arXiv subjects

Adilbek Kairzhan

Publications and source records attributed to Adilbek Kairzhan.

14 recordsLinked to original sources

arXiv2026

Global dynamics for the 1d quartic Klein-Gordon equation with an internal mode

We study a one dimensional nonlinear Klein-Gordon equation with a potential, a localized quadratic nonlinearity, and a non-localized quartic nonlinearity. We assume that the linearized operator has a single discrete eigenvalue below the continuous spectrum, corresponding to an ''internal mode''. This eigenvalue generates localized, time-periodic solutions for the linear equation. Assuming the natural Fermi Golden Rule, we give a global description of the dynamics of the amplitude of the internal mode through the radiation damping mechanism, and prove scattering for the radiation. Our approach is based on the distorted Fourier transform and a collection of refined dispersive decay and smoothing estimates. The corresponding cubic problem is closely related to the question of global-in-space dispersive asymptotics for perturbations of kink solutions in the classical $ϕ^4$ model, which remains a challenging open question. From the perspective of nonlinear estimates, a quartic (non-localized) interaction is essentially sharp for our analysis, and closing the estimates with such a low degree nonlinear term requires a careful and delicate treatment. As a direct application of our analysis, we prove a result on the asymptotic stability of kinks for classes of scalar field models that can be seen as perturbations of the $ϕ^4$ model.

math.AP↗

arXiv2026

Traveling Waves for Nonlocal Derivative Nonlinear Schrödinger Equations: A Variational Characterization

We establish several existence results for traveling-wave solutions of the nonlocal derivative nonlinear Schrödinger equation with general coefficients by variational methods. We study associated minimization problems in the subcritical and critical cases and prove the existence of a minimizer in each case. Finally, we derive Pohozaev-type identities and use them to establish corresponding nonexistence results.

math.AP↗

arXiv2024

A Hamiltonian Dysthe equation for hydroelastic waves in a compressed ice sheet

Nonlinear hydroelastic waves along a compressed ice sheet lying on top of a two-dimensional fluid of infinite depth are investigated. Based on a Hamiltonian formulation of this problem and by applying techniques from Hamiltonian perturbation theory, a Hamiltonian Dysthe equation is derived for the slowly varying envelope of modulated wavetrains. This derivation is further complicated here by the presence of cubic resonances for which a detailed analysis is given. A Birkhoff normal form transformation is introduced to eliminate non-resonant triads while accommodating resonant ones. It also provides a non-perturbative scheme to reconstruct the ice-sheet deformation from the wave envelope. Linear predictions on the modulational instability of Stokes waves in sea ice are established, and implications for the existence of solitary wavepackets are discussed for a range of values of ice compression relative to ice bending. This Dysthe equation is solved numerically to test these predictions. Its numerical solutions are compared to direct simulations of the full Euler system, and very good agreement is observed.

math.AP↗

arXiv2024

Asymptotic stability near the soliton for quartic Klein-Gordon in 1D

We consider the nonlinear focusing Klein-Gordon equation in $1 + 1$ dimensions and the global space-time dynamics of solutions near the unstable soliton. Our main result is a proof of optimal decay, and local decay, for even perturbations of the static soliton originating from well-prepared initial data belonging to a subset of the stable manifold constructed in Bates-Jones (Dynamics reported, 1989) and Kowalczyk-Martel-Muñoz (J. Eur. Math. Soc., 2021). Our results complement those of Kowalczyk-Martel-Muñoz (J. Eur. Math. Soc., 2021) and confirm numerical results of Bizon-Chmaj-Szpak (J. Math. Phys., 2011) when considering nonlinearities $u^p$ with $p \geq 4$. In particular, we provide new information both local and global in space about asymptotically stable perturbations of the soliton under localization assumptions on the data.

math.AP↗

arXiv2024

Interaction between long internal waves and free surface waves in deep water

We consider a density-stratified fluid composed of two immiscible layers separated by a sharp interface. We study the regime of long internal waves interacting with modulated surface wave packets and describe their resonant interaction by a system of equations where the internal wave solves a high-order Benjamin-Ono (BO) equation coupled to a linear Schrödinger equation for the envelope of the free surface. The perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler's equations as described in Benjamin-Bridges [J. Fluid Mech. 333, 1997] and Craig-Guyenne-Kalisch [Comm. Pure Appl. Math. 58, 2005]. We also establish a local wellposedness result for a reduced BO-Schrödinger system using an approach developed by Linares-Ponce-Pilod [J. Diff. Eqs. 250, 2011].

math.AP↗

arXiv2022

A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

This paper is a study of the water wave problem in a two-dimensional domain of infinite depth in the presence of nonzero constant vorticity. A goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear quasi-monochromatic surface gravity waves. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we derive a Hamiltonian Dysthe equation for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational instability of Stokes waves depending on its direction and strength. Our method also provides a non-perturbative procedure to reconstruct the surface elevation from the wave envelope, based on the Birkhoff normal form transformation to eliminate all non-resonant triads. This model is tested against direct numerical simulations of the full Euler equations and against a related Dysthe equation recently derived by Curtis, Carter and Kalisch (J. Fluid Mech. 855, 2018) in the context of constant vorticity. Very good agreement is found for a range of values of the vorticity.

math.AP↗

arXiv2022

Standing waves on quantum graphs

We review evolutionary models on quantum graphs expressed by linear and nonlinear partial differential equations. Existence and stability of the standing waves trapped on quantum graphs are studied by using methods of the variational theory, dynamical systems on a phase plane, and the Dirichlet-to-Neumann mappings.

math.AP↗

arXiv2021

Hamiltonian Dysthe equation for 3D deep-water gravity waves

This article concerns the water wave problem in a three-dimensional domain of infinite depth and examines the modulational regime for weakly nonlinear wavetrains. We use the method of normal form transformations near the equilibrium state to provide a new derivation of the Hamiltonian Dysthe equation describing the slow evolution of the wave envelope. A precise calculation of the third-order normal form allows for a refined reconstruction of the free surface. We test our approximation against direct numerical simulations of the three-dimensional Euler system and against predictions from the classical Dysthe equation, and find very good agreement.

math.AP↗

arXiv2021

Multi-pulse edge-localized states on quantum graphs

Edge-localized stationary states of the focusing nonlinear Schrodinger equation on a general quantum graph are considered in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which approach asymptotically to a composition of N solitons, each sitting on a bounded (pendant, looping, or internal) edge. Not only we prove that such states exist in the limit of large mass, but also we compute the precise Morse index (the number of negative eigenvalues in the corresponding linearized operator). In the case of the edge-localized N-soliton states on the pendant and looping edges, we prove that the Morse index is exactly N. The technical novelty of this work is achieved by avoiding elliptic functions (and related exponentially small scalings) and closing the existence arguments in terms of the Dirichlet-to-Neumann maps for relevant parts of the given graph.

math-ph↗

arXiv2020

Standing waves on a flower graph

A flower graph consists of a half line and $N$ symmetric loops connected at a single vertex with $N \geq 2$ (it is called the tadpole graph if $N = 1$). We consider positive single-lobe states on the flower graph in the framework of the cubic nonlinear Schrodinger equation. The main novelty of our paper is a rigorous application of the period function for second-order differential equations towards understanding the symmetries and bifurcations of standing waves on metric graphs. We show that the positive single-lobe symmetric state (which is the ground state of energy for small fixed mass) undergoes exactly one bifurcation for larger mass, at which point $(N-1)$ branches of other positive single-lobe states appear: each branch has $K$ larger components and $(N-K)$ smaller components, where $1 \leq K \leq N-1$. We show that only the branch with $K = 1$ represents a local minimizer of energy for large fixed mass, however, the ground state of energy is not attained for large fixed mass. Analytical results obtained from the period function are illustrated numerically.

math.AP↗

arXiv2019

Drift of spectrally stable shifted states on star graphs

When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated along the edges with a translational parameter and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable because of the irreversible drift along the incoming edge towards the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph.

math.AP↗

arXiv2018

Orbital instability of standing waves for NLS equation on Star Graphs

We consider a nonlinear Schrödinger (NLS) equation with any positive power nonlinearity on a star graph $Γ$ ($N$ half-lines glued at the common vertex) with a $δ$ interaction at the vertex. The strength of the interaction is defined by a fixed value $α\in \mathbb{R}$. In the recent works of Adami {\it et al.}, it was shown that for $α\neq 0$ the NLS equation on $Γ$ admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for $α<0$ that, in the NLS equation with a subcritical power-type nonlinearity, the unique symmetric standing wave is orbitally stable. In this paper, we analyze stability of standing waves for both $α<0$ and $α>0$. By extending the Sturm theory to Schrödinger operators on the star graph, we give the explicit count of the Morse and degeneracy indices for each standing wave. For $α<0$, we prove that all nonsymmetric standing waves in the NLS equation with any positive power nonlinearity are orbitally unstable. For $α>0$, we prove the orbital instability of all standing waves.

math.AP↗

arXiv2017

Spectral stability of shifted states on star graphs

We consider the nonlinear Schrödinger (NLS) equation with the subcritical power nonlinearity on a star graph consisting of $N$ edges and a single vertex under generalized Kirchhoff boundary conditions. The stationary NLS equation may admit a family of solitary waves parameterized by a translational parameter, which we call the shifted states. The two main examples include (i) the star graph with even $N$ under the classical Kirchhoff boundary conditions and (ii) the star graph with one incoming edge and $N-1$ outgoing edges under a single constraint on coefficients of the generalized Kirchhoff boundary conditions. We obtain the general counting results on the Morse index of the shifted states and apply them to the two examples. In the case of (i), we prove that the shifted states with even $N \geq 4$ are saddle points of the action functional which are spectrally unstable under the NLS flow. In the case of (ii), we prove that the shifted states with the monotone profiles in the $N-1$ outgoing edges are spectrally stable, whereas the shifted states with non-monotone profiles in the $N-1$ outgoing edges are spectrally unstable, the two families intersect at the half-soliton states which are spectrally stable but nonlinearly unstable. Since the NLS equation on a star graph with shifted states can be reduced to the homogeneous NLS equation on a line, the spectral instability of shifted states is due to the perturbations breaking this reduction. We give a simple argument suggesting that the spectrally stable shifted states are nonlinear unstable under the NLS flow due to the perturbations breaking the reduction to the NLS equation on a line.

math.AP↗

arXiv2017

Nonlinear Instability of Half-Solitons on Star Graphs

We consider a half-soliton stationary state of the nonlinear Schrodinger equation with the power nonlinearity on a star graph consisting of N edges and a single vertex. For the subcritical power nonlinearity, the half-soliton state is a degenerate critical point of the action functional under the mass constraint such that the second variation is nonnegative. By using normal forms, we prove that the degenerate critical point is a nonlinear saddle point, for which the small perturbations to the half-soliton state grow slowly in time resulting in the nonlinear instability of the half-soliton state. The result holds for any $N \geq 3$ and arbitrary subcritical power nonlinearity. It gives a precise dynamical characterization of the previous result of Adami {\em et al.}, where the half-soliton state was shown to be a saddle point of the action functional under the mass constraint for $N = 3$ and for cubic nonlinearity.

math.AP↗