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arXiv:1509.09272v1 [math.AP] 30 Sep 2015
On stationary solutions of KdV and mKdV
equations
A.V. Faminskii and A.A. Nikolaev
Abstract Stationary solutions on a bounded interval for an initial-boundary value
problem to Korteweg–de Vries and modified Korteweg–de Vries equation (for the
last one both in focusing and defocusing cases) are constructed. The method of the
study is based on the theory of conservative systems with one degree of freedom.
The obtained solutions turn out to be periodic. Exact relations between the length of
the interval and coefficients of the equations which are necessary and sufficient for
existence of nontrivial solutions are established.
Both Korteweg–de Vries equation (KdV)
ut+aux+uxxx +uux=0
and modified Korteweg–de Vries equation (mKdV)
ut+aux+uxxx ±u2ux=0
(the sign ”+” stands for the focusing case and the sign ”-” – for the defocusing one)
describe propagation of long nonlinear waves in dispersive media. We assume ato
be an arbitrary real constant. If these equations are considered on a bounded interval
(0,L), then for well-posedness of an inital-boundary value problem besides an initial
A.V. Faminskii
Peoples’ Friendship University of Russia, Miklukho-Maklai str. 6, Moscow, 117198, Russia, e-
mail: afaminskii@sci.pfu.edu.ru
A.A. Nikolaev
Peoples’ Friendship University of Russia, Miklukho-Maklai str. 6, Moscow, 117198, Russia, e-
mail: anikolaev.rudn@yandex.ru
1
2 A.V. Faminskii and A.A. Nikolaev
profile one must set certain boundary conditions, for example,
u
x=0=u
x=L=ux
x=L=0
(see [11, 7, 5, 8] and others).
It follows from the results of [9] that such a problem for KdV equation possesses
certain internal dissipation: under some relations between aand Land sufficiently
small initial data solutions decay at large time. Similar properties hold for mKdV
equation. In order to answer the question if the smallness is essential one has to
construct non-decaying solutions. The simplest case of such solutions are stationary
solutions: u=u(x). In this situation the considered equations are reduced to the
following ordinary differential equations:
u′′′ +au′+uu′=0,(1)
u′′′ +au′+u2u′=0,(2)
u′′′ +au′−u2u′=0,(3)
and the boundary conditions — to the following ones:
u(0) = u(L) = u′(L) = 0.(4)
The goal of the present paper is to investigate existence of nontrivial solutions to
these problems under different relations between aand L. The method of the study is
based on the qualitative theory of conservative systems with one degree of freedom
(see, for example, [4]).
The first example of such a solution by this method for equation (1) was con-
structed in the case a=0, L=2 in [10]. In recent paper [6] also for equation (1)
such solutions were constructed for a=1 and L∈(0,2
π
)and exact formulas via
elliptic Jacobi functions were obtained. In the present paper these special functions
are not used.
Lemma 1. If u ∈C3[0,L]is a solution to any problems (1), (4), or (2), (4), or (3),
(4), then it is infinitely smooth and periodic with period L.
Proof. Integrating each of the equations (1)–(3) we obtain that the function usatis-
fies an equation u′′ +F′(u) = 0,F(0) = 0,F∈C∞.(5)
Following [4] introduce a ”full energy” E(x)≡1
2u′(x)2+Fu(x). Then (5) yields
that E′(x)≡0, that is E(x)≡const. By virtue of (4) E(L) = 0, therefore E(0) = 0
and so u′(0) = 0. The end of the proof is obvious. ⊓⊔
Further let a fundamental period for a nontrivial periodic function denotes a min-
imal possible positive value of a period.
By a symbol ua,Tdenote a nontrivial solution to any of considered problems with
the fundamental period T.
On stationary solutions of KdV and mKdV equations 3
Theorem 1. If aL26=4
π
2then there exists a unique solution ua,Lto problem (1),
(4). If aL2=4
π
2such a solution does not exist.
Theorem 2. If aL2<4
π
2then there exists a unique up to the sign solution ua,Lto
problem (2), (4). If aL2≥4
π
2such solutions do not exist.
Theorem 3. If aL2>4
π
2then there exists a unique up to the sign solution ua,Lto
problem (3), (4). If aL2≤4
π
2such solutions do not exist.
Remark 1. If aL26=4
π
n2for certain natural n≥2 then obviously the function u(x)≡
n2ua/n2,L(nx)is a solution to problem (1), (4) with the fundamental period T=L/n.
If aL2<4
π
n2for certain natural nthen the function u(x)≡nua/n2,L(nx)is a solution
to problem (2), (4) with the fundamental period T=L/n. In particular, nontrivial
solutions to problems (1), (4) and (2), (4) exist for any aand positive L. If aL2≤4
π
2
then nontrivial solutions to problem (3), (4) do not exist.
Further for convenience we pass from the segment [0,L]to the segment [−1,1].
For x∈[−1,1]in the case of equation (1) make a substitution y(x)≡L2
4uL
2(x+1),
while in the case of equations (2) and (3) — substitution y(x)≡L
2uL
2(x+1). Then
for b=L2
4athese equations transform respectively to the following ones:
y′′′ +by′+yy′=0,(6)
y′′′ +by′+y2y′=0,(7)
y′′′ +by′−y2y′=0,(8)
and consider periodic solutions to these equations with the fundamental period T=
2 such that y(−1) = y′(−1) = 0.(9)
We apply the following lemma in the spirit of the qualitative theory of conserva-
tive systems with one degree of freedom.
Lemma 2. Consider an initial value problem
y′′ +F′(y) = 0,y(−1) = y′(−1) = 0,(10)
where F ∈C∞, F(0) = 0. Then a nontrivial periodic solution to problem (10) with
the fundamental period T =2exist if and only if F′(0)6=0and there exists y06=0
such that F(y0) = 0, F′(y0)6=0, F(y)<0for y ∈(0,y0)if y0>0, F(y)<0for
y∈(y0,0)if y0<0and
Zy0
0
dy
p−2F(y)=1 if y0>0,Z0
y0
dy
p−2F(y)=1 if y0<0.(11)
Proof. First of all note that similarly to (5) E(x)≡1
2y′(x)2+Fy(x)≡0 if y(x)
is a solution to problem (10). Due to uniqueness of solutions to the initial value
problem the condition F′(0)6=0 is necessary for existence of nontrivial solutions.
4 A.V. Faminskii and A.A. Nikolaev
Consider, for example, the case F′(0)<0. If the function Fis negative ∀y>0
then it is easy to see that there is no periodic solution to problem (10). Therefore,
existence of positive y0such that F(y0) = 0, F(y)<0 for y∈(0,y0)is necessary.
Uniqueness of the solution implies that the function y(x)is even (if exists). Then
it is easy to see that it possesses the following properties: y′(x)>0 for x∈(−1,0),
y′(x)<0 for x∈(0,1),y(0) = y0,y′(0) = 0. Again due to uniqueness F′(y0)6=0.
Therefore, for x∈[0,1]the function y(x)satisfies the following conditions:
dy
dx =−p−2F(y),y(0) = y0,y(1) = 0.
Integrating we obtain that Zy0
0
dy
p−2F(y)=1.
It is easy to see that under these assumptions the desired solution exist. The case
F′(0)>0 is considered in a similar way (then y0<0). ⊓⊔
Now we can prove our theorems.
Proof (Theorem 1). Equation (6) is equivalent to equation
y′′ +by +1
2y2=c(12)
for certain real constant c. Therefore, construction of a solution transforms to search
of a constant csuch for a function
F(y)≡1
6y3+b
2y2−cy =1
6y(y2+3by−6c)≡1
6yF0(y)
the hypothesis of Lemma 2 is satisfied. Note that F′(y) = 1
2y2+by −c. Therefore,
the condition F′(0)6=0 implies that c6=0.
Real simple nonzero roots of the function F0exist if and only if D=9b2+24c>0
and then these roots are expressed by formulas y0=1
2(−3b+√D),y1=−1
2(3b+
√D).
It is easy to see that if c>0 then for any bthe root y0>0, F(y)<0 for y∈(0,y0),
F′(y0)6=0. If c∈(−3b2/8,0)then for b>0 the root y0<0, F(y)<0 for y∈(y0,0),
F′(y0)6=0.
Therefore, we have to find the constant cfor which condition (11) is satisfied.
Note that
−2F(y) = 1
3y(y0−y)(y−y1).
After the change of variable y=y0teach of equations (11) reduces to an equation
I(b,c)≡√3Z1
0
dt
pt(1−t)(y0t−y1)=1.
Since y0t−y1=1
2(√D−3b)t+1
2(√D+3b)it is easy to see that for the fixed bthe
function I(b,c)monotonically decreases. Moreover, lim
c→+∞I(b,c) = 0 and for b>0
On stationary solutions of KdV and mKdV equations 5
lim
c→−3
8b2+0I(b,c) = r2
bZ1
0
dt
√t(1−t)= +∞,lim
c→0I(b,c) = 1
√bZ1
0
dt
pt(1−t)=
π
√b,
for b=0
lim
c→0+0I(b,c) = lim
c→0+0
1
√2cZ1
0
dt
pt(1−t)(t+1)= +∞,
for b<0
lim
c→0+0I(b,c) = 1
p|b|Z1
0
dt
t√1−t= +∞.
Therefore, the desired value of cexists and is unique if b6=
π
2, while for b=
π
2
such a value does not exist. ⊓⊔
Remark 2. The substitution u(x) = a0+v(x−x0)under the appropriate choice of
the parameters a0and x0transforms any periodic solution of equation (1) with the
period Lto solution of an equation v′′′ + (a+a0)v′+vv′=0 satisfying conditions
v(0) = v′(0) = v(L) = v′(L) = 0. Therefore, any solution of equation (1) with the
fundamental period Lcan be expressed in this way by the functions ua+a0,L. Solu-
tions similar to functions ua,Lwere considered also in [13]. In [2] representation of
periodic solutions of equation (1) is given via elliptic Jacobi functions. The advan-
tage of our approach is that it can give transparent description of solutions.
Consider, for example, the case b>0. Then for b∈(0,
π
2)the constructed solu-
tion of problem (6), (9) is an even ”hill” of the height y0=1
2(−3b+√9b2+24c)>
0, while for b>
π
2— an even ”hole” of the depth y0<0. Note that Ic(b,c)<0,
Ib(b,c)<0. Therefore, the equation I(b,c) = 1 determines a smooth decreasing
function c(b). Since I(
π
2,0) = 1 we have that c(
π
2) = 0. Return to equation
(1). Let a>0. If u0=1
2(−3a+√9a2+384cL−2), where c=c(L2a/4), then for
L<2
π
/√athe solution ua,Lto problem (1), (4) is a ”hill” of the height u0>0
and for L>2
π
/√a— a ”hole” of the depth u0<0 (the center in both cases is at
the point L/2). In addition, u0→+∞as L→0, u0→0 as L→2
π
/√a,u0→0 as
L→+∞.
Proof (Theorem 2). Equation (7) is equivalent to equation
y′′ +by +1
3y3=c(13)
for certain real constant c. Let
F(y)≡1
12y4+b
2y2−cy =1
12y(y3+6by −12c)≡1
12yF0(y).
Note that the substitution z(x)≡ −y(x)leads to an equation similar to (13), where c
is replaced by (−c). Therefore, further it is sufficient to assume that c>0 (if c=0
then F′(0) = 0).
Similarly to the proof of Theorem 1 we need to find the roots of the function F0.
We apply Cardano formulas. Let D=8b3+36c2,
6 A.V. Faminskii and A.A. Nikolaev
p=3
q6c+√D,q=3
q6c−√Dif D≥0,
p=3
q6c+ip|D|=p2|b|ei
3arccos(3c/√2|b|3),q=pif D<0.
The the function F0has a real root y0=p+q>0. Moreover, if D>0 there are two
complex conjugate roots with negative real parts, and if D≤0 (it is possible only
for b<0) — two negative real roots y1and y2(y1=y2if D=0).
According to Vi`ete formulas y1+y2=−y0,y1y2=6b−y0y1−y0y2=6b+y2
0
and then
−2F(y) = 1
6y(y0−y)(y2+y0y+y2
0+6b).
After the change of variable y=y0tfirst equation (11) reduces to an equation
I(b,c)≡√6Z1
0
dt
qt(1−t)(y2
0t2+y2
0t+y2
0+6b)
=1.
It is easy to see that for the fixed bthe function y0(c)monotonically increases and
y0(c)→+∞as c→+∞(note that y0=p8|b|cos1
3arccos(3c/p2|b|3)if D<0).
Then for the fixed bthe function I(b,c)monotonically decreases and lim
c→+∞I(b,c) =
0. Moreover, if c→0+0 then y0(c)→0 for b≥0 and y0(c)→p6|b|for b<0.
Therefore,
lim
c→0+0I(b,c) = 1
√bZ1
0
dt
pt(1−t)=
π
√bif b>0,
lim
c→0+0I(b,c) = lim
c→0+0
√6
3
√12cZ1
0
dt
pt(1−t3)= +∞if b=0,
lim
c→0+0I(b,c) = 1
p|b|Z1
0
dt
t√1−t2= +∞if b<0.
Hence, the desired positive value of cexists and is unique if b<
π
2, while for b≥
π
2
such a value does not exist. ⊓⊔
Proof (Theorem 3). Equation (8) is equivalent to equation
y′′ +by −1
3y3=c(14)
for certain real constant c. Let
F(y)≡− 1
12y4+b
2y2−cy =−1
12y(y3−6by +12c)≡ − 1
12yF0(y).
As in the proof of Theorem 2 consider only the case c>0.
Again apply Cardano formulas. Let D=−8b3+36c2,
p=3
q−6c+√D,q=3
q−6c−√Dif D≥0,
On stationary solutions of KdV and mKdV equations 7
p=3
q−6c+ip|D|=√2be i
3
π
+arccos(3c/√2b3),q=pif D<0.
If D>0 then the function F0has a real root y0=p+q<0 and two complex
conjugate roots y1and y2. If D=0 then again the function F0has a real root y0=
p+q<0 and a double real root y1=y2>0. Both these two cases do not satisfy the
hypothesis of Lemma 2 since F′(0)<0.
It remains to consider the case D<0 (it is possible only if b>0), then c∈
(0,√2
3b3/2). Here the function F0has three distinct real roots, where a root y0=
p+q=√8bcos
π
3+1
3arccos(3c/√2b3)>0, a root y1<0, a root y2>y0. We
have that y1+y2=−y0,y2y2=−6b+y2
0and then
−2F(y) = 1
6y(y0−y)(6b−y2
0−y0y−y2).
After the change of variable y=y0tfirst equation (11) reduces to an equation
I(b,c)≡√6Z1
0
dt
qt(1−t)(6b−y2
0(1+t+t2))
=1.
Similarly to the previous theorem for the fixed bthe function y0(c)monotonically
increases, therefore, unlike to the previous theorem the function I(b,c)also mono-
tonically increases. It is easy to see that
lim
c→0+0I(b,c) = 1
√bZ1
0
dt
pt(1−t)=
π
√b,
lim
c→√2
3b3/2−0I(b,c) = r3
bZ1
0
dt
(1−t)pt(t+2)= +∞.
Hence, the desired positive value of cexists and is unique if b>
π
2, while for b≤
π
2
such a value does not exist. ⊓⊔
Remark 3. In [3, 1, 12] periodic solutions of equations (2) and (3) were considered in
the case when the constant c=0 in equations (13) and (14). Therefore, the periodic
solutions constructed in the present paper do not coincide with solutions from that
papers.
Acknowledgements The first author was supported by Project 333, State Assignment in the field
of scientific activity implementation of Russia.
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