Phase-Shifted Nanopteron Solutions to a Singularly Perturbed Korteweg--de Vries Equation

Abstract: We construct nanopteron solutions to the differential equation [ ε^2u^{(4)}+u''-u+u^2 =0, ] which is a singular perturbation of the traveling wave problem $u&#39;&#39;-u+u<sup>2=0$ for the Korteweg--de Vries (KdV) equation. These nanopteron solutions are the superposition of the sech$<sup>2$-type KdV traveling wave profile, a small localized error, and a periodic "ripple." The ripple is parametrized by its amplitude and its phase shift. In the process of constructing the nanopteron, we preselect one of these parameters and then solve for the other, and we provide detailed commentary on how the amplitude and phase shift solution processes differ. Our work revisits several prior constructions of nanopteron solutions for this singularly perturbed KdV equation in light of modern developments of nanopterons for lattice differential equations. In the context of these contemporary methods, the simple structure of the singularly perturbed KdV equation allows it to serve as an accessible template for nanopteron constructions in more complicated problems.

• {This} paper presents a modernized construction of phase-shifted nanopterons for a singularly perturbed KdV equation. This solves for both amplitude and phase-shift simultaneously,.
• {This} construction templates complex settings including water wave and lattice, systems adding system increase, establishing a clear connection between localizations and ripple corrections.
• {This} discovery resolves key limitations of existing non existence-proofs and provides rigorous amplitude estimates and nonlinear scaling relationships between periodic wave - amplitude.

The problem and its context

The paper studies the singularly perturbed fourth-order boundary value problem

$\epsilon^2 u^{(4)} + u'' - u + u^2 = 0,$

which is the traveling wave equation for the fifth-order Korteweg–de Vries (fKdV) equation and arises from the full water wave problem in the small surface tension regime $0 < \beta < 1/3$ with Froude number $F > 1$. At $\epsilon = 0$ the problem reduces to the KdV traveling wave equation, whose solitary wave is $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$, analytic on strips of width less than $\pi$ and exponentially localized. For $\epsilon > 0$, Amick and McLeod proved that no solution vanishing at both infinities exists; instead the problem admits nanopterons—superpositions of a localized core, a small localized error, and a far-field periodic ripple whose amplitude is exponentially small in $\epsilon$.

The paper's contribution is a self-contained, modernized construction of phase-shifted nanopterons that treats both selection mechanisms simultaneously: fixing the ripple's phase shift and solving for its amplitude (as in Amick–Toland and Sun–Shen), and fixing an exponentially small nonzero amplitude and solving for the phase shift (as in Lombardi). The author positions this as a template for nanopteron constructions in Fermi–Pasta–Ulam–Tsingou (FPUT) dimer lattices, where analogous constructions exist but are complicated by two-component nonlocal systems.

Main results

Two theorems constitute the core. In the amplitude-selection theorem, given any phase shift $\theta \in [0,\pi)$ there exist $\alpha_\epsilon$ and analytic functions $0 < \beta < 1/3$0, $0 < \beta < 1/3$1 such that

$0 < \beta < 1/3$2

solves the problem, with the ripple frequency satisfying $0 < \beta < 1/3$3, the localized remainder bounded by $0 < \beta < 1/3$4, and—most notably—the amplitude estimate

$0 < \beta < 1/3$5

This one-power improvement at nonzero phase shift agrees with Sun's formal asymptotics. In the phase-shift-selection theorem, prescribing an amplitude $0 < \beta < 1/3$6 with $0 < \beta < 1/3$7 in any compact subinterval of $0 < \beta < 1/3$8 yields a phase shift $0 < \beta < 1/3$9 with $F > 1$0. A caveat is stated plainly: the selected phase shift may be zero, since the argument does not rule out $F > 1$1; by contrast, Lombardi's normal-form approach guarantees a nonzero phase shift because his solvability condition carries its own asymptotic phase parameter that interacts with the ripple's phase. Nonvanishing of the amplitude itself is not proven here either—it follows only indirectly from Amick and McLeod's nonexistence of solitary waves.

Mechanism: solvability condition and oscillatory integrals

The linearization at zero, $F > 1$2, factors via the critical frequency $F > 1$3 as $F > 1$4. Its range on the strip-analytic space $F > 1$5 is characterized by the single solvability condition $F > 1$6 (one condition suffices because only even solutions are sought; asymmetric nanopterons are excluded by prior work of Benilov, Grimshaw, and Kuznetsova). Lombardi's oscillatory integral estimate gives $F > 1$7, which is what makes exponentially small amplitudes tractable: factors of $F > 1$8 introduced by the ansatz are cancelled by the exponential smallness of $F > 1$9 applied to analytic integrands.

Perturbing directly from $\epsilon = 0$0 yields an overdetermined system—one unknown $\epsilon = 0$1 must satisfy both the differential equation and the solvability condition—which explains why solitary waves fail. Introducing the periodic ripple supplies the second unknown (amplitude or phase shift), closing the problem. Conceptually, the operator controlling the remainder equation is $\epsilon = 0$2 with $\epsilon = 0$3, i.e., the KdV linearization plus the singular perturbation bundled into one component—a structural simplification relative to dimer lattice problems, and closer in spirit to Johnson and Wright's Whitham construction, where the localized core "through a sort of weak resonance, excites a very small amplitude periodic wave."

Construction pipeline

The proof proceeds through several stages. First, even periodic solutions $\epsilon = 0$4 of the nonlinear problem are built via a quantitative Crandall–Rabinowitz-type bifurcation from the simple eigenvalue $\epsilon = 0$5 at frequency $\epsilon = 0$6, formulated abstractly in Hilbert spaces with uniform transversality and coercivity estimates; the coefficients are power series in $\epsilon = 0$7 whose terms are trigonometric polynomials of degree $\epsilon = 0$8, a degree bound essential for strip extension. Second, these periodics are extended analytically to the strip $\epsilon = 0$9 with an asymptotic phase shift built in via the "phase selector" $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$0, using the rescaled amplitude $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$1; the extension requires the exponentially small amplitude, while the exponentially small amplitude requires the extension—a circularity resolved by constructing both simultaneously. Third, a refined leading-order limit replaces $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$2 by $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$3 with $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$4, improving the residual from $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$5 to $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$6 and thereby sharpening the amplitude estimates; the author notes this step is a novelty rather than a necessity here, unlike in the small-mass FPUT and mass-in-mass problems where it was indispensable.

With the ansatz $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$7, the phase shift generates residual terms analyzed order by order in $\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$8; the dominant contributions combine into a selection mechanism

$\sigma(z) = \tfrac{3}{2}\sech^2(z/2)$9

where $\pi$0 admits the explicit expansion $\pi$1 with $\pi$2, uniformly bounded below away from zero for $\pi$3 bounded away from $\pi$4. This lower bound permits division and converts the mechanism into fixed-point equations for $\pi$5 or $\pi$6.

Fixed-point arguments and their asymmetry

A key technical observation is that the invertibility of $\pi$7 follows from viewing it as a perturbation of $\pi$8, which factors the KdV linearization $\pi$9. Although $\epsilon > 0$0 does not converge to $\epsilon > 0$1 in operator norm, its inverse converges effectively to $\epsilon > 0$2—a phenomenon also present in the lattice problems.

The two selection problems then diverge structurally. Lipschitz estimates in amplitude carry a factor $\epsilon > 0$3 (visible already in the toy identity $\epsilon > 0$4), forcing "decay-borrowing" arguments and a mixed-space contraction over a three-space nesting $\epsilon > 0$5 of decaying analytic function spaces. Lipschitz estimates in phase shift lack this factor, so the phase-shift problem needs only an ordinary contraction in $\epsilon > 0$6 once $\epsilon > 0$7 is selected—but $\epsilon > 0$8 itself cannot be obtained by contraction, because the phase-shift map has poor Lipschitz dependence on $\epsilon > 0$9; instead the intermediate value theorem is applied to $\epsilon$0. The author resolves an apparent circularity in choosing the contraction radius $\epsilon$1 (on which the phase-shift map depends) carefully in the appendix.

Comparison with prior approaches

Relative to Amick and Toland, the treatment avoids separating the zero and nonzero phase shift cases, runs a genuine two-component fixed point rather than solving amplitude from the remainder first, and connects the governing operator transparently to the invertible KdV linearization. Relative to Sun and Shen, who decoupled the problem via changes of variables into "KdV coupled to an oscillatory field," the present work keeps the one-component structure intact. Relative to Lombardi, whose general framework subsumes this problem only after a nonlinear normal form change of variables involving doing and undoing a long wave scaling, the construction applies directly and exposes the Lyapunov–Schmidt and bifurcation mechanisms explicitly. The appendix also develops Sun's formal amplitude–phase asymptotics rigorously as an expansion of the selection mechanism:

$\epsilon$2

Limitations and open questions

Several limits are conceded. The nonvanishing of the ripple amplitude and of the selected phase shift are not established within the framework; the latter is attributed to the absence of Lombardi-style Jost-solution solvability conditions carrying an intrinsic asymptotic phase. The proposed remedy—replacing $\epsilon$3 with an integral against an asymptotically sinusoidal Jost function $\epsilon$4 and inverting $\epsilon$5 jointly—is outlined but not carried out. An alternative factorization route inspired by Amick and Toland's regular perturbation $\epsilon$6 of $\epsilon$7 is likewise conjectured rather than executed. The extension program for periodics is shown to hold for vector-valued functions, suggesting exportability to multicomponent systems, but the natural next step—nanopterons in irreducible "KdV coupled to an oscillatory field" systems—remains open, as do the corresponding questions for dimer lattices: whether dimer solitary waves exist at special parameter values, and whether arbitrary phase shifts can be preselected under the nonlocal structure of lattice traveling wave problems.

Conclusion

The paper delivers rigorous, exponentially small existence results for phase-shifted nanopterons of the singularly perturbed KdV traveling wave problem under both selection protocols, with amplitude estimates $\epsilon$8 (zero phase shift) and $\epsilon$9 (nonzero phase shift), and phase shifts of size $\theta \in [0,\pi)$0 at prescribed amplitude. By isolating the roles of the Lyapunov–Schmidt decomposition, Lombardi's oscillatory integral estimates, decay-borrowing, and mixed-space contractions in the simplest possible setting, it provides a coherent reference architecture for nanopteron constructions in more complex water wave, Whitham, and FPUT lattice problems.