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Persistence of a unique periodic wave train in convecting shallow water fluid

  • Sumin Yang and Qian Wen EMAIL logo
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

The coexistence of a traveling pulse and a periodic traveling wave was established in a convecting shallow water model when taking a nonlinear buoyancy term u u x . In this brief communication, we show that the mechanical balance underlying this coexistence is disrupted by a stronger nonlinear dissipation ( u 2 u x ) x , which arises from an enhanced buoyancy term u 2 u x . Consequently, the convecting shallow water model exhibits either a unique periodic wave or a unique solitary wave, each within a fixed range of wave speeds. Furthermore, we show that the wave speed is monotonic with respect to the wave amplitude and is smaller than that observed in the model with the buoyancy term u u x . A numerical study is performed to verify the theoretical study.

Keywords: Abelian integral; solitary wave; periodic traveling wave
MSC 2010: 35B10; 35B20; 35B25

1 Introduction

When modeling real-world problems, it is crucial to consider higher-order truncations or small corrections due to unavoidable external influences. In convective situations, the shallow water model [1], given by the equation

(1) u t + λ 1 u u x + λ 3 u x x x + ε ( λ 2 u x x + λ 4 u x x x x + λ 5 ( u u x ) x ) = 0

is more applicable than the classical KdV model

u t + u u x + u x x x = 0

which was derived under ideal assumptions. In equation (1), ε is sufficiently small and appears alongside λ 2 , λ 4 , and λ 5 as specified in [1]. Equation (1) can model water motion in a variety of dissipative scenarios, such as a shallow liquid layer exposed to the air and heated from the air side, or a shallow water layer undergoing shear (modeling internal waves) [2]. The parameters λ i s can take a wide range of values, from positive to negative, and are generally considered as free parameters [3,4]. For example, when modeling shallow fluid layers under Marangoni-Bénard convection, both λ 4 and λ 5 are positive. The sign of λ 2 is negative if the instability threshold is greater than the diffusion, and positive otherwise. The sign of λ 3 is determined by the air-liquid surface tension. The signs of all coefficients may change when considering the buoyancy effect u u x , refer [3,4] and references therein. In the case of a long layer with much smaller surface tension, the coefficients can be approximated as λ 1 = λ 2 = λ 3 = λ 4 = 1 , and λ 5 = 0 , as demonstrated in Topper and Kawahara’s study [5] and related study on the existence of periodic waves [6,7]. Specifically, Aspe and Depassier [8] derived approximations for the parameters as follows:

(2) λ 1 = 3 2 σ G ( 10 + σ G ) , λ 2 = σ R 2 15 , λ 3 = σ G 2 1 3 + 34 21 σ , λ 4 = 682 σ 3 G + 717 σ 2079 , λ 5 = 8 G ,

where ε is a small parameter that expresses the excess of the Rayleigh number beyond its critical values as ε R 2 , while σ denotes the Prandtl number, and G denotes the Galileo number.

Equation (1) demonstrates numerous variations in various contexts, including delay dissipation [9], the elimination of one or two terms in (1) [10,11], and the extension of the model to incorporate a more potent nonlinear buoyancy term u 2 u x [12–14],

(3) u t + λ 1 u 2 u x + λ 3 u x x x + ε ( λ 2 u x x + λ 4 u x x x x + λ 5 ( u u x ) x ) = 0 .

The fundamental study of equations (1) and (3) focuses on traveling waves, including traveling pulses (solitary waves) and periodic waves (wave trains). For equation (1), the existence of traveling pulses was rigorously proven in [3,15] through dynamical analysis, with numerical verification performed in [4] as a follow-up to [3]. For equation (3), the traveling pulse was constructed using the same method employed in [15], as detailed in [12,13]. Notably, the existence of periodic wave trains in both equations, as well as the coexistence of a periodic wave train and a pulse in equation (3) and their non-coexistence in equation (1), was rigorously derived in [2] through the analysis of a planar dynamical system on a normally hyperbolic manifold.

It is noteworthy that a stronger nonlinear buoyancy u 2 u x must induce a stronger dissipative effect ( u 2 u x ) x , since the perturbation in (1) and (3) is the first-order corrected approximation compared to the zero-order approximation (the unperturbed equation). Therefore, it is reliable to explore the periodic wave trains and pulse of the extended perturbed KdV equation,

(4) u t + λ 1 u 2 u x + λ 3 u x x x + ε ( λ 2 u x x + λ 4 u x x x x + λ 5 ( u 2 u x ) x ) = 0 .

The main purpose of this study is to provide complementary results to [2]. Specifically, we prove that (i) the mechanical balance on the wave coexistence of a traveling pulse and a periodic wave train is lost due to the stronger dissipation ( u 2 u x ) x . The convecting shallow model only possesses a unique periodic wave train within a fixed speed range or a unique pulse with a certain speed. (ii) We assess the impact of the dissipation on the wave speed and compare the new wave range for periodic wave trains with the previous ones by taking the same physical parameters as in [2].

The study is organized as follows: First, we briefly summarize the transformation of the partial differential equation (PDE) model into a planar dynamical system and prove that the Melnikov function for (3) has at most one zero, indicating the uniqueness of either a periodic wave train or a pulse in (4). In Section 3, we conduct a numerical study using the parameters employed in [2] to explore the periodic wave train and investigate its related wave speed. Finally, in Section 4, we present our conclusions.

2 Existence of a unique periodic wave train and pulse

In this section, we concisely illustrate the reduction of the PDE model to a planar dynamical system and study the zeros of the bifurcation function associated with the perturbed Hamiltonian system. We omit extensive computational analysis as our result is a complementary part to [2], and the detailed analysis is the same as the system reduction in [2], except for the terms with λ 5 . Substituting the wave profile z = x − c t into equation (4), direct integration yields a third-order ordinary differential equation (ODE),

(5) − c u + λ 1 3 u 3 + λ 3 d 2 u d z 2 + ε λ 2 d u d z + λ 5 u 2 d u d z + λ 4 d 3 u d z 3 = 0 ,

where the boundary conditions are taken as d u d z , d 2 u d z 2 , d 3 u d z 3 → 0 as z → ∞ . Equation (5) is the standard form of singular perturbation as the higher-order term has a small parameter ε . To explore the associated critical manifold and its hyperbolic property, we can rewrite equation (5) into a first-order ODE system. This ODE system is a three-dimensional dynamical system, which has a projection on a normally hyperbolic manifold M ε to be a perturbed two-dimensional Hamiltonian system,

d u d z = v , d v d z = 1 λ 3 c u − λ 1 3 u 3 + ε λ 3 2 ( − ( λ 2 λ 3 + λ 4 c ) − λ 3 λ 5 u 2 + λ 1 λ 4 u 2 ) v + O ( ε 2 ) .

Taking the parameters α 1 = 3 c λ 1 , α 2 = c 3 λ 1 λ 3 , and k = λ 3 c , and introducing the scaling ε ˜ = 3 c ( λ 1 λ 4 − λ 3 λ 5 ) k ε λ 1 λ 3 2 , u = α 1 u ˜ , v = α 2 v ˜ , and z = k ξ , we have a simpler form by dropping the tildes on the symbols,

(6) d u d z = v , d v d z = u ( 1 − u 2 ) + ε ( β 0 + u 2 ) v ,

and

β 0 = − λ 1 λ 2 λ 3 + λ 1 λ 4 c 3 c ( λ 1 λ 4 − λ 3 λ 5 ) .

System (6) ε = 0 has a Hamiltonian function

H ( u , v ) = v 2 2 − u 2 2 + u 4 4 ≔ v 2 2 + A ( u ) .

The portrait of system (6) ε = 0 is depicted in Figure 1 in [2], and there is a double homoclinic loop. The right branches of this loop correspond to the light solitary wave of equation (3) ε = 0 , while the left branches correspond to the dark solitary wave. We investigate the right branch because the wave motion is investigated on a shallow water layer. Inside the right branch (satisfying H ( 0 , 0 ) = 0 ), the periodic orbits { Γ h } and the right homoclinic loop Γ 0 correspond to periodic wave trains and traveling pulses, respectively, for the shallow water model (3) ε = 0 . The center is defined by H ( 1 , 0 ) = − 1 4 . Under perturbation, most periodic orbits are spirally broken, only a certain number of Γ h persist as limit cycles of (6) or periodic wave trains of (4). Correspondingly, we take the Melnikov function as the main tool [16] to investigate the persisting periodic orbits and the homoclinic loop,

M ( h ) = ∮ Γ h ( β 0 + u 2 ) v d u .

Figure 1 The speed of the unique persisting periodic wave train is an increasing function with respect to the amplitude. (a) c ( h ) c\left(h) is increasing on h ∈ ( − 1 4 , 0 ) h\in \left(-\frac{1}{4},0) and (b) c ( h ( u ) ) c\left(h\left(u)) is increasing on u ∈ ( 1 , 2 ) u\in \left(1,\sqrt{2}) .
Figure 1

The speed of the unique persisting periodic wave train is an increasing function with respect to the amplitude. (a) c ( h ) is increasing on h ∈ ( − 1 4 , 0 ) and (b) c ( h ( u ) ) is increasing on u ∈ ( 1 , 2 ) .

Theorem 1

Consider the shallow water equation (4), there is a unique periodic wave train if

β 0 = − λ 1 λ 2 λ 3 + λ 1 λ 4 c 3 c ( λ 1 λ 4 − λ 3 λ 5 ) ∈ − 1 , − 4 5 .

The wave speed for the persisting wave train is an increasing function with respect to the amplitude h, and the amplitude increases with respect to β 0 . In particular, when β 0 = − 4 5 , there exists a traveling pulse for the shallow water model (4).

Proof

We write

(7) M ( h ) = I 0 ( h ) β 0 + I 2 ( h ) I 0 ( h ) ≔ I 0 ( h ) ( β 0 + r ( h ) ) ,

where I 0 ( h ) = ∮ Γ h v d u , I 2 ( h ) = ∮ Γ h u 2 v d u . First, I 0 ( h ) = ∮ Γ h v d u = ∮ i n t Γ h d u d v > 0 by Green formula. Then, r ( h ) = I 2 ( h ) I 0 ( h ) is well defined.

Next we show r ( h ) is monotonic in h ∈ ( − 1 6 , 0 ) . Let

ℒ i ( u , z ) = u i A ′ ( u ) − z i A ′ ( z ) ,

where i = 0 , 2 , ( u , z ) is on the curve { ( u , z ) : q ˜ ( u , z ) = 2 u 2 + 2 u z + 2 z 2 − 3 u − 3 z = 0 } , q ˜ ( u , z ) is the factor of A ( u ) − A ( z ) = 0 . Then,

d d u ℒ i ( u , z ) = ∂ ∂ u ℒ i ( u , z ) − ∂ ∂ z ℒ i ( u , z ) × ∂ u q ˜ ( u , z ) ∂ z q ˜ ( u , z ) , i = 0 , 2 .

A careful computation on the curve { ( u , z ) : q ˜ ( u , z ) = 0 } gives that

ℒ 0 ( u , z ) ≠ 0 and ℒ 0 ( u , z ) ℒ 2 ( u , z ) ℒ 0 ′ ( u , z ) ℒ 2 ′ ( u , z ) ≠ 0

on { ( u , z ) ∣ 1 < z < 3 2 , 0 < u < 1 } . Therefore, r ( h ) is monotonic for h ∈ ( − 1 6 , 0 ) by the Chebyshev criterion [17]. Furthermore, a direct computation gives

lim h → − 1 6 r ( h ) = 1 and lim h → 0 r ( h ) = 4 5 .

Hence, r ( h ) decreases in the interval ( − 1 6 , 0 ) from 1 to 4 5 monotonically. By (7), when selecting an appropriate β 0 ∈ ( − 1 , − 4 5 ) , M ( h ) must possess a simple root at some h , indicating a unique periodic wave train of equation (4) that exhibits an increasing wave speed corresponding to the wave amplitude,

(8) c = c ( h ) = λ 1 λ 2 λ 3 3 ( λ 1 λ 4 − λ 3 λ 5 ) r ( h ) − λ 1 λ 4 ∈ λ 1 λ 2 λ 3 2 λ 1 λ 4 − 3 λ 3 λ 5 , 5 λ 1 λ 2 λ 3 7 λ 1 λ 4 − 12 λ 3 λ 5 .

Furthermore, β 0 = β 0 ( h ) = − r ( h ) is monotonically increasing, as well as the amplitudes with respect to β 0 ,

h ( β 0 ) = ( − r ( h ) ) − 1 ( β 0 ) .

In particular, M ( 0 ) = 0 when β 0 = − 4 5 . Then, the homoclinic loop Γ 0 persists under the perturbation according to the Poincaré bifurcation theory [16], indicating the existence of a traveling pulse in the shallow model (4) with a wave speed c 0 = 5 λ 1 λ 2 λ 3 7 λ 1 λ 4 − 12 λ 3 λ 5 . This completes the proof.□

Remark 2

(i) Compared to Theorem 4 in [2], the periodic wave train and the traveling pulse cannot coexist for equation (4). The main reason lies in the loss of mechanical balance for wave coexistence, resulting from the same dissipation level of ( u 2 u x ) x and the fourth-order diffusion u x x x x . Within the bifurcation function M ( h ) , only two generating elements can be constructed, whereas the bifurcation function ℳ ( h ) possesses three generating elements, as referenced in equation (16) and the reduced planar system (13) in [2]. The coexistence of waves arises due to three distinct levels of dissipation. (ii) By investigating ℳ * ( h ) in [2], one can obtain the wave speed for the persisting periodic wave train for equation (1) as

c * ∈ λ 1 λ 2 λ 3 λ 1 λ 4 − 2 λ 3 λ 5 , 7 λ 1 λ 2 λ 3 5 λ 1 λ 4 − 12 λ 3 λ 5 .

The speed of the unique persisting periodic wave for equation (4) is smaller than that of the periodic wave train of the same amplitude for equation (1), due to the stronger nonlinearity of buoyancy. (iii) The tool we employ is the associated Melnikov function of the perturbed Hamiltonian system, which has proven highly effective in studying periodic waves in shallow water models such as perturbed Camassa-Holm equations [18,19], generalized Benjamin-Bona-Mahony (BBM) equations [20,21], and reaction-diffusion equations [22].

3 Numerical study

We take the parameters used in [2] to perform the numerical study. In detail, we take Rayleigh number R = 1 0 4 , the water temperature equal to 2 0 ° C , and the layer length L = 1000 m . Then, the viscosity of water ν ≈ 1.0016 and the Prandtl number σ ≈ 7.56 , referring to Wikipedia. Furthermore, the Galilei number G = g L 3 ν 2 ≈ 9.768715103 × 1 0 9 , where g represents the gravitational acceleration. Substituting the above values into (2) and (8), the corresponding estimation

c ∈ ( 16.4530680548 , 23.5043829355 )

can be derived. The portraits of c ( h ) and c ( h ( u ) ) are depicted in Figure 1 and obtained by numerical computation. The numerical study reveals that the wave speed of the unique persistent periodic wave train increases as a function of h , which represents the amplitude of the periodic wave train. Notably, this wave speed is slower than the wave speed c * ranging between 32.90613608 and 46.06859050, as derived for equation (1) in Section 5 of [2]. Furthermore, we set c = 19 to analyze the zeros of M ( h ) and M ( h ( u ) ) . The results indicate that M ( h ) increases from ( − 1 4 , 0 ) to a maximum and then decreases towards the right endpoint, revealing a unique zero. Consequently, we simulate the oscillatory motion of equation (4), as illustrated in Figure 2, which shows the oscillator motion passing through the point u = 1.3328970104 .

Figure 2 Taking c = 19 c=19 to simulate the periodic wave and M {\mathbb{M}} : (a) A root of M ( h ) {\mathbb{M}}\left(h) at h = − 0.0992175028 h=-0.0992175028 ; (b) a root u = 1.3328970104 u=1.3328970104 of M ( h ( u ) ) {\mathbb{M}}\left(h\left(u)) ; (c) a time series indicates that an orbit initializing at u = 1.092 u=1.092 close to a periodic orbit crosses through u = 1.3328970104 u=1.3328970104 ; and (d) a periodic wave crosses through u = 1.3328970104 u=1.3328970104 .
Figure 2

Taking c = 19 to simulate the periodic wave and M : (a) A root of M ( h ) at h = − 0.0992175028 ; (b) a root u = 1.3328970104 of M ( h ( u ) ) ; (c) a time series indicates that an orbit initializing at u = 1.092 close to a periodic orbit crosses through u = 1.3328970104 ; and (d) a periodic wave crosses through u = 1.3328970104 .

4 Conclusion

In this study, we investigated a convecting shallow water model with a stronger nonlinear buoyancy and dissipation. We found that the mechanical balance of the coexistence of a traveling pulse and a periodic wave train is lost. The related wave speed of the unique persisting periodic wave train and the pulse becomes smaller than that of the original model (1). Another interesting problem is the spectral stability and nonlinear stability of the pulse when stronger buoyancy is considered, which will be explored in future study.

The weak dissipations ε ( u x x + u x x x x ) , always referred to as the Kuramoto-Sivashinsky perturbation, have been incorporated into the generalized BBM equation, as reported in [20,21,23], where only a unique periodic traveling wave was established. The Kuramoto-Sivashinsky perturbation has also been considered in the Camassa-Holm equation, as discussed in [18] and the Degasperis-Procesi equation in [24]; however, only a solitary wave was verified in the dissipative in the Camassa-Holm equation and the Degasperis-Procesi equation.

The methodology employed in this study can be adapted to other models beyond KdV-like equations, including the dissipative Camassa-Holm equation and the Degasperis-Procesi equation:

u t − u x x t + 2 k u x + 4 u u x = 3 u x u x x + u u x x x + ε ( α 1 u x x + α 2 ( u u x ) x + u x x x x ) ,

where α 1 and α 2 measure the linear and nonlinear viscosity, respectively. By utilizing a similar analytical approach as outlined above, we can determine which types of nonlinear dissipations affect or do not affect the balance of mechanisms underlying the existence of a unique periodic traveling wave. Moreover, there is ample scope for further research on these models. Beyond periodic waves emerging through Poincaré bifurcation, periodic peakon waves and peakon waves remain unexplored. The study of periodic traveling waves emerging near those periodic peakon waves and the peakon waves deserves further investigation. In this regard, the bifurcation theories and methods presented in [25–27] can serve as instrumental in studying periodic wave solutions.

Acknowledgements

We greatly value and appreciate any comments and suggestions provided by the anonymous reviewers and editors. Qian Wen is a member (Master Student) of the Group of ODE and Dynamical Systems in Guangxi University of Finance and Economics.

  1. Funding information: This work was supported by the Natural Science Foundation of Guangxi (2020JJG110003), the Research Fund of School of Information Engineering, Hangzhou Dianzi University (KYP0223007), and the Innovation Project of Guangxi Graduate Education (YCSW2024152).

  2. Author contributions: The authors are responsible for the complete content of this manuscript and agreed to its submission.

  3. Conflict of interest: No conflict of interest is stated by the authors.

  4. Ethical approval: The conducted research is not related to either human or animals use.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-06-25
Revised: 2024-11-01
Accepted: 2024-12-05
Published Online: 2025-03-19

© 2025 the author(s), published by De Gruyter

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  28. Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
  29. ω ℒ -biprojective and ω ¯ -contractible Banach algebras
  30. Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
  31. Solutions of several general quadratic partial differential-difference equations in ℂ2
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  36. A new family of multivalent functions defined by certain forms of the quantum integral operator
  37. A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
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  42. A study of generalized Mittag-Leffler-type function of arbitrary order
  43. On the approximation of Kantorovich-type Szàsz-Charlier operators
  44. Split quaternion Fourier transforms for two-dimensional real invariant field
  45. Quantum injectivity of G-frames in Hilbert spaces
  46. Some results on disjointly weakly compact sets
  47. On Motzkin sequence spaces via q-analog and compact operators
  48. Existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in ℝ3
  49. Stability analysis of linear time-invariant difference-differential system with constant and distributed delays
  50. The discriminant of quasi m-boundary singularities
  51. Norm constrained empirical portfolio optimization with stochastic dominance: Robust optimization non-asymptotics
  52. Fuzzy stability of multi-additive mappings
  53. On inequalities involving n-polynomial exponential-type convex functions
  54. Singularities of multiplicative spherical Darboux image and multiplicative rectifying developable surface
  55. A golden ratio technique for equilibrium problem in reflexive Banach spaces
  56. A parallel inertial three-step iteration monotone hybrid algorithm for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with graphs applicable to signal recovery problems
  57. Multiple and unique nontrivial solutions for fractional differential equations with singular property and derivatives contained in the nonlinear term
  58. New soliton solutions for a nonlinear complex hyperbolic Schrödinger dynamical equation with a truncated M-fractional derivative
  59. On a generalization of derangement polynomials and numbers
  60. The description of entire solutions of complex PDEs and PDDEs
  61. A modified RMIL conjugate gradient-based projection algorithm for constrained nonlinear equations: application to image denoising
  62. Fast solution strategies for time-space fractional linear complementarity problems governing American options pricing
  63. Existence results for Robin problems involving p(x)-Laplacian-like operators with convection term
  64. On asymptotic behaviors of a specific cubic functional equation and its hyperstability
  65. The description of entire solutions for some class of complex nonlinear partial differential equation (systems) in C 2
  66. Variations in the geometry of the basins of escape in a modified Hénon–Heiles potential
  67. A Rothe method for a viscoelastic contact problem involving time-fractional derivatives in locking materials
  68. Upper and lower solution method for a higher order ϕ-Laplacian BVPs on an infinite interval
  69. Weyl almost periodic functions on time scales and their Fourier series
  70. Integrable system of null curve and Betchov-Da Rios equation
  71. Fekete–Szegö problems for (β, Φ)-spirllike mapping of complex order γ in Banach space
  72. Modulated convergence: a deferred approach
  73. Infinitely many solutions for an instantaneous and non-instantaneous fourth-order differential system with local assumptions
  74. Existence and nonexistence of normalized solutions for the Biharmonic equation with combined nonlinearities
  75. Ekeland’s variational principle for interval-valued functions with an α-level set in Kaleva-Seikkala’s type fuzzy metric spaces
  76. Ekeland’s variational principle for interval-valued functions with an α-level set in Kaleva-Seikkala’s type fuzzy metric spaces
  77. On Kurzweil integral of fuzzy number valued functions with two variables
  78. On split common null point and common fixed point problems for multivalued demicontractive mappings
  79. Approximation by weighted Durrmeyer-type max-product neural network operators
  80. A new predictor-corrector interior-point algorithm for semidefinite optimization
  81. Densities of measures: fine properties and examples
  82. Review Articles
  83. Characterization generalized derivations of tensor products of nonassociative algebras
  84. Special Issue on Differential Equations and Numerical Analysis - Part II
  85. Existence and optimal control of Hilfer fractional evolution equations
  86. Persistence of a unique periodic wave train in convecting shallow water fluid
  87. Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
  88. Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
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  92. Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
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  94. Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
  95. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
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  101. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
  102. Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
  103. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
  104. Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
  105. New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
  106. Special Issue on Approximation Theory and Special Functions 2024 conference
  107. Ulam-type stability for Caputo-type fractional delay differential equations
  108. Faster approximation to multivariate functions by combined Bernstein-Taylor operators
  109. (λ, ψ)-Bernstein-Kantorovich operators
  110. Some special functions and cylindrical diffusion equation on α-time scale
  111. (q, p)-Mixing Bloch maps
  112. Orthogonalizing q-Bernoulli polynomials
  113. On better approximation order for the max-product Meyer-König and Zeller operator
  114. Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
  115. A note on linear compositions of the Mellin convolution operators in the weighted Mellin-Lebesgue spaces
  116. A new perspective on generalized Laguerre polynomials
  117. Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime
  118. Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces
  119. ℐ-αβ-statistical relative uniform convergence for double sequences and its applications
  120. New developments for the Jacobi polynomials
  121. Generalization of Sheffer-λ polynomials
  122. Fractional calculus containing certain bivariate Mittag-Leffler kernel with respect to function
  123. A new type of soft multi rough sets
  124. Special Issue on Variational Methods and Nonlinear PDEs
  125. A note on mean field type equations
  126. Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
  127. Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
  128. Bifurcation, quasi-periodic, and wave solutions to the fractional model of optical fibers in communication systems
  129. Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth
  130. Ground state solutions to singularly perturbed Chern-Simons-Schrödinger systems with a neutral scalar field
  131. Weak solutions to an asymptotic equation of the variational sine-Gordon equation
  132. Multiplicity of positive solutions for a concave-convex fractional elliptic system with critical growth
https://doi.org/10.1515/dema-2025-0104
Keywords for this article
Abelian integral; solitary wave; periodic traveling wave
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  57. Multiple and unique nontrivial solutions for fractional differential equations with singular property and derivatives contained in the nonlinear term
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  66. Variations in the geometry of the basins of escape in a modified Hénon–Heiles potential
  67. A Rothe method for a viscoelastic contact problem involving time-fractional derivatives in locking materials
  68. Upper and lower solution method for a higher order ϕ-Laplacian BVPs on an infinite interval
  69. Weyl almost periodic functions on time scales and their Fourier series
  70. Integrable system of null curve and Betchov-Da Rios equation
  71. Fekete–Szegö problems for (β, Φ)-spirllike mapping of complex order γ in Banach space
  72. Modulated convergence: a deferred approach
  73. Infinitely many solutions for an instantaneous and non-instantaneous fourth-order differential system with local assumptions
  74. Existence and nonexistence of normalized solutions for the Biharmonic equation with combined nonlinearities
  75. Ekeland’s variational principle for interval-valued functions with an α-level set in Kaleva-Seikkala’s type fuzzy metric spaces
  76. Ekeland’s variational principle for interval-valued functions with an α-level set in Kaleva-Seikkala’s type fuzzy metric spaces
  77. On Kurzweil integral of fuzzy number valued functions with two variables
  78. On split common null point and common fixed point problems for multivalued demicontractive mappings
  79. Approximation by weighted Durrmeyer-type max-product neural network operators
  80. A new predictor-corrector interior-point algorithm for semidefinite optimization
  81. Densities of measures: fine properties and examples
  82. Review Articles
  83. Characterization generalized derivations of tensor products of nonassociative algebras
  84. Special Issue on Differential Equations and Numerical Analysis - Part II
  85. Existence and optimal control of Hilfer fractional evolution equations
  86. Persistence of a unique periodic wave train in convecting shallow water fluid
  87. Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
  88. Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
  89. Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
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  92. Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
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  100. Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
  101. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
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  106. Special Issue on Approximation Theory and Special Functions 2024 conference
  107. Ulam-type stability for Caputo-type fractional delay differential equations
  108. Faster approximation to multivariate functions by combined Bernstein-Taylor operators
  109. (λ, ψ)-Bernstein-Kantorovich operators
  110. Some special functions and cylindrical diffusion equation on α-time scale
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  113. On better approximation order for the max-product Meyer-König and Zeller operator
  114. Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
  115. A note on linear compositions of the Mellin convolution operators in the weighted Mellin-Lebesgue spaces
  116. A new perspective on generalized Laguerre polynomials
  117. Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime
  118. Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces
  119. ℐ-αβ-statistical relative uniform convergence for double sequences and its applications
  120. New developments for the Jacobi polynomials
  121. Generalization of Sheffer-λ polynomials
  122. Fractional calculus containing certain bivariate Mittag-Leffler kernel with respect to function
  123. A new type of soft multi rough sets
  124. Special Issue on Variational Methods and Nonlinear PDEs
  125. A note on mean field type equations
  126. Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
  127. Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
  128. Bifurcation, quasi-periodic, and wave solutions to the fractional model of optical fibers in communication systems
  129. Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth
  130. Ground state solutions to singularly perturbed Chern-Simons-Schrödinger systems with a neutral scalar field
  131. Weak solutions to an asymptotic equation of the variational sine-Gordon equation
  132. Multiplicity of positive solutions for a concave-convex fractional elliptic system with critical growth
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