On the linearized log–KdV equation
Abstract
The logarithmic KdV (log–KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the linearized log–KdV equation at the Gaussian solitary waves. By using the decomposition of solutions in the energy space in terms of Hermite functions, we show that the time evolution is related to a Jacobi difference operator with a limit circle at infinity. This exact reduction allows us to characterize both spectral and linear orbital stability of solitary waves. We also introduce a convolution representation of solutions to the log–KdV equation with the Gaussian weight and show that the time evolution in such a weighted space is dissipative with the exponential rate of decay.
1 Introduction
We address the logarithmic Korteweg–de Vries (logKdV) equation derived in the context of solitary waves in granular chains with Hertzian interaction forces [6, 7, 8]:
\hb@xt@.01(1.1) 
The log–KdV equation (LABEL:logKdV) has a twoparameter family of Gaussian solitary waves
\hb@xt@.01(1.2) 
where is a symmetric standing wave given by
\hb@xt@.01(1.3) 
Global solutions to the log–KdV equation (LABEL:logKdV) were constructed in [2] in the energy space
\hb@xt@.01(1.4) 
by a modification of analytic methods available for the log–NLS equation [5] (also reviewed in Section 9.3 in [4]). In the energy space , the following quantities for the momentum and energy,
\hb@xt@.01(1.5) 
and
\hb@xt@.01(1.6) 
are nonincreasing functions of time . Uniqueness, continuous dependence, and energy conservation are established in [2] under the additional condition , which is not satisfied in the neighborhood of the family of Gaussian solitary waves given by (LABEL:solitonorbit) and (LABEL:Gaussian). As a result, orbital stability of the Gaussian solitary waves was not established for the log–KdV equation (LABEL:logKdV), in a sharp contrast with that in the log–NLS equation established in [3].
A possible path towards analysis of orbital stability of Gaussian solitary waves is to study their linear and spectral stability by using the linearized log–KdV equation
\hb@xt@.01(1.7) 
where is the Schrödinger operator with a harmonic potential given by the differential expression
\hb@xt@.01(1.8) 
The linearized log–KdV equation (LABEL:linlogKdV) arises at the formal linearization of the log–KdV equation (LABEL:logKdV) at the perturbation . The Schrödinger operator is the Hessian operator of the second variation of at . Although in (LABEL:energy) is not a functional at , the second variation of is well defined at by
\hb@xt@.01(1.9) 
which is formally conserved in the time evolution of (LABEL:linlogKdV).
With new estimates to be obtained for the linearized log–KdV equation (LABEL:linlogKdV), we may hope to develop an ultimate solution of the outstanding problem on the orbital stability of the Gaussian solitary waves. Indeed, if we set for the solution to the log–KdV equation (LABEL:logKdV), we obtain an equivalent evolution equation
\hb@xt@.01(1.10) 
where the linearized part coincides with (LABEL:linlogKdV) and the nonlinear term is given by
It is clear that the nonlinear term does not behave uniformly in unless decays at least as fast as in (LABEL:Gaussian). On the other hand, if , where is a bounded function in its variables, then , where is analytic in for any . Therefore, obtaining new estimates for the linearized log–KdV equation (LABEL:linlogKdV) in a function space with Gaussian weights may be useful in the nonlinear analysis of the log–KdV equation (LABEL:logKdVw).
The spectrum of in consists of equally spaced simple eigenvalues
which include exactly one negative eigenvalue with the eigenvector (defined without normalization). Therefore, is not convex at in . Nevertheless, is positive in the constrained space
\hb@xt@.01(1.11) 
which corresponds to the fixed value in (LABEL:momentum) at the linearized approximation.
Several results were obtained for the linearized log–KdV equation (LABEL:linlogKdV). In [8], linear orbital stability of Gaussian solitary waves was obtained in the following sense: for every , there exists a unique global solution of the linearized log–KdV equation (LABEL:linlogKdV) which satisfies the following bound
\hb@xt@.01(1.12) 
for some independent positive constant . This result was obtained in [8] from the conservation of in the time evolution of smooth solutions to the linearized log–KdV equation (LABEL:linlogKdV), the symplectic decomposition of the solution , into the translational part and the residual part,
\hb@xt@.01(1.13) 
and the coercivity of in the squared norm in the sense
\hb@xt@.01(1.14) 
for some positive constant . The first two facts are rather standard in energy methods for linear PDEs, whereas the last fact, that is, the inequality (LABEL:coercivityassumed), should not be taken as granted.
In [2], the nonzero spectrum of the linear operator
\hb@xt@.01(1.15) 
was studied by using the Fourier transform that maps the thirdorder differential operator in physical space into a secondorder differential operator in Fourier space. Indeed, the Fourier transform applied to the linearized log–KdV equation (LABEL:linlogKdV) yields the time evolution in the form
\hb@xt@.01(1.16) 
where is the Fourier image of operator given by
\hb@xt@.01(1.17) 
By reducing the eigenvalue problem for to the symmetric Sturm–Liouville form, it was found in [2] that the spectrum of in is purely discrete and consists of a double zero eigenvalue and a symmetric sequence of simple purely imaginary eigenvalues such that
The double zero eigenvalue corresponds to the Jordan block
\hb@xt@.01(1.18) 
whereas the purely imaginary eigenvalues correspond to the eigenfunctions , which are smooth in but decay algebraically as . The Fourier transform of is supported on the halfline and decays like a Gaussian function at infinity. It follows from the spectrum of in that the Gaussian solitary waves are spectral stable.
The eigenfunctions of were also used in [2] for spectral decompositions in the constrained space in order to provide an alternative proof of the linear orbital stability of the Gaussian solitary waves. This alternative technique still relies on the conjecture of the coercivity of in the squared norm, that is, on the inequality (LABEL:coercivityassumed).
Because of the algebraic decay of the eigenfunctions of , it is not clear if a function of that decays like the Gaussian function as can be represented as series of eigenfunctions. Numerical simulations were undertaken in [2] to illustrate that solutions to the linearized log–KdV equation (LABEL:linlogKdV) with Gaussian initial data did not spread out as the time variable evolves. Nevertheless, the solutions exhibited visible radiation at the left slopes.
The present work is developed to obtain new estimates for the linearized log–KdV equation (LABEL:linlogKdV). In the first part of this work, we rely on the basis of Hermite functions in based Sobolev spaces and analyze the discrete operators that replace the differential operators. In the second part, we obtain dissipative estimates on the evolution of the linearized log–KdV equation (LABEL:linlogKdV) by representing solutions in terms of a convolution with the Gaussian solitary wave .
The paper is structured as follows. Section 2 sets up the basic formalism of the Hermite functions and reports useful technical estimates. Section 3 is devoted to the proof of the coercivity bound (LABEL:coercivityassumed). As explained above, this coercivity bound implies linear orbital stability of the Gaussian solitary wave in the constrained space and it is assumed to be granted in [2, 8]. The proof of coercivity relies on the decomposition of in terms of the Hermite functions.
Section 4 is devoted to the analysis of linear evolution expressed in terms of the Hermite functions. It is shown that this evolution reduces to the selfadjoint Jacobi difference operator with the limit circle behavior at infinity. As a result, a boundary condition is needed at infinity in order to define the spectrum of the Jacobi operator and to obtain the normpreserving property of the associated semigroup. Both linear orbital stability and spectral stability of Gaussian solitary waves (LABEL:Gaussian) is equivalently proven by using the Jacobi difference operator.
In Section 5, we give numerical approximations of eigenvalues and eigenvectors of the Jacobi difference equation. We show numerically that there exist subtle differences between the representation of eigenvectors of in the physical space and the representation of these eigenvectors by using decomposition in terms of the Hermite functions.
Section 6 reports weighted estimates for solutions to the linearized log–KdV equation (LABEL:linlogKdV) by using a convolution representation with the Gaussian weight. We show that the convolution representation is invariant under the time evolution of the linearized log–KdV equation (LABEL:linlogKdV), which is expressed by a dissipative operator on a halfline. The semigroup of the fundamental solution in the norm decays to zero exponentially fast as time goes to infinity.
Section 7 concludes the paper with discussions of further prospects.
Notations: We denote with the Sobolev space of times weakly differentiable functions on the real line whose derivatives up to order are in . The norm for in the Sobolev space is equivalent to the norm in the Lebesgue space . We denote with the space of square integrable functions with the weight . The set consists of all nonnegative integers, whereas the set includes only positive integers. The sequence space includes squared summable sequences, whereas contains finite (compactly supported) sequences.
Acknowledgements. The author thanks Gerald Teschl and Thierry Gallay for help on obtaining results reported in Sections 4 and 6, respectively. The research of the author is supported by the NSERC Discovery grant.
2 Preliminaries
We recall definitions of the Hermite functions [1, Chapter 22]:
\hb@xt@.01(2.1) 
where denote the set of Hermite polynomials, e.g.,
Hermite functions satisfy the Schrödinger equation for a quantum harmonic oscillator:
\hb@xt@.01(2.2) 
at equally spaced energy levels. By the Sturm–Liouville theory [12], the set of Hermite functions forms an orthogonal and normalized basis in .
In connection to the selfadjoint operator given by (LABEL:Schrodinger), we obtain the eigenfunctions of , from the correspondence . With proper normalization, we define
\hb@xt@.01(2.3) 
It follows from the wellknown relations for Hermite polynomials
that functions in the sequence satisfy the differential relations
\hb@xt@.01(2.4) 
The following elementary result is needed in further estimates.
Lemma 2.1
Let be given by
Then, there is a positive constant such that
\hb@xt@.01(2.5) 

Proof. We write
\hb@xt@.01(2.6) By Taylor series, for every and every , there is such that
\hb@xt@.01(2.7) Furthermore, we recall Euler’s constant given by the limit
\hb@xt@.01(2.8) Since is bounded as , the estimate (LABEL:Taylor) and the limit (LABEL:Eulergamma) yield the bound
\hb@xt@.01(2.9) for some positive constant . Substituting (LABEL:boundlog) into (LABEL:fmexp) proves the desired bound (LABEL:bounddecay).
The following technical result is needed for the proof of coercivity of the energy function.
Lemma 2.2
Let by defined by . Then, there is a positive constant such that
\hb@xt@.01(2.10) 

Proof. Multiplying the differential relation (LABEL:diffHermite) by and integrating by parts, we obtain
\hb@xt@.01(2.11) Integrating directly, we compute
\hb@xt@.01(2.12) Furthermore, using (LABEL:differenceintegral) at , we also compute
\hb@xt@.01(2.13) Thanks to orthogonality of Hermite functions, the righthand side of (LABEL:differenceintegral) is zero for and the numerical sequence satisfies the recurrence equation
\hb@xt@.01(2.14) starting with the initial values for and in (LABEL:f0) and (LABEL:f1). The recurrence equation (LABEL:differencesecondorder) admits the exact solution
\hb@xt@.01(2.15) Applying the bound (LABEL:bounddecay) of Lemma LABEL:lemmafsequence with , or , yields the bound (LABEL:projectiondecay).
3 Coercivity of the energy function
In order to prove the coercivity bound (LABEL:coercivityassumed), we define the compatible squared norm in space ,
\hb@xt@.01(3.1) 
The second variation is defined by (LABEL:energysecondvariation). The following theorem yields the coercivity bound for the energy function, which was assumed in [2, 8] without a proof.
Theorem 3.1
There exists a constant such that for every satisfying the constraints
\hb@xt@.01(3.2) 
it is true that
\hb@xt@.01(3.3) 
where .

Proof. The upper bound in (LABEL:coercivity) follows trivially from the identity
whereas the lower bound holds if there is a constant such that for every satisfying constraints (LABEL:constraints), it is true that
\hb@xt@.01(3.4) By the spectral theorem, we represent every by
\hb@xt@.01(3.5) where the vector belongs to . It follows from the first constraint in (LABEL:constraints) that . Using the norm in (LABEL:normH1), we obtain
Therefore,
and coercivity (LABEL:coercivityL2) is proved if we can show that is bounded by up to a multiplicative constant. To show this, we use the second constraint in (LABEL:constraints). Since , as it follows from (LABEL:f1), we have
\hb@xt@.01(3.6) By Lemma LABEL:lemmaprojections, there is a positive constant such that
\hb@xt@.01(3.7) which follows from convergence of . Hence, by using Cauchy–Schwarz inequality in (LABEL:projection2), we obtain
so that the bound (LABEL:coercivityL2) follows. The statement of the theorem is proven.
4 Time evolution of the linearized log–KdV equation
The time evolution of the linearized log–KdV equation (LABEL:linlogKdV) is considered in the constrained energy space given by (LABEL:constrainedspace). For a vector , we use the decomposition involving the Hermite functions,
\hb@xt@.01(4.1) 
By using and the differential relations (LABEL:diffHermite), the evolution problem for the vector is written as the lattice differential equation
\hb@xt@.01(4.2) 
It follows from (LABEL:latticec) for that if (so that ), then and for every . If , then it follows from (LABEL:latticec) for that the time evolution of a projection of to (which is proportional to the translational mode ) is given by
\hb@xt@.01(4.3) 
The projection is decoupled from the rest of the system (LABEL:latticec). Therefore, introducing for , we close the evolution system (LABEL:latticec) at the lattice differential equation
\hb@xt@.01(4.4) 
Since , then , so that we can introduce , with the vector . The sequence satisfies the evolution system in the skewsymmetric form
\hb@xt@.01(4.5) 
The evolution system (LABEL:latticea) can be expressed in the symmetric form by using the transformation
\hb@xt@.01(4.6) 
The new sequence satisfies the evolution system written in the operator form
\hb@xt@.01(4.7) 
where is the Jacobi operator defined by
\hb@xt@.01(4.8) 
The Jacobi difference equation is . According to the definition in Section 2.6 of [11], the Jacobi operator is said to have a limit circle at infinity if a solution of with is in for some . By Lemma 2.15 in [11], this property remains true for all . The following lemma shows that this is exactly our case.
Lemma 4.1
The Jacobi operator defined by (LABEL:Jacobif) has a limit circle at infinity.

Proof. Let us consider the case and define a solution of with . The numerical sequence satisfies the recurrence relation
starting with . Then, for even , whereas for odd is given by the exact solution
By Lemma LABEL:lemmafsequence with and , there exists a positive constant such that
\hb@xt@.01(4.9) This guarantees that .
By Lemma 2.16 in [11], the Jacobi operator with the domain
\hb@xt@.01(4.10) 
is selfadjoint if for all , where the discrete Wronskian is given by
\hb@xt@.01(4.11) 
and .
In order to define a selfadjoint extension of the Jacobi operator with the limit circle at infinity, we need to define a boundary condition as follows:
\hb@xt@.01(4.12) 
where is real. By Lemma 2.17 and Theorem 2.18 in [11], the operator , where and
\hb@xt@.01(4.13) 
represents a selfadjoint extension of the Jacobi operator with the limit circle at infinity.
Moreover, by Lemma 2.19 in [11], the real spectrum of in is purely discrete. Since there is at most one linearly independent solution of the Jacobi difference equation thanks to the recurrence relation in (LABEL:Jacobif), each isolated eigenvalue of the real spectrum of is simple.
By Lemma 2.20 in [11], all selfadjoint extensions of are uniquely defined by the choice in (LABEL:boundarycondition) spanned by a linear combination of two linearly independent solutions of . Since the value of plays no role for the Jacobi operator thanks again to the recurrence relation in (LABEL:Jacobif) and the value can be uniquely normalized to , we have a unique choice for given by the solution of with . Combining these facts together, we have obtained the following result.
Lemma 4.2
Let be a unique solution of with determined in the proof of Lemma LABEL:lemmacircle. Then, with the domain in (LABEL:domaincondition) is a unique selfadjoint extension of the Jacobi operator given by (LABEL:Jacobif). Moreover, the spectrum of consists of a countable set of simple real isolated eigenvalues.
The following theorem and corollary provide the linear orbital stability of Gaussian solitary waves (LABEL:Gaussian) expressed by using the decomposition in terms of Hermite functions. The same result was obtained in [2, 8] by using alternative techniques involving either the energy method [8] or the spectral decompositions [2].
Theorem 4.3
For every , there exists a unique solution to the evolution system (LABEL:latticea) for every satisfying .

Proof. The semigroup property of the solution operator both for and associated with the linear system (LABEL:latticef) follows from the result of Lemma LABEL:lemJacobi and the classical semigroup theory [9]. The result is transferred to the sequence by using the transformation (LABEL:transformationa).
Corollary 4.4
For every given by (LABEL:constrainedspace), there exists a unique solution to the linearized log–KdV equation (LABEL:linlogKdV) for every satisfying .

Proof. By using the transformations and for and the decomposition (LABEL:decompositionlinlogKdV), we obtain
\hb@xt@.01(4.14) where is set uniquely in . Recall that is an invariant reduction of the linearized log–KdV equation (LABEL:linlogKdV). The assertion of the corollary follows from Theorem LABEL:theoremlinearevolution and the equivalence (LABEL:equivalenceaenergy).
Remark 4.5
At the first glance, the detached equation (LABEL:projectionc1) might imply that the projection to the translational mode may grow at most linearly as in the energy space with conserved . However, it follows directly from the linearized log–KdV equation (LABEL:linlogKdV) [2] for the solution that
Therefore, we obtain as in the proof of Theorem LABEL:theoremcoercivity that
where both terms are globally bounded for all .
5 Numerical approximations of eigenvalues and eigenvectors
We discuss here numerical approximations of eigenvalues in the spectrum of the selfadjoint operator constructed in Lemma LABEL:lemJacobi. Let denote the simple real eigenvalues of with the ordering
These real eigenvalues of transform to eigenvalues of the spectral problem by using the decomposition (LABEL:decompositionlinlogKdV). Therefore, the result of Lemma LABEL:lemJacobi also provides an alternative proof of the spectral stability of the Gaussian solitary waves (LABEL:Gaussian), which is also established in [2]. However, by comparing the eigenvectors obtained in the two alternative approaches, we will see some sharp differences in the definition of function spaces these eigenvectors belong to.
To proceed with numerical approximations, we note that if is a solution of like in the proof of Lemma LABEL:lemmacircle, then for even . Let us denote for . From the bound (LABEL:boundv), we note that as .
Let be a solution of for . Then, we denote and for . It follows from the definition (LABEL:Jacobif) that and satisfy the coupled system of difference equations:
\hb@xt@.01(5.1) 
starting with . The discrete Wronskian (LABEL:Wronskian) is now explicitly computed as
\hb@xt@.01(5.2) 
Since generally as , by applying Lemma LABEL:lemmafsequence with and to the first equation of system (LABEL:coupleddifference), the limit exists and is generally nonzero. Moreover, the sign alternation of and ensures that the sequence is signdefinite for large enough , so that the limit is actually . This is confirmed in Figure LABEL:figWronskian(a), which shows the Wronskian sequence given by (LABEL:Wronskiannumerics) for .
Computing numerically by truncation of at a sufficiently large , e.g. at , we plot versus on Figure LABEL:figWronskian(b). Oscillations of are observed and the first two zeros of are located at
These values are nicely compared to the first two eigenvalues computed in [2] for :
The numerical approximations confirm that the eigenvalues obtained by using the Jacobi difference equation are the same as the eigenvalues obtained in [2] from the Sturm–Liouville problem derived in the Fourier space. Numerically, we find for the first two zeros of the limiting Wronskian that the decay rate of the sequence remains generic:
but the decay rate of the sequence becomes faster:
Let us now recall the correspondence of eigenvectors of the Jacobi difference equation and eigenvectors of the linearized log–KdV operator . From the previous transformations, we obtain
\hb@xt@.01(5.3) 
where
\hb@xt@.01(5.4) 
are respectively the odd and even components of the eigenvector with respect to . Thanks to the decay of the sequences and , we note that
\hb@xt@.01(5.5) 
but that . Therefore, generally defined by (LABEL:linearizedoperator). Thus, the eigenvector given by (LABEL:decompositionnumerics) does not solve the eigenvalue problem in the classical sense compared to the eigenvectors constructed in [2] with the Fourier transform.
In order to clarify the sense for the eigenvectors given by (LABEL:decompositionnumerics), we denote and project the eigenvalue problem to and . The projection is uniquely found by
\hb@xt@.01(5.6) 
which can also be obtained from the projection equation (LABEL:projectionc1). The component satisfies formally . After separating the even and odd parts of the eigenvalue problem, we obtain the coupled system
\hb@xt@.01(5.7) 
As we have indicated above, it is difficult to prove that each term of the coupled system (LABEL:linearizedformal) belongs to if and are given by the decomposition (LABEL:decompositionnumerics) in terms of the Hermite functions. In order to formulate the coupled problem (LABEL:linearizedformal) rigorously, we would like to show that the components of the eigenvector belong to
\hb@xt@.01(5.8) 
and satisfy the coupled system