travelling wave in one dimension

16.1 Traveling Waves

Learning objectives.

By the end of this section, you will be able to:

• Describe the basic characteristics of wave motion
• Define the terms wavelength, amplitude, period, frequency, and wave speed
• Explain the difference between longitudinal and transverse waves, and give examples of each type
• List the different types of waves

We saw in Oscillations that oscillatory motion is an important type of behavior that can be used to model a wide range of physical phenomena. Oscillatory motion is also important because oscillations can generate waves, which are of fundamental importance in physics. Many of the terms and equations we studied in the chapter on oscillations apply equally well to wave motion ( Figure 16.2 ).

Types of Waves

A wave is a disturbance that propagates, or moves from the place it was created. There are three basic types of waves: mechanical waves, electromagnetic waves, and matter waves.

Basic mechanical wave s are governed by Newton’s laws and require a medium. A medium is the substance mechanical waves propagate through, and the medium produces an elastic restoring force when it is deformed. Mechanical waves transfer energy and momentum, without transferring mass. Some examples of mechanical waves are water waves, sound waves, and seismic waves. The medium for water waves is water; for sound waves, the medium is usually air. (Sound waves can travel in other media as well; we will look at that in more detail in Sound .) For surface water waves, the disturbance occurs on the surface of the water, perhaps created by a rock thrown into a pond or by a swimmer splashing the surface repeatedly. For sound waves, the disturbance is a change in air pressure, perhaps created by the oscillating cone inside a speaker or a vibrating tuning fork. In both cases, the disturbance is the oscillation of the molecules of the fluid. In mechanical waves, energy and momentum transfer with the motion of the wave, whereas the mass oscillates around an equilibrium point. (We discuss this in Energy and Power of a Wave .) Earthquakes generate seismic waves from several types of disturbances, including the disturbance of Earth’s surface and pressure disturbances under the surface. Seismic waves travel through the solids and liquids that form Earth. In this chapter, we focus on mechanical waves.

Electromagnetic waves are associated with oscillations in electric and magnetic fields and do not require a medium. Examples include gamma rays, X-rays, ultraviolet waves, visible light, infrared waves, microwaves, and radio waves. Electromagnetic waves can travel through a vacuum at the speed of light, v = c = 2.99792458 × 10 8 m/s . v = c = 2.99792458 × 10 8 m/s . For example, light from distant stars travels through the vacuum of space and reaches Earth. Electromagnetic waves have some characteristics that are similar to mechanical waves; they are covered in more detail in Electromagnetic Waves .

Matter waves are a central part of the branch of physics known as quantum mechanics. These waves are associated with protons, electrons, neutrons, and other fundamental particles found in nature. The theory that all types of matter have wave-like properties was first proposed by Louis de Broglie in 1924. Matter waves are discussed in Photons and Matter Waves .

Mechanical Waves

Mechanical waves exhibit characteristics common to all waves, such as amplitude, wavelength, period, frequency, and energy. All wave characteristics can be described by a small set of underlying principles.

The simplest mechanical waves repeat themselves for several cycles and are associated with simple harmonic motion. These simple harmonic waves can be modeled using some combination of sine and cosine functions. For example, consider the simplified surface water wave that moves across the surface of water as illustrated in Figure 16.3 . Unlike complex ocean waves, in surface water waves, the medium, in this case water, moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. In Figure 16.3 , the waves causes a seagull to move up and down in simple harmonic motion as the wave crests and troughs (peaks and valleys) pass under the bird. The crest is the highest point of the wave, and the trough is the lowest part of the wave. The time for one complete oscillation of the up-and-down motion is the wave’s period T . The wave’s frequency is the number of waves that pass through a point per unit time and is equal to f = 1 / T . f = 1 / T . The period can be expressed using any convenient unit of time but is usually measured in seconds; frequency is usually measured in hertz (Hz), where 1 Hz = 1 s −1 . 1 Hz = 1 s −1 .

The length of the wave is called the wavelength and is represented by the Greek letter lambda ( λ ) ( λ ) , which is measured in any convenient unit of length, such as a centimeter or meter. The wavelength can be measured between any two similar points along the medium that have the same height and the same slope. In Figure 16.3 , the wavelength is shown measured between two crests. As stated above, the period of the wave is equal to the time for one oscillation, but it is also equal to the time for one wavelength to pass through a point along the wave’s path.

The amplitude of the wave ( A ) is a measure of the maximum displacement of the medium from its equilibrium position. In the figure, the equilibrium position is indicated by the dotted line, which is the height of the water if there were no waves moving through it. In this case, the wave is symmetrical, the crest of the wave is a distance + A + A above the equilibrium position, and the trough is a distance − A − A below the equilibrium position. The units for the amplitude can be centimeters or meters, or any convenient unit of distance.

The water wave in the figure moves through the medium with a propagation velocity v → . v → . The magnitude of the wave velocity is the distance the wave travels in a given time, which is one wavelength in the time of one period, and the wave speed is the magnitude of wave velocity. In equation form, this is

This fundamental relationship holds for all types of waves. For water waves, v is the speed of a surface wave; for sound, v is the speed of sound; and for visible light, v is the speed of light.

Transverse and Longitudinal Waves

We have seen that a simple mechanical wave consists of a periodic disturbance that propagates from one place to another through a medium. In Figure 16.4 (a), the wave propagates in the horizontal direction, whereas the medium is disturbed in the vertical direction. Such a wave is called a transverse wave . In a transverse wave, the wave may propagate in any direction, but the disturbance of the medium is perpendicular to the direction of propagation. In contrast, in a longitudinal wave or compressional wave, the disturbance is parallel to the direction of propagation. Figure 16.4 (b) shows an example of a longitudinal wave. The size of the disturbance is its amplitude A and is completely independent of the speed of propagation v .

A simple graphical representation of a section of the spring shown in Figure 16.4 (b) is shown in Figure 16.5 . Figure 16.5 (a) shows the equilibrium position of the spring before any waves move down it. A point on the spring is marked with a blue dot. Figure 16.5 (b) through (g) show snapshots of the spring taken one-quarter of a period apart, sometime after the end of` the spring is oscillated back and forth in the x -direction at a constant frequency. The disturbance of the wave is seen as the compressions and the expansions of the spring. Note that the blue dot oscillates around its equilibrium position a distance A , as the longitudinal wave moves in the positive x -direction with a constant speed. The distance A is the amplitude of the wave. The y -position of the dot does not change as the wave moves through the spring. The wavelength of the wave is measured in part (d). The wavelength depends on the speed of the wave and the frequency of the driving force.

Waves may be transverse, longitudinal, or a combination of the two. Examples of transverse waves are the waves on stringed instruments or surface waves on water, such as ripples moving on a pond. Sound waves in air and water are longitudinal. With sound waves, the disturbances are periodic variations in pressure that are transmitted in fluids. Fluids do not have appreciable shear strength, and for this reason, the sound waves in them are longitudinal waves. Sound in solids can have both longitudinal and transverse components, such as those in a seismic wave. Earthquakes generate seismic waves under Earth’s surface with both longitudinal and transverse components (called compressional or P-waves and shear or S-waves, respectively). The components of seismic waves have important individual characteristics—they propagate at different speeds, for example. Earthquakes also have surface waves that are similar to surface waves on water. Ocean waves also have both transverse and longitudinal components.

Example 16.1

Wave on a string.

• The speed of the wave can be derived by dividing the distance traveled by the time.
• The period of the wave is the inverse of the frequency of the driving force.
• The wavelength can be found from the speed and the period v = λ / T . v = λ / T .
• The first wave traveled 30.00 m in 6.00 s: v = 30.00 m 6.00 s = 5.00 m s . v = 30.00 m 6.00 s = 5.00 m s .
• The period is equal to the inverse of the frequency: T = 1 f = 1 2.00 s −1 = 0.50 s . T = 1 f = 1 2.00 s −1 = 0.50 s .
• The wavelength is equal to the velocity times the period: λ = v T = 5.00 m s ( 0.50 s ) = 2.50 m . λ = v T = 5.00 m s ( 0.50 s ) = 2.50 m .

Significance

Check your understanding 16.1.

When a guitar string is plucked, the guitar string oscillates as a result of waves moving through the string. The vibrations of the string cause the air molecules to oscillate, forming sound waves. The frequency of the sound waves is equal to the frequency of the vibrating string. Is the wavelength of the sound wave always equal to the wavelength of the waves on the string?

Example 16.2

Characteristics of a wave.

• The amplitude and wavelength can be determined from the graph.
• Since the velocity is constant, the velocity of the wave can be found by dividing the distance traveled by the wave by the time it took the wave to travel the distance.
• The period can be found from v = λ T v = λ T and the frequency from f = 1 T . f = 1 T .
• The distance the wave traveled from time t = 0.00 s t = 0.00 s to time t = 3.00 s t = 3.00 s can be seen in the graph. Consider the red arrow, which shows the distance the crest has moved in 3 s. The distance is 8.00 cm − 2.00 cm = 6.00 cm . 8.00 cm − 2.00 cm = 6.00 cm . The velocity is v = Δ x Δ t = 8.00 cm − 2.00 cm 3.00 s − 0.00 s = 2.00 cm/s . v = Δ x Δ t = 8.00 cm − 2.00 cm 3.00 s − 0.00 s = 2.00 cm/s .
• The period is T = λ v = 8.00 cm 2.00 cm/s = 4.00 s T = λ v = 8.00 cm 2.00 cm/s = 4.00 s and the frequency is f = 1 T = 1 4.00 s = 0.25 Hz . f = 1 T = 1 4.00 s = 0.25 Hz .

Check Your Understanding 16.2

The propagation velocity of a transverse or longitudinal mechanical wave may be constant as the wave disturbance moves through the medium. Consider a transverse mechanical wave: Is the velocity of the medium also constant?

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• Publication date: Sep 19, 2016
• Location: Houston, Texas
• Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction
• Section URL: https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves

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6 Traveling Waves in One Dimension

• Published: August 2015
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A study of traveling waves is presented for a wave traveling in a one-dimensional medium. The generation of a wave, the analytic expression of a traveling wave, the energy in a wave, and other basic characteristics such as amplitude, wave speed, and phase are discussed. It is shown that a superposition of harmonic waves leads to the beat phenomenon in time as well as space. The distinction between phase and group velocity is illustrated. The Fourier integral method is introduced to study the frequency content of pulses, and evolution of dispersive waves. A discussion of the Doppler effect is included in this chapter.

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Waves in Two and Three Dimensions

Michael Fowler, University of Virginia

Introduction

So far, we’ve looked at waves in one dimension, traveling along a string or sound waves going down a narrow tube.  But waves in higher dimensions than one are very familiar — water waves on the surface of a pond, or sound waves moving out from a source in three dimensions. 

It is pleasant to find that these waves in higher dimensions satisfy wave equations which are a very natural extension of the one we found for a string, and — very important — they also satisfy the Principle of Superposition , in other words, if waves meet, you just add the contribution from each wave.  In the next two paragraphs, we go into more detail, but this Principle of Superposition is the crucial lesson.

The Wave Equation and Superposition in One Dimension

For waves on a string, we found Newton’s laws applied to one bit of string gave a differential wave equation,

∂ 2 y ∂ x 2 = 1 v 2 ∂ 2 y ∂ t 2

and it turned out that sound waves in a tube satisfied the same equation .  Before going to higher dimensions, I just want to focus on one crucial feature of this wave equation: it’s linear , which just means that if you find two different solutions y 1 ( x , t )  and y 2 ( x , t )  then y 1 ( x , t ) + y 2 ( x , t )  is also a solution, as we proved earlier. 

This important property is easy to interpret visually : if you can draw two wave solutions, then at each point on the string simply add the displacement y 1 ( x , t )  of one wave to the other y 2 ( x , t )   — you just add the waves together — this also is a solution.  So, for example, as two traveling waves moving along the string in opposite directions meet each other, the displacement of the string at any point at any instant is just the sum of the displacements it would have had from the two waves singly.  This simple addition of the displacements is termed “interference”, doubtless because if the waves meeting have displacement in opposite directions, the string will be displaced less than by a single wave.  It’s also called the Principle of Superposition .

The Wave Equation and Superposition in More Dimensions

What happens in higher dimensions?  Let’s consider two dimensions, for example waves in an elastic sheet like a drumhead.  If the rest position for the elastic sheet is the ( x , y )  plane, so when it’s vibrating it’s moving up and down in the z -direction, its configuration at any instant of time is a function z ( x , y , t ) .

In fact, we could do the same thing we did for the string, looking at the total forces on a little bit and applying Newton’s Second Law.  In this case that would mean taking one little bit of the drumhead, and instead of a small stretch of string with tension pulling the two ends, we would have a small square of the elastic sheet, with tension pulling all around the edge.  Remember that the net force on the bit of string came about because the string was curving around, so the tensions at the opposite ends tugged in slightly different directions, and didn’t cancel.  The ∂ 2 y / ∂ x 2  term measured that curvature, the rate of change of slope. In two dimensions, thinking of a small square of the elastic sheet, things are more complicated.  Visualize the bit of sheet to be momentarily like a tiny patch on a balloon, you’ll see it curves in two directions, and tension forces must be tugging all around the edges.  The total force on the little square comes about because the tension forces on opposite sides are out of line if the surface is curving around, now we have to add two sets of almost-opposite forces from the two pairs of sides.  I’m not going to go through all the math here, but I hope it’s at least plausible that the equation is:

∂ 2 z ∂ x 2 + ∂ 2 z ∂ y 2 = 1 v 2 ∂ 2 z ∂ t 2 .

The physics of this equation is that the acceleration of a tiny bit of the sheet comes from out-of-balance tensions caused by the sheet curving around in both the x - and y -directions, this is why there are the two terms on the left hand side.

Remarkably, this same equation comes out for sound waves and for the electromagnetic waves we now know as radio, microwaves, light, X-rays: so it’s called the Wave Equation .

And, going to three dimensions is easy: add one more term to give

∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 = 1 v 2 ∂ 2 f ∂ t 2 .

This sum of partial differentiations is so common in physics that there’s a shorthand:

∇ 2 f = ( 1 / v 2 ) ∂ 2 f / ∂ t 2 .

Just as we found in one dimension traveling harmonic waves f ( x − v t ) = A sin ( k x − ω t ) ,  with ω = v k ,  you can verify that the three -dimensional equation has harmonic solutions

f ( x , y , z , t ) = A sin ( k x x + k y y + k z z − ω t ) ,

ω = v | k | , where | k | = k x 2 + k y 2 + k z 2 .

In fact, k →  is a vector in the direction the wave is moving.  The electric and magnetic fields in a radio wave or light wave have just this form (or, closer to the source, a very similar equivalent expression for outgoing spheres of waves, rather than plane waves). 

It’s important to realize that this more complicated equation is still a linear equation — the principle of superposition still holds .  If two waves on an elastic sheet, or the surface of a pond, meet each other, the result at any point is given by simply adding the displacements from the individual waves.  (Assuming as always small waves, so the water waves don’t fall apart into foam.)

We’ll begin by thinking about waves propagating freely in two and three dimensions, than later consider waves in restricted areas, such as a drum head.

How Does a Wave Propagate in Two and Three Dimensions?

A one-dimensional wave doesn’t have a choice: it just moves along the line (well, it could get partly reflected by some change in the line and part of it go backwards).  But when we go to higher dimensions, how a wave disturbance starting in some localized region spreads out is far from obvious.  But we can begin by recalling some simple cases: dropping a pebble into still water causes an outward moving circle of ripples.  If we grant that light is a wave, we notice a beam of light changes direction on going from air into glass.  Of course, it’s not immediately evident that light is a wave: we’ll talk a lot more about that later. 

Huygen’s Picture of Wave Propagation                                              

If a point source of light is switched on, the wavefront is an expanding sphere centered at the source.  Huygens suggested that this could be understood if at any instant in time each point on the wavefront was regarded as a source of secondary wavelets, and the new wavefront a moment later was to be regarded as built up from the sum of these wavelets. For a light shining continuously, this process just keeps repeating.

What use is this idea? For one thing, it explains refraction — the change in direction of a wavefront on entering a different medium, such as a ray of light going from air into glass.

If the light moves more slowly in the glass, velocity v instead of c , with v < c , then Huygen’s picture explains Snell’s Law, that the ratio of the sines of the angles to the normal of incident and transmitted beams is constant, and in fact is the ratio c / v . 

This is evident from the diagram below: in the time the wavelet centered at A has propagated to C , that from B has reached D , the ratio of lengths AC / BD being c / v .  But the angles in Snell’s Law are in fact the angles ABC , BCD , and those right-angled triangles have a common hypotenuse BC , from which the Law follows.

Huygens’ picture also provides a ready explanation of what happens when a plane wave front encounters a barrier with one narrow opening: and by narrow, we mean small compared with the wavelength of the wave.  It’s easy to arrange this for water waves: it’s found that on the other side of the barrier, the waves spread out in circular fashion form the small hole.

Two-Slit Interference:  How Young measured the Wavelength of Light

If the slit is wider than a wavelength or so, the pattern gets more complicated, as we would expect from Huygens’ ideas, because now the waves on the far side arise from a line of sources, not what amounts to one point.  To investigate this further, consider the simplest possible next case: a barrier with two small holes in it, so on the far side we’re looking at waves radiating outwards from, effectively, two point sources.

Animation here!

For two synchronized sources generating harmonic waves, at any point in the tank equally distant from the two sources (the central line in the picture above), the waves will add, the water will be maximally disturbed.  For light waves, there will be a maximum in brightness at the center of a screen as shown in the diagram:

For light waves passing through two narrow slits and shining on a screen (on the right) there will be another bright spot at a point P away from the center C 2 of the screen, provided the distances of P from the two slits differ by a whole number of wavelengths :

On the other hand, at a point approximately half way from the center of the screen to P the waves from the two sources will arrive at the screen exactly out of phase: the crest of one will arrive with the trough of the other, they will cancel, and there will be no light.  Evidently, then, we will see on the screen a series of bright areas and dark areas , the brightest spots being at the points where the waves from the two slits arrive exactly in phase. 

There is an animation of this pattern formation here .

This pattern, generated by what is called interference between the waves, and also referred to as a diffraction pattern is historically important, because it was used to establish that light is a wave, by Thomas Young in 1807.  (Recall Newton had believed light was a stream of particles, and that was very widely accepted at the time.) 

Young used the pattern to find the wavelengths of red and violet light.  His method can be understood from the diagram above.  We did the experiment in class with a slit separation of about 0.2 mm., giving bright spots on the screen about 3 cm apart, with a screen 10 m from the slits.

That is to say, in the diagram above we had S 1 S 2 = 0.2 × 10 − 3 m,   C 1 C 2 = 9.5   m,  and we found C 2 P = x = 3  cm .  (within a percent or two).  Looking at the diagram, it’s clear that the angle to P from the slits is very small, in fact it’s x / L = 3.15 × 10 − 3 radians .   So the diagram as drawn is very exaggerated!  

Now, the line S 1 Q is perpendicular to the light rays setting off for P (they are extremely close to parallel).   The angle between S 1 Q and S 1 S 2   is the same as that between C 1 P and C 1 C 2 , that is, 3.15 × 10 − 3 radians .  This means that the lengths S 1 Q and S 1 S 2 are effectively equal, and therefore that

S 2 Q S 1 S 2 = λ d = x L = 3.15 × 10 − 3 .

This is very accurate for such a small angle, and for the data as given here the wavelength of the light λ = 3.15 × 10 − 3 d = 6.3 × 10 − 7 m = 630 nm .

Another Bright Spot

About ten years after Young’s result a French civil engineer, Augustin Fresnel, independently developed a wave theory of light, and gave a more complete mathematical analysis.  This was disputed by the famous French mathematician Simeon Poisson, who pointed out that if the wave theory were true, one could prove mathematically that in the sharp shadow of a small round object, there would be a bright spot in the center, because the waves coming around the circumference all around would add there.  This seemed ridiculous — but French physicist Francois Arago actually did the experiment, and found the spot!  The wave theory of light had arrived.

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It is however possible to have a wave confined to a given space in a medium and still produce a regular wave pattern that is readily discernible amidst the motion of the medium. For instance, if an elastic rope is held end-to-end and vibrated at just the right frequency , a wave pattern would be produced that assumes the shape of a sine wave and is seen to change over time. The wave pattern is only produced when one end of the rope is vibrated at just the right frequency. When the proper frequency is used, the interference of the incident wave and the reflected wave occur in such a manner that there are specific points along the medium that appear to be standing still. Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a standing wave pattern . There are other points along the medium whose displacement changes over time, but in a regular manner. These points vibrate back and forth from a positive displacement to a negative displacement; the vibrations occur at regular time intervals such that the motion of the medium is regular and repeating. A pattern is readily observable.

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(a) ( x – vt ) 2

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Stability, modulation instability and traveling wave solutions of (3+1)dimensional Schrödinger model in physics

• Open access
• Published: 26 June 2024
• Volume 56 , article number  1237 , ( 2024 )

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• Hijaz Ahmad 1 ,
• Kalim U. Tariq 2 &
• S. M. Raza Kazmi 2  

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The nonlinear Schrödinger equation is one of the most important physical model in optical fiber theory for comprehension of the fluctuations of optical bullet development. In this study, the exact bullet solutions for the (3+1)-dimensional Schrödinger equation which demonstrate the bullet behaviours in optical fibers can be accumulated through the Sardar sub-equation method and the unified method. The applied strategies may retrieve several kinds of optical bullet solutions within one frameworks as well as is quite simple and reliable. Mathematica are utilised for describing the dynamics of different wave structures as 3D, 2D, and contour visualisations for a given set of parameters. As a result, we are able to develop a variety of travelling wave structures namely the periodic, singular and V shaped soliton wave solutions. The stability analysis for the derived results is analysed efficiently while the modulation instability for the governing model has also been studied to demonstrate the reliability of the research. The approaches implemented here works perfectly and can be extended to deal with many advanced models in contemporary areas of science and engineering. The solutions attain by using these techniques are robust, unique and straight forward and has applications in different fields of physics, engineering and mathematical science. Specially physical applications of these obtain results are in the transmission of data in optical fibers. We also add the graphics for the better understanding of the attain solutions behaviour.

Avoid common mistakes on your manuscript.

1 Introduction

Nonlinear partial differential equations (NLPDE) are often encountered on establishing the basic principles of life as well as in mathematical examination of an extensive span of issues simply occurring in weather, hydrodynamics, blood plasma, atmosphere crests, computational life sciences, chemical engineering, and also in materials research (Duran et al. 2023 ; Tan and Yin 2021 ). Numerous amazing models serve as examples of how innovative bullet support can be. Nonlinear Schrödinger equations (NLSE) has applications in various fields of sciences specially in water waves, physics and in the transmission of fiber optics (Ahmad and Rani 2023 ; Zhang et al. 2019 ).

The NLSE are the important type NLPDE that is applicable to classical as well as computational physics (Zulfiqar and Ahmad 2020 ). For the bullet solutions of such types of model many types of approaches are used, as the (3+1)dimensional Schrödinger equation can be solved by utilizing the generalized exponential rational function method (Yokus et al. 2022 ). Many kinds of other approaches are used to study various nonlinear dynamical structures more proficiently such as the modified Jacobi elliptic expansion method Gaballah et al. ( 2024 ) , the Sine-Gordon expansion method Kundu et al. ( 2021 ), the Darboux transformation method (Wang et al. 2021 ) the Khater method (Qin et al. 2020 ), the generalized exponential rational function method (Ghanbari and Gómez-Aguilar 2019 ), the Riccati equation method (Akram et al. 2021 ), the auxiliary equation method (Rezazadeh et al. 2019 ), the unified method (Fokas and Lenells 2012 ), the improved F expansion technique (Islam et al. 2019 ), the modified simple equation method (Biswas et al. 2018 ), the exp(- \(\zeta (\xi )\) ) expansion technique (Lakestani and Manafian 2018 ), the exponential rational function method (Aksoy et al. 2016 ), and the Hirota bilinear method (Alhami and Alquran 2022 ). In our study we obtain some new exact bullet solutions by using the modern techniques, also we say that according to my knowledge the methods applied in this model have not use in this model in previous, so our result are unique and newly drawn.

In this article, the (3+1)-dimensional nonlinear Schrödinger equation which shows wave behaviour in optical fibers can be studied by using the Sardar sub-equation method (Cinar et al. 2022 ) and unified method (Akcagil and Aydemir 2018 )

where \({\mathcal {D}} = {\mathcal {D}}(x,y,z,t)\) are a complex function and x , y , z , and t are the independent space parameters and time parameter accordingly. Where p , \(b_i\) , and \(c_i\) denote the constant. Further if put \(c_1\)  =  \(c_2\)  =  \(c_3\)  = 0, then the above Eq. ( 1 ) can be converted to the below (3+1)dimensional Schrödinger equation.

both the Eqs. ( 1 ) and ( 2 ) demonstrate about ultra-short impulses of light spread via extremely nonlinear mediums. Also both these equations are use in the transmission of data in fibers optical. The bright and dark wave solution of Eq. ( 1 ) is attain by Wazwaz and Mehanna ( 2021 ), further the generalized exponential rational function method are used to attain wave solutions (Kumar et al. 2023 ). This model is solved by these techniques in literature which gives bright and dark wave solution as compared to the techniques used in this article that gives robust, unique and power solutions that has application in transmission of fibers so we prefer these methods that applied in this article. The results are interesting and attractive as compared to these previous applied method in literature. The author use two modern techniques that are not applied to this model in previous so one can say that the attain solution are newly made and robust. The work of this paper is novel because the utilized approaches in this article are not used to the given Eq. ( 1 ) in literature so the solution obtain by applying these strategies are unique and novel. The obtain solutions are highly valuable and has applications in different fields of physics and mathematical science, also these solutions are use when dealing about bullet wave solutions of nonlinear problems. The most important application of these results are in transmission of data in optical fibers. As compared to other methods in literature the applied techniques in this article are simple, unique and straight forward thats why one can chose these techniques.

The arrangement of this article is according to the following steps: In Sect.  2 , we briefly discuss the use methods to solve the given model. In Sect.  3 , the bullet wave solution were constructed. Graphics of the obtain solutions are shown in Sect.  4 , Also the stability and modulation instability are included in Sects.  5 and  6 , accordingly. At end Sect.  7 , discuss the conclusion of article.

2 Methodology

Assume the following NLPDE with four independent variables x , y , z , and t is written as

where \({\mathcal {D}}(x,y,z,t)\) denote a unknown function, L express a polynomial of \({\mathcal {D}}={\mathcal {D}}(x,y,z,t)\) along with its partial derivatives.

we get an ODE by substituting Eqs. ( 4 ) in ( 3 ).

2.1 The Sardar sub-equation method

The Sardar sub-equation method description could be given in four stages that follow:

Step 1 : Assuming the given Eq. ( 5 ) solution must be written using the corresponding ansatz:

\(n _i\) represent unknown function, M is a real Number, and \(\phi (\xi )\) fulfils

the equation shown above has a variety of solutions depending on the circumstance.

Case I. If \(\chi _2>0,~~\chi _1=0\) , as a result, we obtain solution as

Case II. If \(\chi _2<0,~~\chi _1=0\) , as a result, we obtain solution as

Case III. If \(\chi _2<0,~~\chi _1=\frac{\chi ^2_2}{4}\) , as a result, we obtain solution as

Case IV. If \(\chi _2>0,~~\chi _1=\frac{\chi ^2_2}{4}\) , as a result, we obtain solution as

Step 2 : Substituting Eq. ( 6 ) along with Eq. ( 7 ) into Eq. ( 5 ), then we put the similar power \(\phi (\xi )\) to zero, yield a system of algebraic equation.

Step 3 : For the solution of Eq. ( 1 ), we solve the system of equation in last step by adding the term obtained from Eq. ( 6 )in to the Eq. ( 7 ) solution.

2.2 The unified method

Step 1 : The unified method description can be given in five stages that follows:

here, \(\phi =\phi (\xi )\) satisfy

and \(\phi '(\xi )=\frac{d\phi }{d\xi }\) , \(n_i, ~m_i ~and ~\omega\) is the constants. The solution to Eq. ( 5 ) could be represented as:

Family 1 : If \(\omega <0\) , then hyperbolic function solution of the form

Family 2 : If \(\omega >0\) , then parabolic function solution of the form

Family 3 : If \(\omega =0\) , then rational solution of the form

Step 2 : At this step we find the positive number M by using homogeneous balancing technique.

Step 3 : Substituting Eq. ( 8 ) along with Eq. ( 9 ) into Eq. ( 5 ), then we put the similar power \(\psi ^{i}(\xi )~(-N\le i\le N)\) to zero, yield a system of algebraic equation, that solved by using latest computing tool like Mathematica.

3 Traveling wave solutions

By utilizing transform we convert PDE into ODE (Wang 2021 ),

which converts Eq ( 1 ) into real and imaginary components consequently

by putting the coefficient of \({\mathcal {D}}'\) zero in above equation, from this we get

3.1 Application to the Sardar sub-equation method

We find M=1 by applying the homogeneous balanced criterion. Then the solution Eq. ( 14 ) is supposed as

by putting the both Eqs. ( 17 ) and ( 7 ) in ( 14 ), from this we attain a algebraic system of equations which are express below as,

Evaluating these systems can result in the following outcome:

Also we put the above results in Eq. ( 17 ), which results:

Case 1:   If \(\chi _2>0,~~\chi _1=0\) as a result we get

Case 2:   If \(\chi _2<0,~~\chi _1=0\) , as a result we get

Case 3:   If \(\chi _2<0,~~\chi _1=\frac{\chi ^2_2}{4}\) , as a result we get

where \(k_1=\frac{i \sqrt{A T} \left( T \exp \left( \frac{\sqrt{-\chi _2} \xi }{2 \sqrt{2}}\right) -A \exp \left( -\frac{\sqrt{-\chi _2} \xi }{2 \sqrt{2}}\right) \right) }{A \exp \left( -\frac{\sqrt{-\chi _2} \xi }{2 \sqrt{2}}\right) +T \exp \left( \frac{\sqrt{-\chi _2} \xi }{2 \sqrt{2}}\right) }\) .

Case 4:   If \(\chi _2>0,~~\chi _1=\frac{\chi ^2_2}{4}\) , as a result we get

where \(\eta _3=-b_1 \eta _1 \sigma _1-b_3 \rho _1 \sigma _1+\eta _1^2+\rho _1^2+\sigma _1^2\) and \(\Upsilon =t \tau _2+\sigma _2 x+\eta _2 y+\rho _2 z\) .

Case 1:   If \(\chi _2>0,~~\chi _1=0\) , as a result we get

If \(\chi _2<0,~~\chi _1=0\) , as a result we get

where \(k_2=\frac{2 i \sqrt{A T}}{A \exp \left( -\sqrt{2} \sqrt{-\chi _2} \xi \right) +T e^{\sqrt{2} \sqrt{-\chi _2} \xi }}\) .

where \(k_3=\frac{2 i \sqrt{A T}}{T e^{\sqrt{2} \sqrt{-\chi _2} \xi }-A \exp \left( -\sqrt{2} \sqrt{-\chi _2} \xi \right) }\) .

where \(k_4=\frac{i \sqrt{A T} \left( T \exp \left( \frac{\sqrt{-\chi _2} \xi }{2 \sqrt{2}}\right) -A \exp \left( -\frac{\sqrt{-\chi _2} \xi }{2 \sqrt{2}}\right) \right) }{A \exp \left( -\frac{\sqrt{-\chi _2} \xi }{2 \sqrt{2}}\right) +T \exp \left( \frac{\sqrt{-\chi _2} \xi }{2 \sqrt{2}}\right) }\) .

\(k_5=\frac{2 \sqrt{A T}}{A \exp \left( -i \sqrt{2} \sqrt{\chi _2} \xi \right) +T e^{i \sqrt{2} \sqrt{\chi _2} \xi }}\) .

where \(k_6=\frac{2 i \sqrt{A T}}{T e^{i \sqrt{2} \sqrt{\chi _2} \xi }-A \exp \left( -i \sqrt{2} \sqrt{\chi _2} \xi \right) }\)

where \(k_7=\frac{T \exp \left( \frac{i \sqrt{\chi _2} \xi }{2 \sqrt{2}}\right) -A \exp \left( -\frac{i \sqrt{\chi _2} \xi }{2 \sqrt{2}}\right) }{A \exp \left( -\frac{i \sqrt{\chi _2} \xi }{2 \sqrt{2}}\right) +T \exp \left( \frac{i \sqrt{\chi _2} \xi }{2 \sqrt{2}}\right) }\) , \(\eta _3=-b_1 \eta _1 \sigma _1-b_3 \rho _1 \sigma _1+\eta _1^2+\rho _1^2+\sigma _1^2\) and \(\Upsilon =t \tau _2+\sigma _2 x+\eta _2 y+\rho _2 z\) .

3.2 Application to the unified method

The balancing number M=1 is calculated by balancing the highest order derivative and nonlinear term, then we demonstrate Eq. ( 14 ) solution as

by putting the both Eqs. ( 47 ) and ( 9 )in Eq. ( 14 ), from this we attain a algebraic system of equations which are express below as,

By we put the above results Eq. ( 47 ), which results:

Case 1:   If \(\omega <0,\) as a result we get

Case 2:   If \(\omega >0,\) as a result we get

Case 3:   If \(\omega =0,\) as a result we get

where \(\Upsilon =t \tau _2+\sigma _2 x+\eta _2 y+\rho _2 z\) ,  \(\xi =t \tau _1+\sigma _1 x+\eta _1 y+\rho _1 z\) , \(\chi =-b_1 \eta _1 \sigma _1-b_3 \rho _1 \sigma _1+\eta _1^2+\rho _1^2+\sigma _1^2\) .

Case 3:   If \(\omega =0\) , as a result we get

4 Stability analysis

In this portion of paper we deals about the stability of the suggested model. Hamiltonian momentum can be symbolize as of Eq. ( 1 ),

here, \( N, {\mathcal{D}}\,\) represent the momentum and electric potential consequently. The major term of soliton stability is defined as

where \(\sigma\) is the frequency of solitons. Now we put pulse solution of Eqs. ( 73 ) to ( 76 ), from this we get a result as,

where   \(\eta _3=-b_1 \eta _1 \sigma _1-b_3 \rho _1 \sigma _1+\eta _1^2+\rho _1^2+\sigma _1^2\) , \(\Upsilon =t \tau _2+\sigma _2 x+\eta _2 y+\rho _2 z\) , and \(\xi =t \tau _1+\sigma _1 x+\eta _1 y+\rho _1 z\) . From the above condition of stability in Eq. ( 77 ) also the outcome of given above framework in Eq. ( 78 ) greater to zero, then we summarize result as Eq. ( 1 ) are stable nonlinear equation.

5 Modulation instability (MI)

MI additional referred by the term sideband instabilities, is an abnormality that develops when deviations to normal vibrations have gotten stronger because of nonlinearity, causing the development of range opposite buckles and ultimately leading to the division of the ripples via a series of beans and lentils. This can be utilised in all areas of irregular visual and fluids dynamics. One utilise MI to verify that these findings are valid. The incidence, frequency and speed have impact on the modulation instability.

Here we discuss the modulation instability of Eq. ( 1 , with the help of given transformation written as

in above the term \(\upsilon\) , \(\Omega\) and S shows the disturbance, disturbance intensity and steady flow correspondingly, by adding Eqs. ( 79 ) in ( 1 ), then we get outcome by linearizing

we suppose the Eq. ( 80 ) solution of the form

\(R _1\) , \(R _1\) are the constant. Then after adding Eqs. ( 81 ) in ( 80 ) we found standard pulses rate.

the above given equation can be solved which gives

The mechanics pertaining to the steady version of stability are illustrated by the aforementioned Eq. ( 82 ). If

then we say that \(\tau _1\) is stable, and if,

it show instability. So from this we write MI gain spectrum \(f(\sigma _1)\) as

While these are further variables that can affect modulation rate, such as the level of complexity strength has a significant impact upon modulation instability. likewise concepts like team velocity and tuning adjustment might have an impact on it.

Plot of modulation instability gain spectrum for various king of variable involved

6 Discussion and results

By applying advanced software like Mathematica 11.0, we create several visualisations of the proposed model in this portion of the article. We may generate a wide variety of graphics using the various sorts of parameters. Figure ( 2 ) demonstrate solution of \(|{\mathcal {D}}_{31}(x,y,z,t)|\) considering many types of parameters that are   \(|{\mathcal {D}}_{31}(x,y,z,t)|\) for variables \(R=0.1,~\omega =1,~y=0.3,~\tau _2=0.5,~p=0.7,~F=0.2,~K=0.61,~\eta _1=0.24,~z=0.32,~\sigma _2=0.21,~\rho _1=0.51,~\sigma _1=0.34,~\tau _1=0.42,~\rho _2=0.33,~b_1=0.43,~\eta _2=0.23,~b_3=0.12.\) and this reveals periodic soliton wave solutions. Exactly same the Figures ( 8 , 9 ) reveals singular and multi periodic soliton wave solutions with multiple variable combinations. For Fig.  3 , it display singular shaped soliton solution with parameters   \(R=0.1,~\omega =-1.9,~y=0.3,~\tau _2=0.5,~p=0.7,~F=0.2,~K=0.61,~\eta _1=0.24,~z=0.32,~\sigma _2=0.21,~\rho _1=0.51,~\sigma _1=0.34,~\tau _1=0.42,~\rho _2=0.33,~b_1=0.43,~\eta _2=0.23,~b_3=0.12.\) , Exactly same the Figures ( 6 , 7 ) reveals singular shaped behaviour of bullet solutions. Figures  4 , depicting the behavioural patterns of V Shaped solitons with values   \(y=0.3,~\tau _2=0.5,~p=-0.7,~\eta _1=0.4,~z=0.32,~\sigma _2=2.21,~\rho _1=0.7,~\sigma _1=2.2,~\tau _1=0.8,~\rho _2=0.33,~b_1=0.12,~\eta _2=0.23,~b_3=0.2,~c_1=0.21,~c_2=0.9,~c_3=0.3,~\chi _2=0.1,~\chi _3=4.1,~T=0.51,~A=0.4.\) . The graphics of modulation instability gain is shown in the Figs.  1 . The combination of 2D, 3D, and contour plotting resulted in the most effective solution practises evaluation. The attained solutions has applications in different fields of mathematical sciences including physics, water waves and in the transmission of optical fibers. In future these solutions are highly valuable for dealings the partial differential problems in many fields of sciences. As compared to the previous results in literature the obtain solutions in this work is distinct, new, powerful and unique.

Graph of \(|{\mathcal {D}}_{31}(x,y,z,t)|\) of ( 1 ) represent the periodic behaviour for   \(R=0.1,~\omega =1,~y=0.3,~\tau _2=0.5,~p=0.7,~F=0.2,~K=0.61,~\eta _1=0.24,~z=0.32,~\sigma _2=0.21,~\rho _1=0.51,~\sigma _1=0.34,~\tau _1=0.42,~\rho _2=0.33,~b_1=0.43,~\eta _2=0.23,~b_3=0.12.\)

Graph of \(|{\mathcal {D}}_{33}(x,y,z,t)|\) of ( 1 ) represent the singular behaviour for   \(R=0.1,~\omega =-1.9,~y=0.3,~\tau _2=0.5,~p=0.7,~F=0.2,~K=0.61,~\eta _1=0.24,~z=0.32,~\sigma _2=0.21,~\rho _1=0.51,~\sigma _1=0.34,~\tau _1=0.42,~\rho _2=0.33,~b_1=0.43,~\eta _2=0.23,~b_3=0.12.\)

Graph of \(|{\mathcal {D}}_{57}(x,y,z,t)|\) of ( 1 ) represent the V shaped behaviour for   \(R=0.1,~\omega =-1.9,~y=0.3,~\tau _2=0.5,~p=0.7,~F=0.2,~K=0.61,~\eta _1=0.24,~z=0.32,~\sigma _2=0.21,~\rho _1=0.51,~\sigma _1=0.34,~\tau _1=0.42,~\rho _2=0.33,~b_1=0.43,~\eta _2=0.23,~b_3=0.12.\)

Graph of \(|{\mathcal {D}}_{13}(x,y,z,t)|\) of ( 1 ) represent the periodic behaviour for   \(y=0.3,~\tau _2=0.5,~p=-0.7,~\eta _1=0.4,~z=0.32,~\sigma _2=2.21,~\rho _1=0.7,~\sigma _1=2.2,~\tau _1=0.8,~\rho _2=0.33,~b_1=0.12,~\eta _2=0.23,~b_3=0.2,~c_1=0.21,~c_2=0.9,~c_3=0.3,~\chi _2=0.1,~\chi _3=4.1,~T=0.51,~A=0.4.\)

Graph of \(|{\mathcal {D}}_{14}(x,y,z,t)|\) of ( 1 ) represent the singular behaviour for   \(y=0.3,~\tau _2=0.5,~p=-0.7,~\eta _1=0.4,~z=0.32,~\sigma _2=0.21,~\rho _1=0.7,~\sigma _1=1.2,~\tau _1=0.8,~\rho _2=0.33,~b_1=0.12,~\eta _2=0.23,~b_3=0.2,~c_1=0.21,~c_2=0.9,~c_3=0.3,~\chi _2=0.1,~\chi _3=0.1,~T=0.51,~A=0.4.\)

Graph of \(|{\mathcal {D}}_{39}(x,y,z,t)|\) of ( 1 ) represent the bright behaviour for   \(R=0.1,~\omega =0.9,~y=0.3,~\tau _2=0.5,~p=0.7,~F=0.2,~K=0.61,~\eta _1=0.24,~z=0.32,~\sigma _2=0.21,~\rho _1=0.51,~\sigma _1=2.34,~\tau _1=0.42,~\rho _2=0.33,~b_1=0.43,~\eta _2=0.23,~b_3=0.12.\)

Graph of \(|{\mathcal {D}}_{16}(x,y,z,t)|\) of ( 1 ) represent the singular periodic behaviour for   \(y=0.3,~\tau _2=0.5,~p=-0.7,~\eta _1=0.4,~z=0.32,~\sigma _2=0.21,~\rho _1=0.7,~\sigma _1=1.1,~\tau _1=0.8,~\rho _2=0.33,~b_1=0.12,~\eta _2=1.23,~b_3=0.2,~c_1=0.21,~c_2=0.9,~c_3=0.3,~\chi _2=1.8,~\chi _3=0.6,~T=0.51,~A=0.4.\)

Graph of \(|{\mathcal {D}}_{21}(x,y,z,t)|\) of ( 1 ) represent the periodic behaviour for   \(y=1.3,~\tau _2=2.5,~p=-0.7,~\eta _1=0.4,~z=0.32,~\sigma _2=3.21,~\rho _1=0.7,~\sigma _1=3.1,~\tau _1=0.8,~\rho _2=0.33,~b_1=0.12,~\eta _2=1.23,~b_3=0.2,~c_1=0.21,~c_2=0.9,~c_3=0.3,~\chi _2=1.8,~\chi _3=0.6,~T=0.51,~A=0.4.\)

7 Conclusion

In this article, we study the (3+1)-dimensional Schrödinger model by employing some well known analytical techniques such as the Sardar sub-equation method and the unified method. A variety of new bullet solutions have been discovered that are periodic shaped, singular shaped, V shaped soliton wave solutions. The stability and modulation instability of the given NLSE have also been discussed to validate the computations. Additionally, we produced a wide variety of graphical representations of the given data using different parameters with the aid of Mathematica. As a result, it can be seen that the used techniques useful for dealing with a variety of other higher dimensional nonlinear evolution models that exist in the fields of hydrodynamics, plasma, mathematics, and other ocean engineering and sciences. similar models may also be solved using similar tools. The solutions attain by using these techniques are robust, unique and straight forward. In future the plan is to utilize these two techniques to solve the different types of nonlinear Schrödinger equation to attain different bullet solutions. Further on this suggested model over plan is to utilize other approaches that gives unique solutions whose have different application in different fields of science. The attained solutions has also applicable in various branches of mathematical science specially in physics, water waves and in the transmission of optical fibers. This work is highly valuable and in future it is useful for understanding the waves behaviours of various nonlinear problems.

Data availability

No datasets were generated or analysed during the current study.

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Hijaz Ahmad

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Kalim U. Tariq: Methodology, conceptualization, supervision, project administration. Hijaz Ahmad: Identification of the research problem, validation, review and editing. S. M. Raza Kazmi: Formal analysis and investigation, writing original draft, visualization.

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Ahmad, H., Tariq, K.U. & Kazmi, S.M.R. Stability, modulation instability and traveling wave solutions of (3+1)dimensional Schrödinger model in physics. Opt Quant Electron 56 , 1237 (2024). https://doi.org/10.1007/s11082-024-07031-0

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Received : 01 December 2023

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DOI : https://doi.org/10.1007/s11082-024-07031-0

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