Skip to main content

Existence and stability of solitary waves for the generalized Korteweg-de Vries equations

Abstract

In this paper, we consider the fractional Korteweg-de Vries equations with general nonlinearities. By studying constrained minimization problems and applying the method of concentration-compactness, we obtain the existence of solitary waves for the generalized Korteweg-de Vries equations under some assumptions of the nonlinear term. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, in the sense that a solution which starts near the set will remain near it for all time.

1 Introduction

This paper is devoted to studying the existence and stability of solitary wave solutions of the generalized Korteweg-de Vries equation

u t + ( f ( u ) ) x ( L ( u ) ) x =0in R,
(1.1)

where f(u) satisfies the following assumption:

  1. (A)

    f(u)C(R,R), lim u 0 f ( u ) | u | =0 and lim | u | f ( u ) | u | γ =0 for some 1<γ<1+4α,

    L ( u ) ˆ (ξ)= | ξ | 2 α u ˆ (ξ),

0<α1, the Fourier transform Fψ(ξ)= ψ ˆ (ξ)= 1 ( 2 π ) 1 2 R u(x) e i ξ x dx.

When f(u)= 1 2 u 2 and α=1, equation (1.1) is the well-known Korteweg-de Vries equation, introduced by Korteweg and de Vries in 1895 (cf. [1]). The existence and stability of solitary waves of the Korteweg-de Vries equation is considered by Benjamin in [2]. Recently, in [3], Pelinovsky obtained a Korteweg-de Vries equation with a forcing term, which is a simple analytical model of tsunami generation by submarine landslides.

Here, we consider the generalized Korteweg-de Vries equation (1.1). Let F(u)= 0 u f(s)ds. Since the functionals

E(u)= + [ 1 2 u L ( u ) F ( u ) ] dx

and

Q(u)= 1 2 + u 2 dx

are two conserved quantities with (1.1), for studying the existence of solitary wave solutions to (1.1), by the variational methods, the solitary wave solutions to equation (1.1) will be founded as minimizers of

where q>0. Denote the set of minimizers of the problem ( I q ) by

Inspired by the methods used in [4, 5], by studying the problem ( I q ), we obtain the existence of solitary waves for equation (1.1) with some special nonlinearities f(u)= 1 p u p , where 1<p<1+4α, and general nonlinearities satisfying the assumption (A). Moreover, we prove that the set G q of minimizers is a stable set for the initial value problem of equation (1.1) in the sense that a solution which starts near the set will remain near it for all time. In order to obtain those results, we have to overcome one main difficulty: the minimization problem ( I q ) is given in the unbounded domain which results in the loss of compactness. As is done in [4, 6], we overcome the difficulty of loss of compactness by the method of concentration-compactness introduced by Lions in [7, 8] for solving some minimization problems in unbounded domains.

Now we give our main results.

Theorem 1.1 Suppose that α=1 and f(u) satisfies condition (A) and I q 0 <0 for some q 0 >0. Then there exists 0< q q 0 such that G q is not empty. Moreover, if { u n } is a minimizing sequence for the problem ( I q ), then there exist a sequence { y n }R and g G q such that { u n (+ y n )} contains a subsequence converging strongly in H 1 (R) to g, and

lim n + inf g G q u n g=0,

where is the norm of H 1 (R).

Theorem 1.2 Under the assumptions of Theorem  1.1, the set G q is H 1 (R)-stable with respect to equation (1.1), i.e., for any ε>0, there exists δ>0 such that if

inf g G q u 0 g<δ,

then the solution u(x,t) to equation (1.1) with initial data u 0 satisfies

inf g G q u ( t , ) g <ε

for any t[0,T).

Theorem 1.3 Suppose that 0<α<1 and f(u) satisfies condition (A) and I q 0 <0 for some q 0 >0. Then there exists 0< q q 0 such that G q is not empty. Moreover, if { u n } is a minimizing sequence for the problem ( I q ), then there exists a sequence { y n }R and g G q such that { u n (+ y n )} contains a subsequence converging strongly in H α (R) to g, and

lim n + inf g G q u n g α , 2 =0,

where α , 2 is the norm of H α (R) given in Section  5.

Theorem 1.4 Under the assumptions of Theorem  1.1, the set G q is H α (R)-stable with respect to equation (1.1), i.e., for any ε>0, there exists δ>0 such that if

inf g G q u 0 g α , 2 <δ,

then the solution u(x,t) to equation (1.1) with initial data u 0 satisfies

inf g G q u ( t , ) g α , 2 <ε

for any t[0,T).

The paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we study the existence and stability of solitary waves of equation (1.1) with some special nonlinearities f(u)= 1 p u p . Section 4 is devoted to studying equation (1.1) with general nonlinearities f(u) satisfying the assumption (A). We shall consider the existence and stability of solitary waves of equation (1.1) with 0<α<1 in Section 5.

2 Some preliminaries

At first, we give some notations. The set of all integers and the set of all real numbers are written as and , respectively. And all the integrals will be taken over unless specified. L p (R) denotes the usual Lebesgue space with the norm | | p given by

| | p = ( | u | p d x ) 1 p for 1p<+.

The Sobolev space H 1 (R) is defined by

H 1 (R):= { u : u L 2 ( R )  and  u x L 2 ( R ) } ,

whose norm is given by

= ( ( | u x | 2 + | u | 2 ) d x ) 1 2 .

Now, we give Lemma 2.1 and Lemma 2.2 which will be used to study the behavior of the minimizing sequence for the problem ( I q ). Lemma 2.2 is due to Lions [7, 8].

Lemma 2.1 Suppose that B>0 and δ>0 are given. Then there exists η=η(B,δ) such that if u H 1 (R) with uB and | u | p + 1 δ, then

sup y R y 1 2 y + 1 2 | u | p + 1 dxη.

Proof We have

j Z j 1 2 j + 1 2 [ ( u x ) 2 + u 2 ] dx= u 2 B 2 | u | p + 1 p + 1 | u | p + 1 p + 1 = j Z B 2 | u | p + 1 p + 1 j 1 2 j + 1 2 | u | p + 1 dx.

Therefore, there exists some j 0 Z such that

j 0 1 2 j 0 + 1 2 [ ( u x ) 2 + u 2 ] dx B 2 | u | p + 1 p + 1 j 0 1 2 j 0 + 1 2 | u | p + 1 dx.

Applying the Sobolev embedding theorem [9], there exists a constant A such that

( j 0 1 2 j 0 + 1 2 [ ( u x ) 2 + u 2 ] d x ) 1 p + 1 A ( j 0 1 2 j 0 + 1 2 [ ( u x ) 2 + u 2 ] d x ) 1 2 A B | u | p + 1 p + 1 2 ( j 0 1 2 j 0 + 1 2 | u | p + 1 d x ) 1 2 .

Thus, we obtain

j 0 1 2 j 0 + 1 2 | u | p + 1 dx ( | u | p + 1 p + 1 2 A B ) 2 ( p + 1 ) p 1 δ ( p + 1 ) 2 p 1 ( A B ) 2 ( p + 1 ) p 1 .

Taking η= δ ( p + 1 ) 2 p 1 ( A B ) 2 ( p + 1 ) p 1 , it follows that

sup y R y 1 2 y + 1 2 | u | p + 1 dx j 0 1 2 j 0 + 1 2 | u | p + 1 dxη.

 □

Lemma 2.2 Let { u n } be a bounded sequence in H 1 (R) such that

sup y R y 1 y + 1 | u n | 2 dx0 as n+

for some r>0. Then u n 0 in L s (R) for 2<s<.

Proof Let 2<s<. Without loss of generality, we may assume r=1. It follows from the interpolation inequalities that

| u | L s ( B ( y , 1 ) ) A | u | L 2 ( B ( y , 1 ) ) s + 2 2 u H 1 ( B ( y , 1 ) ) s 2 2 ,

where A>0 is a constant independent of u.

Covering by a family of intervals ( y i 1, y i +1) such that each point of is contained in at most two such intervals and summing this inequality over this family of intervals, we get

| u | L s s 2A ( sup y R y 1 y + 1 | u | 2 d x ) s + 2 4 u s 2 2 .

Since { u n } is bounded in H 1 (R) and sup y R y 1 y + 1 | u n | 2 dx0 as n+, applying the above inequality, we know that u n 0 in L s (R) for 2<s<. □

Next, we establish a convergence result that will be used in the proof of Theorem 1.3.

Lemma 2.3 Let fC(R,R) and suppose that

| f ( t ) | C ( | t | + | t | p 1 ) for all tR,
(2.1)

where 1< p 1 <. If u n u 0 in H 1 (R) and u n u 0 a.e. on , then

lim n [ + F ( u n ) d x + F ( u 0 ) d x + F ( u n u 0 ) d x ] =0,

where F(u)= 0 u f(s)ds.

Proof Let R>0. Applying the mean value theorem, we have

+ F ( u n ) d x = | x | < R F ( u n ) d x + | x | R F ( u 0 + ( u n u 0 ) ) d x = | x | < R F ( u n ) d x + | x | R [ F ( u n u 0 ) d x + f ( u n u 0 + θ u 0 ) u 0 ] d x ,

where 0<θ<1 is dependent on x and R. Now we write

| + F ( u n ) d x + F ( u 0 ) d x + F ( u n u 0 ) d x | | | x | < R [ F ( u n ) F ( u 0 ) ] d x | + | | x | R F ( u 0 ) d x | + | | x | < R F ( u n u 0 ) d x | + | | x | R f ( u n u 0 + θ u 0 ) u 0 d x | .
(2.2)

It follows from (2.1), the mean value theorem and the Hölder inequality that

| | x | < R [ F ( u n ) F ( u 0 ) ] d x | = | | x | < R f ( u 0 + θ ( u n u 0 ) ) ( u n u 0 ) d x | C | x | < R | u 0 + θ ( u n u 0 ) | | u n u 0 | d x + C | x | < R | u 0 + θ ( u n u 0 ) | p 1 | u n u 0 | d x C | x | < R | u 0 | | u n u 0 | d x + C | x | < R | u n u 0 | 2 d x + ε | x | < R | u 0 | p 1 | u n u 0 | d x + C ε | x | < R | u n u 0 | p 1 + 1 d x C ( | x | < R | u 0 | 2 d x ) 1 2 ( | x | < R | u n u 0 | 2 d x ) 1 2 + C | x | < R | u n u 0 | 2 d x + ε ( | x | < R | u 0 | 2 p 1 d x ) 1 2 ( | x | < R | u n u 0 | 2 d x ) 1 2 + C ε | x | < R | u n u 0 | p 1 + 1 d x
(2.3)

and

| | x | < R F ( u n u 0 ) d x | C | x | < R | u n u 0 | 2 dx+C | x | < R | u n u 0 | p 1 + 1 dx.
(2.4)

Since the embedding H 1 (R) L loc s (R) (2s<) is compact, the inequalities (2.3) and (2.4) imply that

| | x | < R [ F ( u n ) F ( u 0 ) ] d x | 0as n,
(2.5)
| | x | < R F ( u n u 0 ) d x | 0as n.
(2.6)

Similarly, by the Hölder inequality, the Sobolev embedding theorem and (2.1), we get

| | x | R f ( u n u 0 + θ u 0 ) u 0 d x | C | x | R | u n u 0 + θ u 0 | | u 0 | d x + C | x | R | u n u 0 + θ u 0 | p 1 | u 0 | d x C ( | x | R | u n | 2 d x ) 1 2 ( | x | R | u 0 | 2 d x ) 1 2 + C | x | R | u 0 | 2 d x + ε ( | x | R | u n | 2 p 1 d x ) 1 2 ( | x | R | u 0 | 2 d x ) 1 2 + C ε | x | R | u 0 | p 1 + 1 d x C u n ( | x | R | u 0 | 2 d x ) 1 2 + C | x | R | u 0 | 2 d x + ε u n p 1 ( | x | R | u 0 | 2 d x ) 1 2 + C ε | x | R | u 0 | p 1 + 1 d x .

Since { u n } is bounded in H 1 (R), we see that

| | x | R f ( u n u 0 + θ u 0 ) u 0 d x | 0as R.
(2.7)

Hence, combining (2.2), (2.5), (2.6) and (2.7), we obtain

+ F( u n )dx + F( u 0 )dx + F( u n u 0 )dx0as n.

 □

3 The case of special nonlinearity

In this section, we only consider the case of α=1 and f(u)= 1 p u p , where 1<p<5. Correspondingly,

E(u)= + [ 1 2 ( u x ) 2 1 p 1 p + 1 u p + 1 ] dx.

At first, we commence by studying some properties of the functional I q :(0,+)R and the behavior of the minimizing sequences for the problem ( I q ).

Lemma 3.1 For any q>0,

  1. (i)

    < I q <0;

  2. (ii)

    If { u n } is a minimizing sequence for the problem ( I q ), there exists a constant B>0 such that u n B for all n;

  3. (iii)

    If { u n } is a minimizing sequence for the problem ( I q ), there exist a positive constant δ and a sequence { y n } of real numbers such that

    y n 1 2 y n + 1 2 | u n | p + 1 dxδ

for sufficiently large n.

Proof (i) Choose any function u H 1 (R) such that Q(u)=q and u p + 1 dx0. For any θ>0, define u θ (x)= θ u(θx). Then we have

Q( u θ )=q

and

E( u θ )= 1 2 θ 2 ( u x ) 2 dx 1 p ( p + 1 ) θ p 1 2 u p + 1 dx.

For 1<p<5, by taking θ>0 sufficiently small, we get I q E( u θ )<0.

Next we prove I q >. Let u H 1 (R) such that Q(u)=q. By the Sobolev embedding theorems and interpolation inequalities, we get

| u p + 1 d x | | u | p + 1 p + 1 A | u | P 1 2 ( P + 1 ) P + 1 A u p 1 2 | u | 2 p + 3 2 ,
(3.1)

where A denotes various constants which are independent of u. Using the Young inequality, we derive from (3.1)

| u p + 1 d x | ε u 2 + A ε | u | 2 2 p + 6 5 p ε u 2 + A ε , q ,

where ε>0 is arbitrary and A ε , q depends on ε and q, but not on u. Therefore,

E ( u ) = E ( u ) + Q ( u ) Q ( u ) = 1 2 u 2 1 p ( p + 1 ) u p + 1 d x 1 2 u 2 d x 1 2 u 2 ε p ( p + 1 ) u 2 1 p ( p + 1 ) A ε , q q .

Choosing ε< p ( p + 1 ) 2 , we obtain the lower bound of the functional E

E(u) 1 p ( p + 1 ) A ε , q q,

which implies I q 1 p ( p + 1 ) A ε , q q>.

  1. (ii)

    Let { u n } be a minimizing sequence for the problem ( I q ). Then, by (3.1), we have

    1 2 u n 2 = E ( u n ) + Q ( u n ) + 1 p ( p + 1 ) u n p + 1 d x sup n E ( u n ) + q + 1 p ( p + 1 ) | u n | p + 1 p + 1 A + q + A u p 1 2 | u | 2 p + 3 2 A ( 1 + u p 1 2 ) ,

where A denotes various constants which are independent of n. Since 1<p<5, the existence of the desired bound B follows.

  1. (iii)

    Let { u n } be a minimizing sequence for the problem ( I q ). Then we claim: there exists a constant η>0 such that | u n | p + 1 η for all sufficiently large n. We argue by contradiction: if no such η>0 exists, then lim inf n | u n | p + 1 dx0. Hence

    I q = lim n E( u n ) lim inf n | u n | p + 1 dx0,

which contradicts (i). So, the claim is achieved.

Combining (ii) and Lemma 2.1, there exist a positive constant δ and a sequence { y n } of real numbers such that

y n 1 2 y n + 1 2 | u n | p + 1 dxδ

for sufficiently large n. The proof of Lemma 3.1 is completed. □

The next lemma will establish a subadditivity inequality which will be a crucial step in the proof of the existence minimizer for the problem ( I q ).

Lemma 3.2 For all q 1 , q 2 >0, I q 1 + q 2 < I q 1 + I q 2 .

Proof For given u H 1 (R), | u | 2 2 = q 1 , θ>0, let u θ (x)= θ 2 p + 1 u( θ p 1 p + 1 x), where θ= ( q 2 q 1 ) p + 1 5 p . Then it follows that

Q( u θ )= q 2 q 1 Q(u)= q 2 ,

and

E( u θ )= ( q 2 q 1 ) p + 3 5 p E(u).

Hence we get

I q 2 =inf { ( q 2 q 1 ) p + 3 5 p E ( u ) : Q ( u ) = q 1 } = ( q 2 q 1 ) p + 3 5 p I q 1 .
(3.2)

Now, from (3.2) and Lemma 3.1, we obtain for all q 1 , q 2 >0,

I q 1 + q 2 = ( q 2 + q 1 ) p + 3 5 p I 1 < ( q 2 p + 3 5 p + q 1 p + 3 5 p ) I 1 = I q 1 + I q 2 .

 □

Now we formulate the following two theorems, which are special cases corresponding to Theorem 1.1 and Theorem 1.2, and give their proof with the aim of Lemma 3.1 and Lemma 3.2.

Theorem 3.1 Let α=1 and f(u)= 1 p u p , where 1<p<5. For any q>0, the set G q is not empty. Moreover, if { u n } is a minimizing sequence for the problem ( I q ), then there exist a sequence { y n }R and g G q such that { u n (+ y n )} contains a subsequence converging strongly in H 1 (R) to g, and

lim n + inf g G q u n g=0.

Theorem 3.2 Let α=1 and f(u)= 1 p u p , where 1<p<5. For any q>0, the set G q is H 1 (R)-stable with respect to equation (1.1), i.e., for any ε>0, there exists δ>0 such that if

inf g G q u 0 g<δ,

then the solution u(x,t) to equation (1.1) with initial data u 0 satisfies

inf g G q u ( t , ) g <ε

for any t[0,T).

Proof of Theorem 3.1 From (3.2), it is easy to check that I q is continuous on (0,). Let { u n } be a minimizing sequence for the problem ( I q ). By Lemma 3.1, there exist a positive constant δ and a sequence { y n } of real numbers such that

y n 1 2 y n + 1 2 | u n | p + 1 dxδ

for sufficiently large n.

Let us define v n = u n (x+ y n ). Hence Q( V n )=Q( u n )=q, E( v n )=E( u n ) I q , as n, and

1 2 1 2 | v n | p + 1 dx= y n 1 2 y n + 1 2 | u n | p + 1 dxδ>0.
(3.3)

Since { v n } is bounded in H 1 (R), by Lemma 3.1, we may assume going, if necessary, to a subsequence

v n g in  H 1 ( R ) , v n g in  L loc p + 1 ( R ) , v n g a.e. on  R .
(3.4)

Hence, by (3.3), we get g0. And applying the Brezis-Lieb lemma [10], we have

| v n | 2 2 = | v n g | 2 2 + | g | 2 2 ,
(3.5)
| v n | p + 1 p + 1 = | v n g | p + 1 p + 1 + | g | p + 1 p + 1 .
(3.6)

Now we show that Q(g)= 1 2 g 2 dx=q. In the contrary case, 0<Q(g)=λ<q. By (3.5), we obtain lim n q n = lim n Q( v n g)=qλ. Then it follows from (3.5) and (3.6) that

I q = E ( v n ) + o ( 1 ) 1 2 v n 2 1 2 | v n | 2 2 1 p ( p + 1 ) | v n | p + 1 p + 1 + o ( 1 ) = E ( v n g ) + E ( g ) + o ( 1 ) I q n + I λ + o ( 1 ) .
(3.7)

Since I q is continuous on (0,), letting n, we get I q I q λ + I λ , which contradicts Lemma 3.2. Therefore Q(g)=q. It then follows from (3.5) that

v n gin  L 2 (R).
(3.8)

Applying the interpolation inequality, (3.4) and (3.8), we get

v n gin  L p + 1 (R).
(3.9)

Using the weak low semi-continuity of the norm in H 1 (R), we know that

I q 1 2 v n 2 1 2 | v n | 2 2 1 p ( p + 1 ) | v n | p + 1 p + 1 + o ( 1 ) 1 2 g 2 1 2 | v n g | 2 2 1 2 | g | 2 2 1 p ( p + 1 ) | v n g | p + 1 p + 1 1 p ( p + 1 ) | g | p + 1 p + 1 + o ( 1 ) = E ( g ) 1 2 | v n g | 2 2 1 p ( p + 1 ) | v n g | p + 1 p + 1 + o ( 1 ) .

Letting n, by (3.8) and (3.9), we obtain E(g) I q . On the other hand, it follows from Q(g)=q that E(g) I q . Therefore E(g)= I q , which implies that g is a minimizer of the problem ( I q ) (i.e., G q ). Then it follows from (3.7), (3.8) and (3.9) that

v n = u n (+ y n )gin  H 1 (R).

We prove lim n + inf g G q u n g=0 with an argument by contradiction. Assume that there exist ε 0 >0 and a subsequence { u n k } of { u n } such that

inf g G q u n k g ε 0 >0
(3.10)

for all n k . With the result of the above proof, we obtain that there exist a subsequence of { u n k }, denoted again by { u n k }, { y n k }R and g G q such that

u n k (+ y n k )gin  H 1 (R).

Since g( y n k ) G q ,

u n k g ( y n k ) = u n k ( + y n k ) g 0as  n k +,

which contradicts (3.10). □

An immediate consequence of Theorem 3.1 is that G q forms a stable set for the initial-value problem for equation (1.1).

Proof of Theorem 3.2 We prove Theorem 3.2 with an argument by contradiction. Assume that the set G q is not H 1 (R)-stable. Then there exist ε 0 >0, { ψ n } H 1 (R) and a sequence of times { t n } such that

inf g G q ψ n g< 1 n ,
(3.11)

and

inf g G q u n ( , t n ) g ε 0
(3.12)

for all n, where u n (x,t) solves equation (1.1) with u n (x,0)= ψ n .

Equation (3.11) implies that

E( ψ n ) I q ,Q( ψ n )q.

Choose { μ n }R such that Q( μ n ψ n )=q for all n. Thus μ n 1 as n+. Hence the sequence v n = μ n u n (, t n ) satisfies Q( v n )=q and

lim n E( v n )= lim n E ( u n ( , t n ) ) = lim n E( ψ n )= I q .

Therefore { v n } is a minimizing sequence for the problem ( I q ). By Theorem 1.1, there exists { g n k } G q such that

v n k g n k < ε 0 2
(3.13)

for sufficiently large n k . Since μ n 1 and u n (, t n ) is bounded, we derive from (3.12) and (3.13)

ε 0 u n k ( , t n k ) g n k u n k ( , t n k ) μ n k u n k ( , t n k ) + μ n k u n k ( , t n k ) g n k ( | μ n k 1 | ) u n k ( , t n k ) + ε 0 2 3 4 ε 0
(3.14)

for sufficiently large n k . (3.14) is a contradiction. Therefore, the set G q is H 1 (R)-stable with respect to equation (1.1). □

4 The case for more general nonlinearities

In this section, we consider (1.1) with α=1 and more general nonlinearities f satisfying condition (A). At first, we study the properties of the functional I q :(0,)R and the minimizing sequence of the problem ( I q ).

Lemma 4.1

  1. (i)

    For any q>0, I q is finite and continuous on (0,). Moreover, each minimizing sequence for ( I q ) is bounded;

  2. (ii)

    I q 0 for any q>0.

Proof (i) According to assumption (A), we observe that for each ε>0 there exists C ε >0 such that

| F ( u ) | ε | u | 2 +ε | u | γ + 1 + C ε | u | α ,
(4.1)

where 2<α<γ+1. By the Sobolev embedding theorems and interpolation inequalities, we obtain

+ | u | α dxA u α 2 2 | u | 2 α + 2 2
(4.2)

and

+ | u | γ + 1 dxA u γ 1 2 | u | 2 γ + 3 2 ,
(4.3)

where A>0 is independent of u. Then using the Young inequality, we can derive from (4.2) and (4.3) that for all η>0, there exists C η >0 such that

+ | u | α dxη u 2 + C η | u | 2 2 ( α + 2 ) 6 α
(4.4)

and

+ | u | γ + 1 dxη u 2 + C η | u | 2 2 γ + 6 5 γ .
(4.5)

Let u H 1 (R) such that Q(u)=q. It follows from (4.1), (4.4) and (4.5) that

E ( u ) = 1 2 u 2 1 2 + u 2 d x + F ( u ) d x 1 2 u 2 1 2 | u | 2 2 ε | u | 2 2 ε η u 2 ε C η | u | 2 2 γ + 6 5 γ C ε η u 2 C ε C η | u | 2 2 ( α + 2 ) 6 α ( 1 2 ε η C ε η ) u 2 C ε , η , q ,
(4.6)

where C ε , η , q is a positive constant dependent only on ε and η for given q>0. Choosing ε>0 and η>0 such that 1 2 εη C ε η>0, we see that I q >.

Since | θ u | 2 2 = θ 2 | u | 2 2 for θ>0, it is easy to check that I q is continuous on (0,).

Let { u n } be a minimizing sequence for the problem ( I q ). From (4.6) and the fact that I q is finite, we know that { u n } is bounded in H 1 (R).

  1. (ii)

    For given u H 1 (R) such that Q(u)=q, let u θ (x)= 1 θ u( x θ ) for θ>0. We obtain that

    | u θ | 2 2 = | u | 2 2 =q
    (4.7)

and

E( u θ )= 1 2 θ 2 + ( u x ) 2 dxθ + F ( 1 θ u ) dx.
(4.8)

Combining (4.1), (4.7) and (4.8), we obtain

I q E( u θ )0as θ+.

 □

Lemma 4.2 Suppose that I q 0 <0 for some q 0 >0. Then the following two properties hold:

  1. (i)

    I q q is non-increasing on (0,+) and lim q 0 + I q q =0;

  2. (ii)

    there exists q 1 q 0 such that

    I q q > I q 0 q 0 for q(0, q 1 ).

Proof First we observe that if σ>0 and β>0 with Q(u)=β and u σ (x)=u( 1 σ x), then Q( u σ )=σβ and

E( u σ )= 1 2 σ + ( u x ) 2 dxσ + F(u)dx.

Consequently, for q 1 >0 and q 2 >0, we have

I q 2 =inf { q 1 2 q 2 + ( u x ) 2 d x q 2 q 1 + F ( u ) d x , Q ( u ) = q 1 } .

If q 1 > q 2 >0, then for each ε>0, there exists u H 1 (R) with Q(u)= q 1 such that

I q 2 +ε> 1 2 q 1 q 2 + ( u x ) 2 dx q 2 q 1 + F(u)dx> q 2 q 1 E(u) q 2 q 1 I q 1 .
(4.9)

This inequality yields I q 2 q 2 I q 1 q 1 for 0< q 2 < q 1 .

Since I q 0 for all q>0, we see that

lim q 0 + I q q =A0.

We claim that A=0. Letting ε= q 2 , 0<q q 0 , from (4.9), there exists u ( q ) H 1 (R) with Q( u ( q ) )= q 0 such that

I q + q 2 1 2 q 0 q + ( u x ( q ) ) 2 d x q q 0 + F ( u ( q ) ) d x q q 0 [ 1 2 + ( u x ( q ) ) 2 d x + F ( u ( q ) ) d x ] .
(4.10)

It follows from (4.6) and (4.10) that

I q + q 2 q q 0 ( C 1 ( q 0 ) + ( u x ( q ) ) 2 d x C 2 ( q 0 ) ) ,

where C 1 ( q 0 )>0 and C 2 ( q 0 )>0 are constants independent of q 0 . Hence we obtain q 0 2 C 1 ( q 0 ) + ( u ( q ) ) x 2 dx C 2 ( q 0 ), which implies

+ ( u x ( q ) ) 2 dx C 3 ( q 0 ),
(4.11)

where C 3 ( q 0 ) is dependent only on q 0 . Combining (4.1), (4.4), (4.5) and (4.11), we also get

| + F ( u ( q ) ) d x | C 4 ( q 0 ),
(4.12)

where C 4 ( q 0 ) is dependent only on q 0 .

We claim that

+ ( u x ( q ) ) 2 dx0as q 0 + .
(4.13)

Indeed, if there exist ε 0 >0 and q n 0 + such that + ( u x ( q n ) ) 2 dx ε 0 , then by (4.11) and (4.12), we obtain

I q n q n + q n q 0 ε 0 2 1 q n 2 1 q 0 C 4 ( q 0 )+as  q n 0 + ,

which contradicts lim q 0 + I q q =A0. Therefore (4.13) is achieved and this implies that lim q 0 + + F( u ( q ) )dx=0 and, consequently,

I q q +q 1 q 0 + F ( u ( q ) ) dx0as q 0 + .

This shows that lim q 0 + I q q =0.

  1. (2)

    We observe that lim q 0 + I q q =0> I q 0 q 0 , which implies (ii). □

Then we establish a subadditivity inequality similar to Lemma 3.2 with the aim of Lemma 4.2.

Lemma 4.3 Suppose that I q 0 <0 for some q 0 >0. Then there exists 0< q q 0 such that I q < I q q + I q for 0<q< q .

Proof According to Lemma 4.2, the set

{ q 1 | q 1 q 0  and  I q q > I q 0 q 0  or each  q ( 0 , q 1 ) }

is nonempty. We define

q =sup { q 1 | q 1 q 0  and  I q q > I q 0 q 0  for each  q ( 0 , q 1 ) } .
(4.14)

It follows from the continuity of I q and lim q 0 + I q q =0 that 0< q q 0 ,

I q = q q 0 I q 0 <0,
(4.15)
I q > q q 0 I q 0 for all q ( 0 , q ) .
(4.16)

Therefore,

I q = q q 0 I q 0 = q q q 0 I q 0 + q q 0 I q 0 < I q q + I q ,

for all q(0, q ). □

Now we give the proof of Theorem 1.1 with the aim of Lemma 4.1, Lemma 4.2 and Lemma 4.3. Since the proof of Theorem 1.1 is similar to that of Theorem 3.1, we only give the sketch of the proof.

Proof of Theorem 1.1 Let { u n } be a minimizing sequence of I q , where q is defined in (4.14). Since { u n } is bounded, we may assume

u n u in  H 1 ( R ) , u n u a.e. on  R .

First, we consider the case u=0. In this case, by Lemma 2.2, either

  1. (a)

    u n 0 in L s (R) for 2<s<, or

  2. (b)

    there exists a sequence { y n }R such that

    υ n (x)= u n (x+ y n )g0in  H 1 (R).

In the case (a), combining Lemma 2.2 and condition (A), we obtain

lim n + F( u n )dx=0

and, consequently,

I q = lim n E( u n )= lim n [ 1 2 + ( u n x ) 2 d x + F ( u n ) d x ] 0,

which contradicts Lemma 4.1. Hence (b) holds. Then it follows from Lemma 2.3 and Lemma 4.3 that g is the minimizer for the problem ( I q ) (i.e., G q ) and the result of Theorem 1.1 holds. The proof is similar to that of Theorem 3.1, we omit the details.

If u0, we repeat the previous argument in the case (b) to obtain the result of Theorem 1.1. □

Proof of Theorem 1.2 Theorem 1.2 is an immediate result of Theorem 1.1. We can prove it with an argument similar to that of Theorem 3.2. Here, we omit the details of the proof. □

5 The case for 0<α<1

In this section, we only consider the case of 0<α<1, i.e., we consider the existence and stability of solitary waves for the fractional Korteweg-de Vries equations with general nonlinearities. At first, we give the definition of H α (R). The fractional order Sobolev space H α (R) is defined by

H α (R):= { u ; u : R C , u L 2 ( R )  and  F 1 [ ( 1 + | ξ | 2 ) α 2 F u ] L 2 ( R ) } ,

whose norm is given by

α , 2 = | F 1 [ ( 1 + | ξ | 2 ) α 2 F ] | 2 .

Since the functionals

E(u)= + [ 1 2 | ( ) α 2 u | 2 F ( u ) ] dx

and

Q(u)= 1 2 + u 2 dx

are two conserved quantities with (1.1), for studying the existence of solitary wave solutions to (1.1), by the variational methods, the solitary wave solutions to the equation (1.1) will be founded as minimizers of

where q>0. Denote the set of minimizers of the problem ( I q ) by

Similar to Lemma 4.1 and Lemma 4.3, we obtain the following two lemmas.

Lemma 5.1

  1. (i)

    I q and I q are finite and continuous on (0,+); moreover, for any q>0, each minimizing sequence for the problem ( I q ) or ( I q ) is bounded in H α (R);

  2. (ii)

    I q 0 for any q>0.

Lemma 5.2 If I q 0 <0 for some q 0 >0, then there exists q 1 , 0< q 1 q 0 , such that

I q 1 < I q 1 q + I q for 0<q< q 1 .

Applying the above two lemmas and commutator estimates [[5], Lemma 2.5], we prove Theorem 1.3 and Theorem 1.4 by similar steps to those given in Section 4. Here we omit the details of Theorem 1.3 and Theorem 1.4.

References

  1. Korteweg DJ, de Vries G: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 1895, 39: 422–443.

    Article  Google Scholar 

  2. Benjamin TB: The stability of solitary waves. Proc. R. Soc. Lond. Ser. A 1972, 328: 153–183. 10.1098/rspa.1972.0074

    Article  MathSciNet  Google Scholar 

  3. Pelinovsky E: Analytical models of tsunami generation by submarine landslides. Nato Science Series 21. In Submarine Landslides and Tsunamis. Kluwer Academic, Dordrecht; 2003:111–128.

    Chapter  Google Scholar 

  4. Albert JP: Concentration compactness and the stability of solitary-wave solutions to nonlocal equations. Contemporary Mathematics 221. In Applied Analysis. Am. Math. Soc., Providence; 1999:1–29.

    Google Scholar 

  5. Guo B, Huang D: Existence and stability of standing waves for nonlinear fractional Schrödinger equations. J. Math. Phys. 2005., 53: Article ID 083702

    Google Scholar 

  6. Cazenave T, Lions PL: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 1982, 85: 549–561. 10.1007/BF01403504

    Article  MathSciNet  Google Scholar 

  7. Lions PL: The concentration-compactness principle in the calculus of variation. The locally compact case. I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 109–145.

    Google Scholar 

  8. Lions PL: The concentration-compactness principle in the calculus of variation. The locally compact case. II. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 223–283.

    Google Scholar 

  9. Adams RA: Sobolev Space. Academic Press, New York; 1975.

    Google Scholar 

  10. Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.

    Book  Google Scholar 

Download references

Acknowledgements

The work was supported in part by Special Fund of Fundamental Scientific Research Business Expense for Higher School of Central Government (projects for young teachers, No. ZY20110226) and the National Natural Science Foundation of China (No. 41276020).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mingli Hong.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Hong, M. Existence and stability of solitary waves for the generalized Korteweg-de Vries equations. Bound Value Probl 2013, 121 (2013). https://doi.org/10.1186/1687-2770-2013-121

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2013-121

Keywords