}^{4}+aD_{x}^{2}+D_{t}^{2})f\cdot f=0 .
Starting with the ansatz of perturbative form (6) and determining(10)\omegaandA , we find thata blowing up breather solution exists for \sigma>0 and there is no time‐localized breather

solution for \sigma\leq 0 . For \sigma>0 , we have a singular breather solution (7) and (8) with\omega=k\sqrt{\sigma-k^{2} , a=\sqrt{\frac{\sigma-4k^{2} {\sigma-k^{2} },

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where k is a constant satisfying 4k^{2}<\sigma . This breather is a finite‐time blowing up solution
but still localized in time, that is, singularities appear only in a finite interval around t=0

and the vanishing condition (3) is satisfied. This solution is shown in Fig. 4.

(a)t=\pm 2.5(b)t=\pm 228

(c)

(d)

t=\pm 2.2

t=0

Figure 4: (x, u) ‐plots of finite‐time blowing up breather solution with k= \frac{1}{3} for Boussinesq II equationwith \sigma=+1.

For \sigma=+1 , by taking the limit karrow 0 in the above blowing up breather, we get thefinite‐time blowing up rational rogue wave solution,

f=t^{2}+x^{2}-3, u=4 \frac{t^{2}-x^{2}-3}{(t^{2}+x^{2}-3)^{2} ,which is localized in time and shown in Fig. 5. For \sigma=-1 , although the breather solution(a)Figure 5:t=\pm 2.5(b)t=\pm 175

(c)

(d)

t=\pm 15t=0(x, u) ‐plots of finite‐time blowing up rational rogue wave solution for Boussinesq II equation

with \sigma=+1.

doesn’t exist, we have a rational solution. In fact substituting (9) into the bilinear formof Boussinesq II equation (10) and determining A and B , we get a singular solution,

f=t^{2}-x^{2}-3, u=-4 \frac{t^{2}+x^{2}-3}{(t^{2}-x^{2}-3)^{2} .

This solution is not localized in time and describes the annihilation and creation of a pairof singularities (Fig. 6). There is no rational solution of the form of (9) for \sigma=0.
3

Determinant structure of breather and rational solutions

For the Boussinesq I equation, the Akhmediev breathers can be superposed and the N‐breather solution is given in terms of the Gram determinant,

f=1 \leq ij\leq Nd,et(\int_{-\infty}^{x}(e^{\xi_{\dot{x} +e^{\eta_{x} \cdot) (e^{\xi_{j}^{*} +e^{\eta_{j}^{*} )dx)

=1\leqij\leqNd,et(\frac{e^{\xi_{}+\xi_{j}^{*} {p_{\dot{i}+p_{j}*+ \frac{e^{\xi_{x}+\eta_{j}^{*} {p_{i}+p_{j}+\frac{e^{\eta_{i}+\xi_{j}^{*} {p_{i}+p_{j}*+\frac{e^{\eta_{i}+\eta_{j}^{*} {p_{\dot{i}+p_{j}*)

,

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(a)

(b)

t=\pm 10(c)t=\pm 5(d)t=\pm 175

t=0

Figure 6: (x, u) ‐plots of singular rational solution for Boussinesq II equation with \sigma=-1.

\xi_{j}=p_{j}x+i\sqrt{3}p_{\dot{j}}^{2}t+\xi_{j0}, \eta_{j}=p_{j}^{*}x+ i\sqrt{3}p_{j}^{*2}t+\eta_{j0},where p_{j}, \xi_{j0} and \eta_{j0} are arbitrary complex constants satisfying the reduction condition,p_{j}^{2}+p_{j}p_{j}^{*}+p_{\dot{j} ^{*2}=- \frac{\sigma}{4},N.

and {\rm Re} p_{j}>0 for j=1,2,

Here

*

denotes complex conjugate. Since f is thedeterminant of positive definite Hermitian matrix,uin (5) gives regular solution. It

is straightforward to prove that the above f actually satisfies the bilinear Boussinesq I

equation (4) by using the Laplace expansion technique.
The higher order rational solutions are also expressed in the determinant form,

f=|\begin{ar y}{l } m_{10} m_{1} \cdots m_{1,M-} m_{1,M+} m_{1,M+3} \cdots m_{1,2N- M1} m_{30} m_{31} \cdots m_{3,M-1} m_{3,M+1} m_{3,M+} \cdots m_{3,2N- M1} \vdots \vdots \vdots \vdots \vdots \vdots m_{2N-1,0} m_{2N-1,} \cdots m_{2N-1,M } m_{2N-1,M+} m_{2N-1,M+3} \cdots m_{2N-1,2N-M1} \end{ar y}|,m_{ij}=A_{i}B_{j} \frac{e^{(p+q)x+\sqrt{-3\epsilon}(p^{2}-q^{2})t} {p+q} p=q=\sqrt{-\frac{\sigma}{12}}A_{i}= \sum_{k=0}^{i}\frac{a_{ik} {(i-k)!}(p\partial_{p})^{i-k} , (i=1,3, \cdots , 2N-1),B_{j}=\{ begin{ar y}{l \sum_{l=0}^{j\frac{b_jl}{(j-l)!}(q\parti l_{q})^{j-l}, (j=M+1,M+3,\cdots ,2N-M1), \frac{1}\dot{j}!(q\parti l_{q})^{j}, (j=0,1 \cdots,M-1), \end{ar y}a_{ik}= \sum_{r=0}^{k}\frac{3^{r+1}-1}{(r+2)!}a_{i+2,k-r} , (i=1,3, \cdots , 2N -3;0\leq k\leq i)b_{jl}= \sum_{s=0}^{l}\frac{3^{s+1}-1}{(s+2)!}b_{j+2,l-s} (j=M+1, M+3 2N-M-3;0\leq l\leq j),,,where \epsilon is the sign \pm 1 in the left‐hand side of eq. (2) and 0\leq M\leq N . Here a_{2N-1,k}(0\leq k\leq 2N-1) and b_{2N-M-1,l}(0\leq l\leq 2N-M-1) are arbitrary constants, however
they are not independent and we can take a_{2N-1,0}=1, a_{2N-1,2k}=0(1\leq k\leq N-1) ,133
b_{2N-M-1,0}=1, b_{2N-M-1,2l}=0(1 \leq l\leq\frac{2N-M-1}{2}) without loss of generality. The
coefficients a_{ik}(1\leq i\leq 2N-3) and b_{jl}(M+1\leq j\leq 2N-M-3) are recursively definedin the above way. The elements m_{ij} of determinant are given by the parameter derivatives.
We can prove that the above f actually satisfies the bilinear form of Boussinesq equationin the same way with the case of NLS equation [5].

It should be pointed out that the above f is written in the form of two component
determinant, that is, we have N\cross M matrix (m_{2i-1,j-1})_{1\leq i\leq N,1\leq j\leq M} on left, and N\cross(N-M) matrix (m_{2i-1,2j-M-1})_{1\leq i\leq N,M+1\leq j\leq N} on right in the determinant. For soliton
equations of complex variables such as NLS equation, the rational solutions are expressedby the single component Gram determinant which corresponds to the case of M=0 . Thisis because we need M=0 to satisfy the complex conjugate condition. For real variable
equations such as Boussinesq equation, the determinant solutions are not restricted by
the conjugate condition and we have a free parameter M . Thus a wider class of rational
solutions is obtained when the reality condition of f is satisfied. However it is still unclear
whether those solutions are regular and whether they are localized in time.

4

Other equations

The bilinear method can be applied to nonintegrable equations also. In general the bilinear
equation,P(D)f\cdot f=0,
where P(D) is a polynomial of D_{x}, D_{y}, D_{t} , . . ., admits at least 2‐soliton solution and1‐breather solution of the form,

f=1+e^{\xi}+e^{\xi^{*}}+Ae^{\xi+\xi^{*}} \xi=px+qy+\omega t+
Thus it might be possible to find regular and time‐localized breather solutions among theabove f . One simple example is the bilinear equation,(D_{x}^{4}+\sigma D_{y}^{2}-D_{t}^{2})f\cdot f=0,
which is transformed tou_{tt}=u_{xxxx}+3(u^{2})_{xx}+\sigma u_{yy},through the variable transformation (5). This equation admits the breather solution,

far ow\wedge\sqrt{\frac{4k^{4}-\sigma l^{2} {k^{4}-\sigma l^{2} }\cosh(\sqrt{k^ {4}-\sigma l^{2} t)-\cos(kx+ly),

where k and l are constants satisfying k^{4}>\sigma l^{2} , and if \sigma=-1 , we have the rational
solution,farrow-t^{2}+(kx+y)^{2}+3k^{4}.

Another example of bilinear equation,(D_{x}^{4}+D_{x}^{2}D_{y}^{2}-D_{t}^{2})f\cdot f=0,
is transformed tou_{tt}=u_{xxxx}+3(u^{2})_{xx}+u_{xxyy}+(2v_{x}^{2}+uv_{y})_{xx},u_{y}=v_{xx},134by the variable transformations (5) and v=(2\log f)_{y} . This equation has the time‐
localized breather solution,

farrow\wedge 2\cosh(k\sqrt{k^{2}+l^{2}}t)-\cos(kx+ly),

and we also find another breather type solution,

f-2\cosh(\sqrt{2}ky)\cosh(k^{2}t)-\sinh(\sqrt{2}ky)\cos(kx),by straightforward calculation. The bilinear formalism provides a simple and useful way
to construct these kinds of solutions and the equations admitting such solutions simulta‐
neously.

5

Concluding remarks

According to the signs of coefficients, there are various types of solutions of the Boussi‐

nesq eq. (2), such as the Akhmediev breather, rational rogue wave, finite‐time blowing up
breather, finite‐time blowing up rational rogue wave and solutions of interacting singular‐ities. Only some cases of the equations and solutions are relevant for the nonlinear water
waves, but the Boussinesq equations may appear in some other contexts of physics. It
might be interesting to study interpretations of those solutions in various physical systems.
The multi Akhmediev breathers and higher order rational solutions are presented in
the determinant form. The multi Akhmediev breather solutions are regular and describe
time‐localized excitation of waves. Investigating properties of the higher order rational
solutions including regularity and time‐localization may be a future work.
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